the dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · the...
TRANSCRIPT
![Page 1: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/1.jpg)
The dynamics of holomorphic correspondences oncompact Riemann surfaces
Gautam Bharali
Indian Institute of Science
2016 Indo-French Conference: 11–23, 2016
Institute of Mathematical Sciences, Chennai
January 11, 2016
Gautam Bharali Dynamics of holomorphic correspondences
![Page 2: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/2.jpg)
Holomorphic correspondences
Let X1, X2 be compact k-dim’l. complex manifolds.
A holomorphiccorrespondence from X1 to X2 is just an analytic k-chain
Γ =N∑j=1
mjΓj , (which means
I Γ1, . . . ,ΓN : distinct irred. complex subvarieties of X1 ×X2 of dim. k;
I mj ’s are +ve integers)
with the following properties: for each Γj ,
π1|Γj& π2|Γj
are
surjective ∀j; and
for x ∈ X, the set
∪1≤j≤N(π−1
1 x ∩ Γj)
is
finite.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 3: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/3.jpg)
Holomorphic correspondences
Let X1, X2 be compact k-dim’l. complex manifolds. A holomorphiccorrespondence from X1 to X2 is just an analytic k-chain
Γ =
N∑j=1
mjΓj , (which means
I Γ1, . . . ,ΓN : distinct irred. complex subvarieties of X1 ×X2 of dim. k;
I mj ’s are +ve integers)
with the following properties: for each Γj ,
π1|Γj& π2|Γj
are
surjective ∀j; and
for x ∈ X, the set
∪1≤j≤N(π−1
1 x ∩ Γj)
is
finite.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 4: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/4.jpg)
Holomorphic correspondences
Let X1, X2 be compact k-dim’l. complex manifolds. A holomorphiccorrespondence from X1 to X2 is just an analytic k-chain
Γ =
N∑j=1
mjΓj , (which means
I Γ1, . . . ,ΓN : distinct irred. complex subvarieties of X1 ×X2 of dim. k;
I mj ’s are +ve integers)
with the following properties: for each Γj ,
π1|Γj& π2|Γj
are
surjective ∀j; and
for x ∈ X, the set
∪1≤j≤N(π−1
1 x ∩ Γj)
is
finite.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 5: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/5.jpg)
Holomorphic correspondences
Let X1, X2 be compact k-dim’l. complex manifolds. A holomorphiccorrespondence from X1 to X2 is just an analytic k-chain
Γ =
N∑j=1
mjΓj , (which means
I Γ1, . . . ,ΓN : distinct irred. complex subvarieties of X1 ×X2 of dim. k;
I mj ’s are +ve integers)
with the following properties: for each Γj ,
π1|Γj& π2|Γj
are
surjective ∀j; and
for x ∈ X, the set
∪1≤j≤N(π−1
1 x ∩ Γj)
is
finite.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 6: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/6.jpg)
Holomorphic correspondences
Let X1, X2 be compact k-dim’l. complex manifolds. A holomorphiccorrespondence from X1 to X2 is just an analytic k-chain
Γ =
N∑j=1
mjΓj , (which means
I Γ1, . . . ,ΓN : distinct irred. complex subvarieties of X1 ×X2 of dim. k;
I mj ’s are +ve integers)
with the following properties: for each Γj ,
π1|Γj& π2|Γj
are
surjective ∀j; and
for x ∈ X, the set
∪1≤j≤N(π−1
1 x ∩ Γj)
is
finite.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 7: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/7.jpg)
Holomorphic correspondences
Let X1, X2 be compact k-dim’l. complex manifolds. A holomorphiccorrespondence from X1 to X2 is just an analytic k-chain
Γ =
N∑j=1
mjΓj , (which means
I Γ1, . . . ,ΓN : distinct irred. complex subvarieties of X1 ×X2 of dim. k;
I mj ’s are +ve integers)
with the following properties: for each Γj ,
π1|Γj& π2|Γj
are
surjective ∀j; and
for x ∈ X, the set
∪1≤j≤N(π−1
1 x ∩ Γj)
is
finite.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 8: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/8.jpg)
Holomorphic correspondences
Let X1, X2 be compact k-dim’l. complex manifolds. A holomorphiccorrespondence from X1 to X2 is just an analytic k-chain
Γ =
N∑j=1
mjΓj , (which means
I Γ1, . . . ,ΓN : distinct irred. complex subvarieties of X1 ×X2 of dim. k;
I mj ’s are +ve integers)
with the following properties: for each Γj ,
π1|Γj& π2|Γj
are
surjective ∀j; and
for x ∈ X, the set
∪1≤j≤N(π−1
1 x ∩ Γj)
is
finite.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 9: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/9.jpg)
Why should we care ?
In this talk, X1 = X2 (= X, say), both Riemann surfaces. Acorrespondence can be composed with itself (which we’ll define).Wish to study the dynamical system that arises.
Why would one want to do this ?
By the de Franchis Theorem, there are only finitely many holomorphicself-maps on a compact hyperbolic Riemann surface. I.e., no interestingholomorphic dynamics!
But not too hard to show that there are infinitely many holomorphiccorrespondences Γ on a compact hyperbolic Riemann surface; evensatisfying
dtop(Γ ) > dtop(†Γ ).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 10: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/10.jpg)
Why should we care ?
In this talk, X1 = X2 (= X, say), both Riemann surfaces. Acorrespondence can be composed with itself (which we’ll define).Wish to study the dynamical system that arises.
Why would one want to do this ?
By the de Franchis Theorem, there are only finitely many holomorphicself-maps on a compact hyperbolic Riemann surface. I.e., no interestingholomorphic dynamics!
But not too hard to show that there are infinitely many holomorphiccorrespondences Γ on a compact hyperbolic Riemann surface; evensatisfying
dtop(Γ ) > dtop(†Γ ).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 11: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/11.jpg)
Why should we care ?
In this talk, X1 = X2 (= X, say), both Riemann surfaces. Acorrespondence can be composed with itself (which we’ll define).Wish to study the dynamical system that arises.
Why would one want to do this ?
By the de Franchis Theorem, there are only finitely many holomorphicself-maps on a compact hyperbolic Riemann surface.
I.e., no interestingholomorphic dynamics!
But not too hard to show that there are infinitely many holomorphiccorrespondences Γ on a compact hyperbolic Riemann surface; evensatisfying
dtop(Γ ) > dtop(†Γ ).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 12: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/12.jpg)
Why should we care ?
In this talk, X1 = X2 (= X, say), both Riemann surfaces. Acorrespondence can be composed with itself (which we’ll define).Wish to study the dynamical system that arises.
Why would one want to do this ?
By the de Franchis Theorem, there are only finitely many holomorphicself-maps on a compact hyperbolic Riemann surface. I.e., no interestingholomorphic dynamics!
But not too hard to show that there are infinitely many holomorphiccorrespondences Γ on a compact hyperbolic Riemann surface; evensatisfying
dtop(Γ ) > dtop(†Γ ).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 13: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/13.jpg)
Why should we care ?
In this talk, X1 = X2 (= X, say), both Riemann surfaces. Acorrespondence can be composed with itself (which we’ll define).Wish to study the dynamical system that arises.
Why would one want to do this ?
By the de Franchis Theorem, there are only finitely many holomorphicself-maps on a compact hyperbolic Riemann surface. I.e., no interestingholomorphic dynamics!
But not too hard to show that there are infinitely many holomorphiccorrespondences Γ on a compact hyperbolic Riemann surface; evensatisfying
dtop(Γ ) > dtop(†Γ ).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 14: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/14.jpg)
Point of entry
In 2006, Dinh–Sibony proved a result that, paraphrased for holomorphiccorrespondences, is:
Result. Let Γ be a holomorphic correspondence on a k-dim’l. compactKahler manifold (X,ω) and assume that dtop(Γ ) > dtop(
†Γ ). Suppose∫X ω
k = 1. Then, ∃µΓ − a Borel prob. measure on X − that satisfiesΓ ∗(µΓ ) = dtop(Γ )µΓ , and
1
dtop(Γ )n(Γn)∗(ωk)
weak∗−−−→ µΓ as measures, as n→∞.
Clarifications:
The pullback Γ ∗(ωk) is carried out in the sense of currents.
dtop(Γ ) :=∑
1≤j≤N mjdegree(π2).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 15: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/15.jpg)
Point of entry
In 2006, Dinh–Sibony proved a result that, paraphrased for holomorphiccorrespondences, is:
Result. Let Γ be a holomorphic correspondence on a k-dim’l. compactKahler manifold (X,ω) and assume that dtop(Γ ) > dtop(
†Γ ). Suppose∫X ω
k = 1. Then,
∃µΓ − a Borel prob. measure on X − that satisfiesΓ ∗(µΓ ) = dtop(Γ )µΓ , and
1
dtop(Γ )n(Γn)∗(ωk)
weak∗−−−→ µΓ as measures, as n→∞.
Clarifications:
The pullback Γ ∗(ωk) is carried out in the sense of currents.
dtop(Γ ) :=∑
1≤j≤N mjdegree(π2).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 16: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/16.jpg)
Point of entry
In 2006, Dinh–Sibony proved a result that, paraphrased for holomorphiccorrespondences, is:
Result. Let Γ be a holomorphic correspondence on a k-dim’l. compactKahler manifold (X,ω) and assume that dtop(Γ ) > dtop(
†Γ ). Suppose∫X ω
k = 1. Then, ∃µΓ − a Borel prob. measure on X − that satisfiesΓ ∗(µΓ ) = dtop(Γ )µΓ , and
1
dtop(Γ )n(Γn)∗(ωk)
weak∗−−−→ µΓ as measures, as n→∞.
Clarifications:
The pullback Γ ∗(ωk) is carried out in the sense of currents.
dtop(Γ ) :=∑
1≤j≤N mjdegree(π2).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 17: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/17.jpg)
Point of entry
In 2006, Dinh–Sibony proved a result that, paraphrased for holomorphiccorrespondences, is:
Result. Let Γ be a holomorphic correspondence on a k-dim’l. compactKahler manifold (X,ω) and assume that dtop(Γ ) > dtop(
†Γ ). Suppose∫X ω
k = 1. Then, ∃µΓ − a Borel prob. measure on X − that satisfiesΓ ∗(µΓ ) = dtop(Γ )µΓ , and
1
dtop(Γ )n(Γn)∗(ωk)
weak∗−−−→ µΓ as measures, as n→∞.
Clarifications:
The pullback Γ ∗(ωk) is carried out in the sense of currents.
dtop(Γ ) :=∑
1≤j≤N mjdegree(π2).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 18: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/18.jpg)
Point of entry
In 2006, Dinh–Sibony proved a result that, paraphrased for holomorphiccorrespondences, is:
Result. Let Γ be a holomorphic correspondence on a k-dim’l. compactKahler manifold (X,ω) and assume that dtop(Γ ) > dtop(
†Γ ). Suppose∫X ω
k = 1. Then, ∃µΓ − a Borel prob. measure on X − that satisfiesΓ ∗(µΓ ) = dtop(Γ )µΓ , and
1
dtop(Γ )n(Γn)∗(ωk)
weak∗−−−→ µΓ as measures, as n→∞.
Clarifications:
The pullback Γ ∗(ωk) is carried out in the sense of currents.
dtop(Γ ) :=∑
1≤j≤N mjdegree(π2).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 19: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/19.jpg)
Point of entry
In 2006, Dinh–Sibony proved a result that, paraphrased for holomorphiccorrespondences, is:
Result. Let Γ be a holomorphic correspondence on a k-dim’l. compactKahler manifold (X,ω) and assume that dtop(Γ ) > dtop(
†Γ ). Suppose∫X ω
k = 1. Then, ∃µΓ − a Borel prob. measure on X − that satisfiesΓ ∗(µΓ ) = dtop(Γ )µΓ , and
1
dtop(Γ )n(Γn)∗(ωk)
weak∗−−−→ µΓ as measures, as n→∞.
Clarifications:
The pullback Γ ∗(ωk) is carried out in the sense of currents.
dtop(Γ ) :=∑
1≤j≤N mjdegree(π2).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 20: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/20.jpg)
Early questions
Some natural questions arise that may be tractable when k = 1.
When k = 1, X = CP1 and Γ = graph(f), f a rational map, thensupp(µΓ ) is the Julia set− denoted J(f) − of f .
J(f) is the complement of the Fatou set of f , i.e. CP1 \F (f).
Problem 1. Describe the set F (Γ ) on which the orbits of Γ areinsensitive to small perturbations of initial condition.
Problem 2. Describe the complex geometry of the components ofF (Γ ) in terms analogous to classical complex dynamics.
In classical complex dynamics, a crucial part of studying geometricstructure is the fact that J(f) ∪F (f) = CP1.
I Theorem A will address Problem 1 above.
I Theorem B (time permitting) will address the issue in the box above.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 21: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/21.jpg)
Early questions
Some natural questions arise that may be tractable when k = 1.
When k = 1, X = CP1 and Γ = graph(f), f a rational map, thensupp(µΓ ) is the Julia set− denoted J(f) − of f .
J(f) is the complement of the Fatou set of f , i.e. CP1 \F (f).
Problem 1. Describe the set F (Γ ) on which the orbits of Γ areinsensitive to small perturbations of initial condition.
Problem 2. Describe the complex geometry of the components ofF (Γ ) in terms analogous to classical complex dynamics.
In classical complex dynamics, a crucial part of studying geometricstructure is the fact that J(f) ∪F (f) = CP1.
I Theorem A will address Problem 1 above.
I Theorem B (time permitting) will address the issue in the box above.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 22: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/22.jpg)
Early questions
Some natural questions arise that may be tractable when k = 1.
When k = 1, X = CP1 and Γ = graph(f), f a rational map, thensupp(µΓ ) is the Julia set− denoted J(f) − of f .
J(f) is the complement of the Fatou set of f , i.e. CP1 \F (f).
Problem 1. Describe the set F (Γ ) on which the orbits of Γ areinsensitive to small perturbations of initial condition.
Problem 2. Describe the complex geometry of the components ofF (Γ ) in terms analogous to classical complex dynamics.
In classical complex dynamics, a crucial part of studying geometricstructure is the fact that J(f) ∪F (f) = CP1.
I Theorem A will address Problem 1 above.
I Theorem B (time permitting) will address the issue in the box above.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 23: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/23.jpg)
Early questions
Some natural questions arise that may be tractable when k = 1.
When k = 1, X = CP1 and Γ = graph(f), f a rational map, thensupp(µΓ ) is the Julia set− denoted J(f) − of f .
J(f) is the complement of the Fatou set of f , i.e. CP1 \F (f).
Problem 1. Describe the set F (Γ ) on which the orbits of Γ areinsensitive to small perturbations of initial condition.
Problem 2. Describe the complex geometry of the components ofF (Γ ) in terms analogous to classical complex dynamics.
In classical complex dynamics, a crucial part of studying geometricstructure is the fact that J(f) ∪F (f) = CP1.
I Theorem A will address Problem 1 above.
I Theorem B (time permitting) will address the issue in the box above.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 24: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/24.jpg)
Early questions
Some natural questions arise that may be tractable when k = 1.
When k = 1, X = CP1 and Γ = graph(f), f a rational map, thensupp(µΓ ) is the Julia set− denoted J(f) − of f .
J(f) is the complement of the Fatou set of f , i.e. CP1 \F (f).
Problem 1. Describe the set F (Γ ) on which the orbits of Γ areinsensitive to small perturbations of initial condition.
Problem 2. Describe the complex geometry of the components ofF (Γ ) in terms analogous to classical complex dynamics.
In classical complex dynamics, a crucial part of studying geometricstructure is the fact that J(f) ∪F (f) = CP1.
I Theorem A will address Problem 1 above.
I Theorem B (time permitting) will address the issue in the box above.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 25: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/25.jpg)
Early questions
Some natural questions arise that may be tractable when k = 1.
When k = 1, X = CP1 and Γ = graph(f), f a rational map, thensupp(µΓ ) is the Julia set− denoted J(f) − of f .
J(f) is the complement of the Fatou set of f , i.e. CP1 \F (f).
Problem 1. Describe the set F (Γ ) on which the orbits of Γ areinsensitive to small perturbations of initial condition.
Problem 2. Describe the complex geometry of the components ofF (Γ ) in terms analogous to classical complex dynamics.
In classical complex dynamics, a crucial part of studying geometricstructure is the fact that J(f) ∪F (f) = CP1.
I Theorem A will address Problem 1 above.
I Theorem B (time permitting) will address the issue in the box above.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 26: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/26.jpg)
Early questions
Some natural questions arise that may be tractable when k = 1.
When k = 1, X = CP1 and Γ = graph(f), f a rational map, thensupp(µΓ ) is the Julia set− denoted J(f) − of f .
J(f) is the complement of the Fatou set of f , i.e. CP1 \F (f).
Problem 1. Describe the set F (Γ ) on which the orbits of Γ areinsensitive to small perturbations of initial condition.
Problem 2. Describe the complex geometry of the components ofF (Γ ) in terms analogous to classical complex dynamics.
In classical complex dynamics, a crucial part of studying geometricstructure is the fact that J(f) ∪F (f) = CP1.
I Theorem A will address Problem 1 above.
I Theorem B (time permitting) will address the issue in the box above.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 27: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/27.jpg)
Early questions
Some natural questions arise that may be tractable when k = 1.
When k = 1, X = CP1 and Γ = graph(f), f a rational map, thensupp(µΓ ) is the Julia set− denoted J(f) − of f .
J(f) is the complement of the Fatou set of f , i.e. CP1 \F (f).
Problem 1. Describe the set F (Γ ) on which the orbits of Γ areinsensitive to small perturbations of initial condition.
Problem 2. Describe the complex geometry of the components ofF (Γ ) in terms analogous to classical complex dynamics.
In classical complex dynamics, a crucial part of studying geometricstructure is the fact that J(f) ∪F (f) = CP1.
I Theorem A will address Problem 1 above.
I Theorem B (time permitting) will address the issue in the box above.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 28: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/28.jpg)
Composing two holomorphic correspondences
Given a holomorphic correspondence Γ , we denote by
|Γ | := ∪Nj=1Γj
the set underlying Γ . Now, |Γ | is a relation on X.
If Γ 1 and Γ 2 are correspondences on X, we view Γ 2 Γ 1 as essentially theclassical composition of two relations. Denote the latter operation by ? :
|Γ 2| ? |Γ 1| := (x, z) ∈ X ×X : ∃y s.t.(x, y) ∈ |Γ 1|, (y, z) ∈ |Γ 2|. (∗)
To code the k-chain data into the above “composition” we need to dosome work. . .
Gautam Bharali Dynamics of holomorphic correspondences
![Page 29: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/29.jpg)
Composing two holomorphic correspondences
Given a holomorphic correspondence Γ , we denote by
|Γ | := ∪Nj=1Γj
the set underlying Γ . Now, |Γ | is a relation on X.
If Γ 1 and Γ 2 are correspondences on X, we view Γ 2 Γ 1 as essentially theclassical composition of two relations. Denote the latter operation by ? :
|Γ 2| ? |Γ 1| := (x, z) ∈ X ×X : ∃y s.t.(x, y) ∈ |Γ 1|, (y, z) ∈ |Γ 2|. (∗)
To code the k-chain data into the above “composition” we need to dosome work. . .
Gautam Bharali Dynamics of holomorphic correspondences
![Page 30: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/30.jpg)
Composing two holomorphic correspondences
Given a holomorphic correspondence Γ , we denote by
|Γ | := ∪Nj=1Γj
the set underlying Γ . Now, |Γ | is a relation on X.
If Γ 1 and Γ 2 are correspondences on X, we view Γ 2 Γ 1 as essentially theclassical composition of two relations. Denote the latter operation by ? :
|Γ 2| ? |Γ 1| := (x, z) ∈ X ×X : ∃y s.t.(x, y) ∈ |Γ 1|, (y, z) ∈ |Γ 2|. (∗)
To code the k-chain data into the above “composition” we need to dosome work. . .
Gautam Bharali Dynamics of holomorphic correspondences
![Page 31: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/31.jpg)
Composing two holomorphic correspondences
Given a holomorphic correspondence Γ , we denote by
|Γ | := ∪Nj=1Γj
the set underlying Γ . Now, |Γ | is a relation on X.
If Γ 1 and Γ 2 are correspondences on X, we view Γ 2 Γ 1 as essentially theclassical composition of two relations. Denote the latter operation by ? :
|Γ 2| ? |Γ 1| := (x, z) ∈ X ×X : ∃y s.t.(x, y) ∈ |Γ 1|, (y, z) ∈ |Γ 2|. (∗)
To code the k-chain data into the above “composition” we need to dosome work. . .
Gautam Bharali Dynamics of holomorphic correspondences
![Page 32: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/32.jpg)
Composing two holomorphic correspondences, cont’d.
To begin with, we now use an alternative representation:
Γ 1 =∑′
1≤j≤L1
Γ•1, j , Γ 2 =∑′
1≤j≤L2
Γ•2, j ,
primed sums indicating that the irred. subvarieties Γ•s, j , j = 1, . . . , Ls,s = 1, 2, are not necessarily distinct and repeated according to thecoeffs. ms, j .
We then define
Γ 2 Γ 1 :=
L1∑j=1
L2∑l=1
∑S∈S(j, l)
νSS,
where :
S(j, l) := set of distinct irred. components of Γ•2, l ? Γ•1, j .
Gautam Bharali Dynamics of holomorphic correspondences
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Composing two holomorphic correspondences, cont’d.
To begin with, we now use an alternative representation:
Γ 1 =∑′
1≤j≤L1
Γ•1, j , Γ 2 =∑′
1≤j≤L2
Γ•2, j ,
primed sums indicating that the irred. subvarieties Γ•s, j , j = 1, . . . , Ls,s = 1, 2, are not necessarily distinct and repeated according to thecoeffs. ms, j .
We then define
Γ 2 Γ 1 :=
L1∑j=1
L2∑l=1
∑S∈S(j, l)
νSS,
where :
S(j, l) := set of distinct irred. components of Γ•2, l ? Γ•1, j .
Gautam Bharali Dynamics of holomorphic correspondences
![Page 34: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/34.jpg)
Composing two holomorphic correspondences, cont’d.
To begin with, we now use an alternative representation:
Γ 1 =∑′
1≤j≤L1
Γ•1, j , Γ 2 =∑′
1≤j≤L2
Γ•2, j ,
primed sums indicating that the irred. subvarieties Γ•s, j , j = 1, . . . , Ls,s = 1, 2, are not necessarily distinct and repeated according to thecoeffs. ms, j .
We then define
Γ 2 Γ 1 :=
L1∑j=1
L2∑l=1
∑S∈S(j, l)
νSS,
where :
S(j, l) := set of distinct irred. components of Γ•2, l ? Γ•1, j .
Gautam Bharali Dynamics of holomorphic correspondences
![Page 35: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/35.jpg)
Composing two holomorphic correspondences, cont’d.To understand the coefficient νS , consider the following:
Example. Take the correspondencesΓ 1 = Γ 2 = the completion in CP1 × CP1 of (z, w) ∈ C2 : w2 = z2 + 1 =: Γ.
Pick z0 ∈ C and consider its 2-orbit:
z0 //
##
√z20 + 1 //
&&
√z20 + 2
−√
z20 + 1
&&
−√
z20 + 2
√z20 + 2 −
√z20 + 2
Two distinct occurences of y indicated by (∗) associated to the point(z0,
√z2
0 + 2 ) ∈ Γ ? Γ.
νS := generic no. of y’s− as (x, z) varies through S− for which thememberships given in (∗) hold.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 36: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/36.jpg)
Composing two holomorphic correspondences, cont’d.To understand the coefficient νS , consider the following:
Example. Take the correspondencesΓ 1 = Γ 2 = the completion in CP1 × CP1 of (z, w) ∈ C2 : w2 = z2 + 1 =: Γ.Pick z0 ∈ C and consider its 2-orbit:
z0 //
##
√z20 + 1 //
&&
√z20 + 2
−√
z20 + 1
&&
−√
z20 + 2
√z20 + 2 −
√z20 + 2
Two distinct occurences of y indicated by (∗) associated to the point(z0,
√z2
0 + 2 ) ∈ Γ ? Γ.
νS := generic no. of y’s− as (x, z) varies through S− for which thememberships given in (∗) hold.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 37: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/37.jpg)
Composing two holomorphic correspondences, cont’d.To understand the coefficient νS , consider the following:
Example. Take the correspondencesΓ 1 = Γ 2 = the completion in CP1 × CP1 of (z, w) ∈ C2 : w2 = z2 + 1 =: Γ.Pick z0 ∈ C and consider its 2-orbit:
z0 //
##
√z20 + 1 //
&&
√z20 + 2
−√
z20 + 1
&&
−√
z20 + 2
√z20 + 2 −
√z20 + 2
Two distinct occurences of y indicated by (∗) associated to the point(z0,
√z2
0 + 2 ) ∈ Γ ? Γ.
νS := generic no. of y’s− as (x, z) varies through S− for which thememberships given in (∗) hold.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 38: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/38.jpg)
Composing two holomorphic correspondences, cont’d.To understand the coefficient νS , consider the following:
Example. Take the correspondencesΓ 1 = Γ 2 = the completion in CP1 × CP1 of (z, w) ∈ C2 : w2 = z2 + 1 =: Γ.Pick z0 ∈ C and consider its 2-orbit:
z0 //
##
√z20 + 1 //
&&
√z20 + 2
−√
z20 + 1
&&
−√
z20 + 2
√z20 + 2 −
√z20 + 2
Two distinct occurences of y indicated by (∗) associated to the point(z0,
√z2
0 + 2 ) ∈ Γ ? Γ.
νS := generic no. of y’s− as (x, z) varies through S− for which thememberships given in (∗) hold.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 39: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/39.jpg)
Composing two holomorphic correspondences, cont’d.To understand the coefficient νS , consider the following:
Example. Take the correspondencesΓ 1 = Γ 2 = the completion in CP1 × CP1 of (z, w) ∈ C2 : w2 = z2 + 1 =: Γ.Pick z0 ∈ C and consider its 2-orbit:
z0 //
##
√z20 + 1 //
&&
√z20 + 2
−√
z20 + 1
&&
−√
z20 + 2
√z20 + 2 −
√z20 + 2
Two distinct occurences of y indicated by (∗) associated to the point(z0,
√z2
0 + 2 ) ∈ Γ ? Γ.
νS := generic no. of y’s− as (x, z) varies through S− for which thememberships given in (∗) hold.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 40: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/40.jpg)
Statement of main results
Recall that when X = CP1 and Γ is the graph of a rational map f ,
dtop(†Γ ) = 1, µΓ exists and supp(µΓ ) equals the Julia set J(f).
For various reasons (e.g., see the Dinh–Sibony theorem) supp(µΓ ) is anatural analogue of the Julia set for general correspondences.
The following theorem provides a relationship between supp(µΓ ) and theFatou set F (Γ ).
Theorem A (B., 2014)
Let X be a compact Riemann surface and let Γ be a holomorphiccorrespondence on X such that dtop(Γ ) > dtop(
†Γ ). Let µΓ denote theDinh–Sibony measure associated to Γ . Then, the Fatou set of Γ , F (Γ ),is disjoint from supp(µΓ ).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 41: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/41.jpg)
Statement of main results
Recall that when X = CP1 and Γ is the graph of a rational map f ,dtop(
†Γ ) = 1, µΓ exists and supp(µΓ ) equals the Julia set J(f).
For various reasons (e.g., see the Dinh–Sibony theorem) supp(µΓ ) is anatural analogue of the Julia set for general correspondences.
The following theorem provides a relationship between supp(µΓ ) and theFatou set F (Γ ).
Theorem A (B., 2014)
Let X be a compact Riemann surface and let Γ be a holomorphiccorrespondence on X such that dtop(Γ ) > dtop(
†Γ ). Let µΓ denote theDinh–Sibony measure associated to Γ . Then, the Fatou set of Γ , F (Γ ),is disjoint from supp(µΓ ).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 42: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/42.jpg)
Statement of main results
Recall that when X = CP1 and Γ is the graph of a rational map f ,dtop(
†Γ ) = 1, µΓ exists and supp(µΓ ) equals the Julia set J(f).
For various reasons (e.g., see the Dinh–Sibony theorem) supp(µΓ ) is anatural analogue of the Julia set for general correspondences.
The following theorem provides a relationship between supp(µΓ ) and theFatou set F (Γ ).
Theorem A (B., 2014)
Let X be a compact Riemann surface and let Γ be a holomorphiccorrespondence on X such that dtop(Γ ) > dtop(
†Γ ). Let µΓ denote theDinh–Sibony measure associated to Γ . Then, the Fatou set of Γ , F (Γ ),is disjoint from supp(µΓ ).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 43: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/43.jpg)
Statement of main results
Recall that when X = CP1 and Γ is the graph of a rational map f ,dtop(
†Γ ) = 1, µΓ exists and supp(µΓ ) equals the Julia set J(f).
For various reasons (e.g., see the Dinh–Sibony theorem) supp(µΓ ) is anatural analogue of the Julia set for general correspondences.
The following theorem provides a relationship between supp(µΓ ) and theFatou set F (Γ ).
Theorem A (B., 2014)
Let X be a compact Riemann surface and let Γ be a holomorphiccorrespondence on X such that dtop(Γ ) > dtop(
†Γ ). Let µΓ denote theDinh–Sibony measure associated to Γ . Then, the Fatou set of Γ , F (Γ ),is disjoint from supp(µΓ ).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 44: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/44.jpg)
Statement of main results
Recall that when X = CP1 and Γ is the graph of a rational map f ,dtop(
†Γ ) = 1, µΓ exists and supp(µΓ ) equals the Julia set J(f).
For various reasons (e.g., see the Dinh–Sibony theorem) supp(µΓ ) is anatural analogue of the Julia set for general correspondences.
The following theorem provides a relationship between supp(µΓ ) and theFatou set F (Γ ).
Theorem A (B., 2014)
Let X be a compact Riemann surface and let Γ be a holomorphiccorrespondence on X such that dtop(Γ ) > dtop(
†Γ ). Let µΓ denote theDinh–Sibony measure associated to Γ .
Then, the Fatou set of Γ , F (Γ ),is disjoint from supp(µΓ ).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 45: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/45.jpg)
Statement of main results
Recall that when X = CP1 and Γ is the graph of a rational map f ,dtop(
†Γ ) = 1, µΓ exists and supp(µΓ ) equals the Julia set J(f).
For various reasons (e.g., see the Dinh–Sibony theorem) supp(µΓ ) is anatural analogue of the Julia set for general correspondences.
The following theorem provides a relationship between supp(µΓ ) and theFatou set F (Γ ).
Theorem A (B., 2014)
Let X be a compact Riemann surface and let Γ be a holomorphiccorrespondence on X such that dtop(Γ ) > dtop(
†Γ ). Let µΓ denote theDinh–Sibony measure associated to Γ . Then, the Fatou set of Γ , F (Γ ),is disjoint from supp(µΓ ).
Gautam Bharali Dynamics of holomorphic correspondences
![Page 46: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/46.jpg)
The Fatou set: preliminary notations/notions
Fix a compact Riemann surface X and a correspondence Γ on it.
N -path:(z0, . . . , zN ; α1, . . . , αN ) ∈ XN+1 × [1 . . L]N : (zj−1, zj) ∈ Γ•αj
, j ≤ N.
PN(z0) = set of all N -paths starting at z0.
Given a multi-index α ∈ [1 . . L]j
Γ•α := (x0, . . . , xj) ∈ Xj+1 : (xi−1, xi) ∈ Γ•αi, 1 ≤ i ≤ j.
Basic idea underlying the Fatou set:A point z0 belongs to the Fatou set if there exists a nbhd. U 3 z0 such that for
every infinite path (z0, z1, z2, . . . ;α1, α2, α3, . . . ), each sequence of analytic
germs of Γ•(α1,...,αn) at (z0, z1, . . . , zn), n = 1, 2, 3, . . . , determined by lifting U
into these varieties admits a subsequence that converges to an analytic set.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 47: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/47.jpg)
The Fatou set: preliminary notations/notions
Fix a compact Riemann surface X and a correspondence Γ on it.
N -path:(z0, . . . , zN ; α1, . . . , αN ) ∈ XN+1 × [1 . . L]N : (zj−1, zj) ∈ Γ•αj
, j ≤ N.
PN(z0) = set of all N -paths starting at z0.
Given a multi-index α ∈ [1 . . L]j
Γ•α := (x0, . . . , xj) ∈ Xj+1 : (xi−1, xi) ∈ Γ•αi, 1 ≤ i ≤ j.
Basic idea underlying the Fatou set:A point z0 belongs to the Fatou set if there exists a nbhd. U 3 z0 such that for
every infinite path (z0, z1, z2, . . . ;α1, α2, α3, . . . ), each sequence of analytic
germs of Γ•(α1,...,αn) at (z0, z1, . . . , zn), n = 1, 2, 3, . . . , determined by lifting U
into these varieties admits a subsequence that converges to an analytic set.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 48: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/48.jpg)
The Fatou set: preliminary notations/notions
Fix a compact Riemann surface X and a correspondence Γ on it.
N -path:(z0, . . . , zN ; α1, . . . , αN ) ∈ XN+1 × [1 . . L]N : (zj−1, zj) ∈ Γ•αj
, j ≤ N.
PN(z0) = set of all N -paths starting at z0.
Given a multi-index α ∈ [1 . . L]j
Γ•α := (x0, . . . , xj) ∈ Xj+1 : (xi−1, xi) ∈ Γ•αi, 1 ≤ i ≤ j.
Basic idea underlying the Fatou set:A point z0 belongs to the Fatou set if there exists a nbhd. U 3 z0 such that for
every infinite path (z0, z1, z2, . . . ;α1, α2, α3, . . . ), each sequence of analytic
germs of Γ•(α1,...,αn) at (z0, z1, . . . , zn), n = 1, 2, 3, . . . , determined by lifting U
into these varieties admits a subsequence that converges to an analytic set.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 49: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/49.jpg)
The Fatou set: preliminary notations/notions
Fix a compact Riemann surface X and a correspondence Γ on it.
N -path:(z0, . . . , zN ; α1, . . . , αN ) ∈ XN+1 × [1 . . L]N : (zj−1, zj) ∈ Γ•αj
, j ≤ N.
PN(z0) = set of all N -paths starting at z0.
Given a multi-index α ∈ [1 . . L]j
Γ•α := (x0, . . . , xj) ∈ Xj+1 : (xi−1, xi) ∈ Γ•αi, 1 ≤ i ≤ j.
Basic idea underlying the Fatou set:A point z0 belongs to the Fatou set if there exists a nbhd. U 3 z0 such that for
every infinite path (z0, z1, z2, . . . ;α1, α2, α3, . . . ), each sequence of analytic
germs of Γ•(α1,...,αn) at (z0, z1, . . . , zn), n = 1, 2, 3, . . . , determined by lifting U
into these varieties admits a subsequence that converges to an analytic set.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 50: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/50.jpg)
The Fatou set: preliminary notations/notions
Fix a compact Riemann surface X and a correspondence Γ on it.
N -path:(z0, . . . , zN ; α1, . . . , αN ) ∈ XN+1 × [1 . . L]N : (zj−1, zj) ∈ Γ•αj
, j ≤ N.
PN(z0) = set of all N -paths starting at z0.
Given a multi-index α ∈ [1 . . L]j
Γ•α := (x0, . . . , xj) ∈ Xj+1 : (xi−1, xi) ∈ Γ•αi, 1 ≤ i ≤ j.
Basic idea underlying the Fatou set:A point z0 belongs to the Fatou set if there exists a nbhd. U 3 z0 such that
for
every infinite path (z0, z1, z2, . . . ;α1, α2, α3, . . . ), each sequence of analytic
germs of Γ•(α1,...,αn) at (z0, z1, . . . , zn), n = 1, 2, 3, . . . , determined by lifting U
into these varieties admits a subsequence that converges to an analytic set.
Gautam Bharali Dynamics of holomorphic correspondences
![Page 51: The dynamics of holomorphic correspondences on compact ...infr2016/talks/bharali.pdf · The dynamics of holomorphic correspondences on compact Riemann surfaces Gautam Bharali Indian](https://reader033.vdocument.in/reader033/viewer/2022060418/5f15ab6b99a02659c911ee03/html5/thumbnails/51.jpg)
The Fatou set: preliminary notations/notions
Fix a compact Riemann surface X and a correspondence Γ on it.
N -path:(z0, . . . , zN ; α1, . . . , αN ) ∈ XN+1 × [1 . . L]N : (zj−1, zj) ∈ Γ•αj
, j ≤ N.
PN(z0) = set of all N -paths starting at z0.
Given a multi-index α ∈ [1 . . L]j
Γ•α := (x0, . . . , xj) ∈ Xj+1 : (xi−1, xi) ∈ Γ•αi, 1 ≤ i ≤ j.
Basic idea underlying the Fatou set:A point z0 belongs to the Fatou set if there exists a nbhd. U 3 z0 such that for
every infinite path (z0, z1, z2, . . . ;α1, α2, α3, . . . ), each sequence of analytic
germs of Γ•(α1,...,αn) at (z0, z1, . . . , zn), n = 1, 2, 3, . . . , determined by lifting U
into these varieties admits
a subsequence that converges to an analytic set.
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The Fatou set: preliminary notations/notions
Fix a compact Riemann surface X and a correspondence Γ on it.
N -path:(z0, . . . , zN ; α1, . . . , αN ) ∈ XN+1 × [1 . . L]N : (zj−1, zj) ∈ Γ•αj
, j ≤ N.
PN(z0) = set of all N -paths starting at z0.
Given a multi-index α ∈ [1 . . L]j
Γ•α := (x0, . . . , xj) ∈ Xj+1 : (xi−1, xi) ∈ Γ•αi, 1 ≤ i ≤ j.
Basic idea underlying the Fatou set:A point z0 belongs to the Fatou set if there exists a nbhd. U 3 z0 such that for
every infinite path (z0, z1, z2, . . . ;α1, α2, α3, . . . ), each sequence of analytic
germs of Γ•(α1,...,αn) at (z0, z1, . . . , zn), n = 1, 2, 3, . . . , determined by lifting U
into these varieties admits a subsequence that converges to an analytic set.
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The Fatou set: secondary notations
A useful map:
π(k)j : Xk+1 −→ X, π
(k)j : (z0, z1, . . . , zk) 7−→ zj , 0 ≤ j ≤ k.
Let z0 ∈ X and U 3 x0 be an open nbhd. Denote paths in PN (z0) byZ ≡ (Z;α). Then
Z [j] := (z0, . . . , zj ; α1, . . . , αj), 1 ≤ j ≤ Npre(Z) := Z [N−1], N ≥ 2.
Next, we define sequences of analytic germs:
S (U,Z) := set of irred. components of
Γ•α ∩ (U ×X) containing Z, if Z ∈P1(z0),
S (U,Z) := set of irred. components of
Γ•α ∩(U×
(XN−1k=1 π
(k)k (S)
)×X
)containing Z, if Z ∈PN (z0), N ≥ 2.
where Fundamental
S is an element of S (U, pre(Z)), compositional
relation
Gautam Bharali Dynamics of holomorphic correspondences
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The Fatou set: secondary notations
A useful map:
π(k)j : Xk+1 −→ X, π
(k)j : (z0, z1, . . . , zk) 7−→ zj , 0 ≤ j ≤ k.
Let z0 ∈ X and U 3 x0 be an open nbhd. Denote paths in PN (z0) byZ ≡ (Z;α).
Then
Z [j] := (z0, . . . , zj ; α1, . . . , αj), 1 ≤ j ≤ Npre(Z) := Z [N−1], N ≥ 2.
Next, we define sequences of analytic germs:
S (U,Z) := set of irred. components of
Γ•α ∩ (U ×X) containing Z, if Z ∈P1(z0),
S (U,Z) := set of irred. components of
Γ•α ∩(U×
(XN−1k=1 π
(k)k (S)
)×X
)containing Z, if Z ∈PN (z0), N ≥ 2.
where Fundamental
S is an element of S (U, pre(Z)), compositional
relation
Gautam Bharali Dynamics of holomorphic correspondences
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The Fatou set: secondary notations
A useful map:
π(k)j : Xk+1 −→ X, π
(k)j : (z0, z1, . . . , zk) 7−→ zj , 0 ≤ j ≤ k.
Let z0 ∈ X and U 3 x0 be an open nbhd. Denote paths in PN (z0) byZ ≡ (Z;α). Then
Z [j] := (z0, . . . , zj ; α1, . . . , αj), 1 ≤ j ≤ Npre(Z) := Z [N−1], N ≥ 2.
Next, we define sequences of analytic germs:
S (U,Z) := set of irred. components of
Γ•α ∩ (U ×X) containing Z, if Z ∈P1(z0),
S (U,Z) := set of irred. components of
Γ•α ∩(U×
(XN−1k=1 π
(k)k (S)
)×X
)containing Z, if Z ∈PN (z0), N ≥ 2.
where Fundamental
S is an element of S (U, pre(Z)), compositional
relation
Gautam Bharali Dynamics of holomorphic correspondences
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The Fatou set: secondary notations
A useful map:
π(k)j : Xk+1 −→ X, π
(k)j : (z0, z1, . . . , zk) 7−→ zj , 0 ≤ j ≤ k.
Let z0 ∈ X and U 3 x0 be an open nbhd. Denote paths in PN (z0) byZ ≡ (Z;α). Then
Z [j] := (z0, . . . , zj ; α1, . . . , αj), 1 ≤ j ≤ Npre(Z) := Z [N−1], N ≥ 2.
Next, we define sequences of analytic germs:
S (U,Z) := set of irred. components of
Γ•α ∩ (U ×X) containing Z, if Z ∈P1(z0),
S (U,Z) := set of irred. components of
Γ•α ∩(U×
(XN−1k=1 π
(k)k (S)
)×X
)containing Z, if Z ∈PN (z0), N ≥ 2.
where Fundamental
S is an element of S (U, pre(Z)), compositional
relation
Gautam Bharali Dynamics of holomorphic correspondences
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The Fatou set: secondary notations
A useful map:
π(k)j : Xk+1 −→ X, π
(k)j : (z0, z1, . . . , zk) 7−→ zj , 0 ≤ j ≤ k.
Let z0 ∈ X and U 3 x0 be an open nbhd. Denote paths in PN (z0) byZ ≡ (Z;α). Then
Z [j] := (z0, . . . , zj ; α1, . . . , αj), 1 ≤ j ≤ Npre(Z) := Z [N−1], N ≥ 2.
Next, we define sequences of analytic germs:
S (U,Z) := set of irred. components of
Γ•α ∩ (U ×X) containing Z, if Z ∈P1(z0),
S (U,Z) := set of irred. components of
Γ•α ∩(U×
(XN−1k=1 π
(k)k (S)
)×X
)containing Z, if Z ∈PN (z0), N ≥ 2.
where
Fundamental
S is an element of S (U, pre(Z)), compositional
relation
Gautam Bharali Dynamics of holomorphic correspondences
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The Fatou set: secondary notations
A useful map:
π(k)j : Xk+1 −→ X, π
(k)j : (z0, z1, . . . , zk) 7−→ zj , 0 ≤ j ≤ k.
Let z0 ∈ X and U 3 x0 be an open nbhd. Denote paths in PN (z0) byZ ≡ (Z;α). Then
Z [j] := (z0, . . . , zj ; α1, . . . , αj), 1 ≤ j ≤ Npre(Z) := Z [N−1], N ≥ 2.
Next, we define sequences of analytic germs:
S (U,Z) := set of irred. components of
Γ•α ∩ (U ×X) containing Z, if Z ∈P1(z0),
S (U,Z) := set of irred. components of
Γ•α ∩(U×
(XN−1k=1 π
(k)k (S)
)×X
)containing Z, if Z ∈PN (z0), N ≥ 2.
where Fundamental
S is an element of S (U, pre(Z)), compositional
relation
Gautam Bharali Dynamics of holomorphic correspondences
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The Fatou set
Definition
Given a path Z ∈PN (z0), the list (A1, . . . ,AN ;U) is called an analytic branchof Γ along Z if U is a connected open nbhd. of z0 and
(a) Aj ∈ S (U,Z [j]), j = 1, . . . , N , and each Aj+1 is related to Aj ,j = 1, . . . , N − 1 by the fundamental compositional relation;
(b) [Aj ]p is irreducible ∀p ∈ Aj , j = 1, . . . , N .
Definition (The Fatou set)
A point z0 is said to belong to the Fatou set of Γ if there exists a singleconnected open nbhd. U 3 z0 such that for each n ∈ Z+, each Z ∈Pn(z0)admits an analytic branch (A1, . . . ,An;U) of Γ along Z, and such that the set
F (z0) :=π
(n)0 ×π(n)
n (An) : n ∈ Z+, Z ∈Pn(z0), and (A1, . . . ,An;U)
is an analytic branch of Γ along Z ,
viewed as a set comprising currents of integration, is relatively compact in thespace of (1, 1)-currents on U ×X.
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The Fatou set
Definition
Given a path Z ∈PN (z0), the list (A1, . . . ,AN ;U) is called an analytic branchof Γ along Z if U is a connected open nbhd. of z0 and
(a) Aj ∈ S (U,Z [j]), j = 1, . . . , N , and each Aj+1 is related to Aj ,j = 1, . . . , N − 1 by the fundamental compositional relation;
(b) [Aj ]p is irreducible ∀p ∈ Aj , j = 1, . . . , N .
Definition (The Fatou set)
A point z0 is said to belong to the Fatou set of Γ if there exists a singleconnected open nbhd. U 3 z0 such that for each n ∈ Z+, each Z ∈Pn(z0)admits an analytic branch (A1, . . . ,An;U) of Γ along Z, and such that the set
F (z0) :=π
(n)0 ×π(n)
n (An) : n ∈ Z+, Z ∈Pn(z0), and (A1, . . . ,An;U)
is an analytic branch of Γ along Z ,
viewed as a set comprising currents of integration, is relatively compact in thespace of (1, 1)-currents on U ×X.
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The Fatou set
Definition
Given a path Z ∈PN (z0), the list (A1, . . . ,AN ;U) is called an analytic branchof Γ along Z if U is a connected open nbhd. of z0 and
(a) Aj ∈ S (U,Z [j]), j = 1, . . . , N , and each Aj+1 is related to Aj ,j = 1, . . . , N − 1 by the fundamental compositional relation;
(b) [Aj ]p is irreducible ∀p ∈ Aj , j = 1, . . . , N .
Definition (The Fatou set)
A point z0 is said to belong to the Fatou set of Γ if there exists a singleconnected open nbhd. U 3 z0 such that for each n ∈ Z+, each Z ∈Pn(z0)admits an analytic branch (A1, . . . ,An;U) of Γ along Z, and such that the set
F (z0) :=π
(n)0 ×π(n)
n (An) : n ∈ Z+, Z ∈Pn(z0), and (A1, . . . ,An;U)
is an analytic branch of Γ along Z ,
viewed as a set comprising currents of integration, is relatively compact in thespace of (1, 1)-currents on U ×X.
Gautam Bharali Dynamics of holomorphic correspondences
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The Fatou set
Definition
Given a path Z ∈PN (z0), the list (A1, . . . ,AN ;U) is called an analytic branchof Γ along Z if U is a connected open nbhd. of z0 and
(a) Aj ∈ S (U,Z [j]), j = 1, . . . , N , and each Aj+1 is related to Aj ,j = 1, . . . , N − 1 by the fundamental compositional relation;
(b) [Aj ]p is irreducible ∀p ∈ Aj , j = 1, . . . , N .
Definition (The Fatou set)
A point z0 is said to belong to the Fatou set of Γ if there exists a singleconnected open nbhd. U 3 z0 such that for each n ∈ Z+, each Z ∈Pn(z0)admits an analytic branch (A1, . . . ,An;U) of Γ along Z, and
such that the set
F (z0) :=π
(n)0 ×π(n)
n (An) : n ∈ Z+, Z ∈Pn(z0), and (A1, . . . ,An;U)
is an analytic branch of Γ along Z ,
viewed as a set comprising currents of integration, is relatively compact in thespace of (1, 1)-currents on U ×X.
Gautam Bharali Dynamics of holomorphic correspondences
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The Fatou set
Definition
Given a path Z ∈PN (z0), the list (A1, . . . ,AN ;U) is called an analytic branchof Γ along Z if U is a connected open nbhd. of z0 and
(a) Aj ∈ S (U,Z [j]), j = 1, . . . , N , and each Aj+1 is related to Aj ,j = 1, . . . , N − 1 by the fundamental compositional relation;
(b) [Aj ]p is irreducible ∀p ∈ Aj , j = 1, . . . , N .
Definition (The Fatou set)
A point z0 is said to belong to the Fatou set of Γ if there exists a singleconnected open nbhd. U 3 z0 such that for each n ∈ Z+, each Z ∈Pn(z0)admits an analytic branch (A1, . . . ,An;U) of Γ along Z, and such that the set
F (z0) :=π
(n)0 ×π(n)
n (An) : n ∈ Z+, Z ∈Pn(z0), and (A1, . . . ,An;U)
is an analytic branch of Γ along Z ,
viewed as a set comprising currents of integration, is relatively compact in thespace of (1, 1)-currents on U ×X.
Gautam Bharali Dynamics of holomorphic correspondences
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The iterative tree
Suppose, for z0 ∈ X, ∃U 3 z0, a connected nbhd. of z0, such that for eachn ∈ Z+, each Z ∈Pn(z0) admits an analytic branch (A1, . . . ,AN ;U) ofΓ along Z. We can define an infinite tree τ(Γ,U) as follows.
V (τ(Γ,U)) :=⋃n∈Z+
⋃Z∈Pn(z0)
S (U,Z),
E(τ(Γ,U)) is defined by the condition
there is an edge between A ,B ∈ V (τ(Γ,U)
⇐⇒ A ∈ S (U,Z) for some Z ∈Pn(z0), n ≥ 2, and
B ∈ S (U, pre(Z)).
Such a tree is called the iterative tree at z0.
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The iterative tree
Suppose, for z0 ∈ X, ∃U 3 z0, a connected nbhd. of z0, such that for eachn ∈ Z+, each Z ∈Pn(z0) admits an analytic branch (A1, . . . ,AN ;U) ofΓ along Z. We can define an infinite tree τ(Γ,U) as follows.
V (τ(Γ,U)) :=⋃n∈Z+
⋃Z∈Pn(z0)
S (U,Z),
E(τ(Γ,U)) is defined by the condition
there is an edge between A ,B ∈ V (τ(Γ,U)
⇐⇒ A ∈ S (U,Z) for some Z ∈Pn(z0), n ≥ 2, and
B ∈ S (U, pre(Z)).
Such a tree is called the iterative tree at z0.
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The iterative tree
Suppose, for z0 ∈ X, ∃U 3 z0, a connected nbhd. of z0, such that for eachn ∈ Z+, each Z ∈Pn(z0) admits an analytic branch (A1, . . . ,AN ;U) ofΓ along Z. We can define an infinite tree τ(Γ,U) as follows.
V (τ(Γ,U)) :=⋃n∈Z+
⋃Z∈Pn(z0)
S (U,Z),
E(τ(Γ,U)) is defined by the condition
there is an edge between A ,B ∈ V (τ(Γ,U)
⇐⇒ A ∈ S (U,Z) for some Z ∈Pn(z0), n ≥ 2, and
B ∈ S (U, pre(Z)).
Such a tree is called the iterative tree at z0.
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The Fatou set: very basic properties
Unlike the case with rational maps, F (Γ ) and supp(µΓ ) do not, ingeneral, partition X under the condition
dtop(Γ ) > dtop(†Γ ) ≥ 2.
This follows from certain computer experiments by Shaun Bullett from the1990s, read together with an entropy estimate of Dinh–Sibony.
This is the motivation of Theorem B, which we shall see soon.
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The Fatou set: very basic properties
Unlike the case with rational maps, F (Γ ) and supp(µΓ ) do not, ingeneral, partition X under the condition
dtop(Γ ) > dtop(†Γ ) ≥ 2.
This follows from certain computer experiments by Shaun Bullett from the1990s, read together with an entropy estimate of Dinh–Sibony.
This is the motivation of Theorem B, which we shall see soon.
Gautam Bharali Dynamics of holomorphic correspondences
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The Fatou set: very basic properties
Unlike the case with rational maps, F (Γ ) and supp(µΓ ) do not, ingeneral, partition X under the condition
dtop(Γ ) > dtop(†Γ ) ≥ 2.
This follows from certain computer experiments by Shaun Bullett from the1990s, read together with an entropy estimate of Dinh–Sibony.
This is the motivation of Theorem B, which we shall see soon.
Gautam Bharali Dynamics of holomorphic correspondences
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Preliminaries to proving Theorem A
Formal principle behind the pull-back of a current:
For a k-dim’l. manifold X and a (p, p)-current
Γ ∗(S) := (π1)∗ (π∗2(S) ∧ [Γ ]) ,
whenever the intersection of π∗2(S) with [Γ ] makes sense.
So, for instance, viewing a smooth (k, k)-form Ω as a current, and a testfunction as a (0, 0)-form,
〈Γ ∗(Ω), ϕ〉 :=
N∑j=1
mj
∫reg(Γj)
(π1|Γj
)∗ϕ(π2|Γj
)∗Ω.
• comes from dualising (π1)∗,
• is the interpretation of “(π∗2(Ω) ∧ [Γ ])” in this case.
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Preliminaries to proving Theorem A
Formal principle behind the pull-back of a current:
For a k-dim’l. manifold X and a (p, p)-current
Γ ∗(S) := (π1)∗ (π∗2(S) ∧ [Γ ]) ,
whenever the intersection of π∗2(S) with [Γ ] makes sense.
So, for instance, viewing a smooth (k, k)-form Ω as a current, and a testfunction as a (0, 0)-form,
〈Γ ∗(Ω), ϕ〉 :=
N∑j=1
mj
∫reg(Γj)
(π1|Γj
)∗ϕ(π2|Γj
)∗Ω.
• comes from dualising (π1)∗,
• is the interpretation of “(π∗2(Ω) ∧ [Γ ])” in this case.
Gautam Bharali Dynamics of holomorphic correspondences
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Preliminaries to proving Theorem A
Formal principle behind the pull-back of a current:
For a k-dim’l. manifold X and a (p, p)-current
Γ ∗(S) := (π1)∗ (π∗2(S) ∧ [Γ ]) ,
whenever the intersection of π∗2(S) with [Γ ] makes sense.
So, for instance, viewing a smooth (k, k)-form Ω as a current, and a testfunction as a (0, 0)-form,
〈Γ ∗(Ω), ϕ〉 :=
N∑j=1
mj
∫reg(Γj)
(π1|Γj
)∗ϕ(π2|Γj
)∗Ω.
• comes from dualising (π1)∗,
• is the interpretation of “(π∗2(Ω) ∧ [Γ ])” in this case.
Gautam Bharali Dynamics of holomorphic correspondences
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Preliminaries to proving Theorem A
Formal principle behind the pull-back of a current:
For a k-dim’l. manifold X and a (p, p)-current
Γ ∗(S) := (π1)∗ (π∗2(S) ∧ [Γ ]) ,
whenever the intersection of π∗2(S) with [Γ ] makes sense.
So, for instance, viewing a smooth (k, k)-form Ω as a current, and a testfunction as a (0, 0)-form,
〈Γ ∗(Ω), ϕ〉 :=
N∑j=1
mj
∫reg(Γj)
(π1|Γj
)∗ϕ(π2|Γj
)∗Ω.
• comes from dualising (π1)∗,
• is the interpretation of “(π∗2(Ω) ∧ [Γ ])” in this case.
Gautam Bharali Dynamics of holomorphic correspondences
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The proof of Theorem A
Assume F (Γ ) 6= ∅. Nothing to prove otherwise.
Pick a z0 in F (Γ ), and let U 3 z0 be as given by the definition. It sufficesto show that for any non-negative function ϕ ∈ C(X;R) withsupp(ϕ) ⊂ U ,
∫X ϕdµΓ = 0.
Let ωX denote the normalized Kahler form associated to the hyperbolicmetric. Call dtop(Γ ) =: d1 and dtop(
†Γ ) =: d0. Easy to show that
〈(Γn)∗(ωX), ϕ〉 =∑
Z∈Pn(z0)
∑A∈S (U,Z)
∫reg(A )
(π1|Γj
)∗ϕ(π2|Γj
)∗ωX ,
where A = π(n)0 ×π
(n)n (A ). Thus:
d−n1 |〈(Γn)∗(ωX), ϕ〉| ≤ d−n1 sup |ϕ|∑
Z∈Pn(z0)
∑A∈S (U,Z)
∫reg(A )
(π2|Γj
)∗ωX ,
Gautam Bharali Dynamics of holomorphic correspondences
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The proof of Theorem A
Assume F (Γ ) 6= ∅. Nothing to prove otherwise.
Pick a z0 in F (Γ ), and let U 3 z0 be as given by the definition. It sufficesto show that for any non-negative function ϕ ∈ C(X;R) withsupp(ϕ) ⊂ U ,
∫X ϕdµΓ = 0.
Let ωX denote the normalized Kahler form associated to the hyperbolicmetric. Call dtop(Γ ) =: d1 and dtop(
†Γ ) =: d0. Easy to show that
〈(Γn)∗(ωX), ϕ〉 =∑
Z∈Pn(z0)
∑A∈S (U,Z)
∫reg(A )
(π1|Γj
)∗ϕ(π2|Γj
)∗ωX ,
where A = π(n)0 ×π
(n)n (A ). Thus:
d−n1 |〈(Γn)∗(ωX), ϕ〉| ≤ d−n1 sup |ϕ|∑
Z∈Pn(z0)
∑A∈S (U,Z)
∫reg(A )
(π2|Γj
)∗ωX ,
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The proof of Theorem A
Assume F (Γ ) 6= ∅. Nothing to prove otherwise.
Pick a z0 in F (Γ ), and let U 3 z0 be as given by the definition. It sufficesto show that for any non-negative function ϕ ∈ C(X;R) withsupp(ϕ) ⊂ U ,
∫X ϕdµΓ = 0.
Let ωX denote the normalized Kahler form associated to the hyperbolicmetric. Call dtop(Γ ) =: d1 and dtop(
†Γ ) =: d0.
Easy to show that
〈(Γn)∗(ωX), ϕ〉 =∑
Z∈Pn(z0)
∑A∈S (U,Z)
∫reg(A )
(π1|Γj
)∗ϕ(π2|Γj
)∗ωX ,
where A = π(n)0 ×π
(n)n (A ). Thus:
d−n1 |〈(Γn)∗(ωX), ϕ〉| ≤ d−n1 sup |ϕ|∑
Z∈Pn(z0)
∑A∈S (U,Z)
∫reg(A )
(π2|Γj
)∗ωX ,
Gautam Bharali Dynamics of holomorphic correspondences
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The proof of Theorem A
Assume F (Γ ) 6= ∅. Nothing to prove otherwise.
Pick a z0 in F (Γ ), and let U 3 z0 be as given by the definition. It sufficesto show that for any non-negative function ϕ ∈ C(X;R) withsupp(ϕ) ⊂ U ,
∫X ϕdµΓ = 0.
Let ωX denote the normalized Kahler form associated to the hyperbolicmetric. Call dtop(Γ ) =: d1 and dtop(
†Γ ) =: d0. Easy to show that
〈(Γn)∗(ωX), ϕ〉 =∑
Z∈Pn(z0)
∑A∈S (U,Z)
∫reg(A )
(π1|Γj
)∗ϕ(π2|Γj
)∗ωX ,
where A = π(n)0 ×π
(n)n (A ).
Thus:
d−n1 |〈(Γn)∗(ωX), ϕ〉| ≤ d−n1 sup |ϕ|∑
Z∈Pn(z0)
∑A∈S (U,Z)
∫reg(A )
(π2|Γj
)∗ωX ,
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The proof of Theorem A
Assume F (Γ ) 6= ∅. Nothing to prove otherwise.
Pick a z0 in F (Γ ), and let U 3 z0 be as given by the definition. It sufficesto show that for any non-negative function ϕ ∈ C(X;R) withsupp(ϕ) ⊂ U ,
∫X ϕdµΓ = 0.
Let ωX denote the normalized Kahler form associated to the hyperbolicmetric. Call dtop(Γ ) =: d1 and dtop(
†Γ ) =: d0. Easy to show that
〈(Γn)∗(ωX), ϕ〉 =∑
Z∈Pn(z0)
∑A∈S (U,Z)
∫reg(A )
(π1|Γj
)∗ϕ(π2|Γj
)∗ωX ,
where A = π(n)0 ×π
(n)n (A ). Thus:
d−n1 |〈(Γn)∗(ωX), ϕ〉| ≤ d−n1 sup |ϕ|∑
Z∈Pn(z0)
∑A∈S (U,Z)
∫reg(A )
(π2|Γj
)∗ωX ,
Gautam Bharali Dynamics of holomorphic correspondences
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The proof of Theorem AHence
d−n1 |〈(Γn)∗(ωX), ϕ〉| ≤ C d−n1
∑Z∈Pn(z0)
∑A∈S (U,Z)
Vol(A ). (∗∗)
At this stage, we need a new tool:
Bishop’s Compactness Theorem (in the style of Harvey–Schiffman).Let (X1, ω1) and (X2, ω2) be compact k-dim’l. Kahler manifolds, and letU be a relatively compact open subset of X1. Let F be a family ofreduced, irreducible, analytic subsets of U ×X2 of pure dimensionp : 1 ≤ p ≤ k. Then, F is compact in the space of currents of bidimension(p, p) if & only if
(a) The volumes of the sets in F are uniformly bounded; and
(b) Given a compact K ⊂ U , there ∃CK > 0 such that, for A ,B ∈ F ,A ∩ (K ×X2) and B ∩ (K ×X2) are no farther than CK in theHausdorff metric.
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The proof of Theorem AHence
d−n1 |〈(Γn)∗(ωX), ϕ〉| ≤ C d−n1
∑Z∈Pn(z0)
∑A∈S (U,Z)
Vol(A ). (∗∗)
At this stage, we need a new tool:
Bishop’s Compactness Theorem (in the style of Harvey–Schiffman).Let (X1, ω1) and (X2, ω2) be compact k-dim’l. Kahler manifolds, and letU be a relatively compact open subset of X1. Let F be a family ofreduced, irreducible, analytic subsets of U ×X2 of pure dimensionp : 1 ≤ p ≤ k. Then, F is compact in the space of currents of bidimension(p, p) if & only if
(a) The volumes of the sets in F are uniformly bounded; and
(b) Given a compact K ⊂ U , there ∃CK > 0 such that, for A ,B ∈ F ,A ∩ (K ×X2) and B ∩ (K ×X2) are no farther than CK in theHausdorff metric.
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The proof of Theorem AHence
d−n1 |〈(Γn)∗(ωX), ϕ〉| ≤ C d−n1
∑Z∈Pn(z0)
∑A∈S (U,Z)
Vol(A ). (∗∗)
At this stage, we need a new tool:
Bishop’s Compactness Theorem (in the style of Harvey–Schiffman).Let (X1, ω1) and (X2, ω2) be compact k-dim’l. Kahler manifolds, and letU be a relatively compact open subset of X1. Let F be a family ofreduced, irreducible, analytic subsets of U ×X2 of pure dimensionp : 1 ≤ p ≤ k.
Then, F is compact in the space of currents of bidimension(p, p) if & only if
(a) The volumes of the sets in F are uniformly bounded; and
(b) Given a compact K ⊂ U , there ∃CK > 0 such that, for A ,B ∈ F ,A ∩ (K ×X2) and B ∩ (K ×X2) are no farther than CK in theHausdorff metric.
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The proof of Theorem AHence
d−n1 |〈(Γn)∗(ωX), ϕ〉| ≤ C d−n1
∑Z∈Pn(z0)
∑A∈S (U,Z)
Vol(A ). (∗∗)
At this stage, we need a new tool:
Bishop’s Compactness Theorem (in the style of Harvey–Schiffman).Let (X1, ω1) and (X2, ω2) be compact k-dim’l. Kahler manifolds, and letU be a relatively compact open subset of X1. Let F be a family ofreduced, irreducible, analytic subsets of U ×X2 of pure dimensionp : 1 ≤ p ≤ k. Then, F is compact in the space of currents of bidimension(p, p) if & only if
(a) The volumes of the sets in F are uniformly bounded; and
(b) Given a compact K ⊂ U , there ∃CK > 0 such that, for A ,B ∈ F ,A ∩ (K ×X2) and B ∩ (K ×X2) are no farther than CK in theHausdorff metric.
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The proof of Theorem AHence
d−n1 |〈(Γn)∗(ωX), ϕ〉| ≤ C d−n1
∑Z∈Pn(z0)
∑A∈S (U,Z)
Vol(A ). (∗∗)
At this stage, we need a new tool:
Bishop’s Compactness Theorem (in the style of Harvey–Schiffman).Let (X1, ω1) and (X2, ω2) be compact k-dim’l. Kahler manifolds, and letU be a relatively compact open subset of X1. Let F be a family ofreduced, irreducible, analytic subsets of U ×X2 of pure dimensionp : 1 ≤ p ≤ k. Then, F is compact in the space of currents of bidimension(p, p) if & only if
(a) The volumes of the sets in F are uniformly bounded; and
(b) Given a compact K ⊂ U , there ∃CK > 0 such that, for A ,B ∈ F ,A ∩ (K ×X2) and B ∩ (K ×X2) are no farther than CK in theHausdorff metric.
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The proof of Theorem AHence
d−n1 |〈(Γn)∗(ωX), ϕ〉| ≤ C d−n1
∑Z∈Pn(z0)
∑A∈S (U,Z)
Vol(A ). (∗∗)
At this stage, we need a new tool:
Bishop’s Compactness Theorem (in the style of Harvey–Schiffman).Let (X1, ω1) and (X2, ω2) be compact k-dim’l. Kahler manifolds, and letU be a relatively compact open subset of X1. Let F be a family ofreduced, irreducible, analytic subsets of U ×X2 of pure dimensionp : 1 ≤ p ≤ k. Then, F is compact in the space of currents of bidimension(p, p) if & only if
(a) The volumes of the sets in F are uniformly bounded; and
(b) Given a compact K ⊂ U , there ∃CK > 0 such that, for A ,B ∈ F ,A ∩ (K ×X2) and B ∩ (K ×X2) are no farther than CK in theHausdorff metric.
Gautam Bharali Dynamics of holomorphic correspondences
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The proof of Theorem A
Thus, from (∗∗), we have that
d−n1 |〈(Γn)∗(ωX), ϕ〉| ≤ C
(d0
d1
)n,
whence ∫X
ϕdµΓ = limn→∞
d−n1 〈(Γn)∗(ωX), ϕ〉 = 0.
Hence the result.
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The proof of Theorem A
Thus, from (∗∗), we have that
d−n1 |〈(Γn)∗(ωX), ϕ〉| ≤ C
(d0
d1
)n,
whence ∫X
ϕdµΓ = limn→∞
d−n1 〈(Γn)∗(ωX), ϕ〉 = 0.
Hence the result.
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Theorem B: preliminaries
Observe: (∗∗) suggests that one could allow the volumes of branches togrow at a certain exponential rate. This motivates the following:
We say that most analytic branches of Γ around z0 converge if there exista connected nbhd. U 3 z0 that admits an iterative tree τ(Γ,U), and anε ∈ (0, 1) such that there is a connected subtree τ(Γ,U), and so that
The n-th generation of τ(Γ,U) contains at least (1− εn) dn0 verticesfrom the n-th generation of the iterative tree; and
The family
F (z0) := π(n)0 ×π(n)
n (An) : n ∈ Z+, and An ∈ Vn(τ(Γ,U))
is relatively compact in the space of (1, 1)-currents on U ×X.
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Theorem B: preliminaries
Observe: (∗∗) suggests that one could allow the volumes of branches togrow at a certain exponential rate. This motivates the following:
We say that most analytic branches of Γ around z0 converge if there exista connected nbhd. U 3 z0 that admits an iterative tree τ(Γ,U), and anε ∈ (0, 1) such that
there is a connected subtree τ(Γ,U), and so that
The n-th generation of τ(Γ,U) contains at least (1− εn) dn0 verticesfrom the n-th generation of the iterative tree; and
The family
F (z0) := π(n)0 ×π(n)
n (An) : n ∈ Z+, and An ∈ Vn(τ(Γ,U))
is relatively compact in the space of (1, 1)-currents on U ×X.
Gautam Bharali Dynamics of holomorphic correspondences
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Theorem B: preliminaries
Observe: (∗∗) suggests that one could allow the volumes of branches togrow at a certain exponential rate. This motivates the following:
We say that most analytic branches of Γ around z0 converge if there exista connected nbhd. U 3 z0 that admits an iterative tree τ(Γ,U), and anε ∈ (0, 1) such that there is a connected subtree τ(Γ,U), and so that
The n-th generation of τ(Γ,U) contains at least (1− εn) dn0 verticesfrom the n-th generation of the iterative tree; and
The family
F (z0) := π(n)0 ×π(n)
n (An) : n ∈ Z+, and An ∈ Vn(τ(Γ,U))
is relatively compact in the space of (1, 1)-currents on U ×X.
Gautam Bharali Dynamics of holomorphic correspondences
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Theorem B: preliminaries
Observe: (∗∗) suggests that one could allow the volumes of branches togrow at a certain exponential rate. This motivates the following:
We say that most analytic branches of Γ around z0 converge if there exista connected nbhd. U 3 z0 that admits an iterative tree τ(Γ,U), and anε ∈ (0, 1) such that there is a connected subtree τ(Γ,U), and so that
The n-th generation of τ(Γ,U) contains at least (1− εn) dn0 verticesfrom the n-th generation of the iterative tree; and
The family
F (z0) := π(n)0 ×π(n)
n (An) : n ∈ Z+, and An ∈ Vn(τ(Γ,U))
is relatively compact in the space of (1, 1)-currents on U ×X.
Gautam Bharali Dynamics of holomorphic correspondences
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Theorem B: preliminaries
Observe: (∗∗) suggests that one could allow the volumes of branches togrow at a certain exponential rate. This motivates the following:
We say that most analytic branches of Γ around z0 converge if there exista connected nbhd. U 3 z0 that admits an iterative tree τ(Γ,U), and anε ∈ (0, 1) such that there is a connected subtree τ(Γ,U), and so that
The n-th generation of τ(Γ,U) contains at least (1− εn) dn0 verticesfrom the n-th generation of the iterative tree; and
The family
F (z0) := π(n)0 ×π(n)
n (An) : n ∈ Z+, and An ∈ Vn(τ(Γ,U))
is relatively compact in the space of (1, 1)-currents on U ×X.
Gautam Bharali Dynamics of holomorphic correspondences
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Theorem B
Theorem B (B., 2015)
Let X be a compact Riemann surface. Let Γ and µΓ be as in Theorem A.Suppose the postcritical set of Γ is disjoint from supp(µΓ ).
Define:
F (Γ ) := the largest open subset of X consisting of points z0 ∈ X such
that most analytic branches of Γ around z0 converge.
Then, F (Γ )C = supp(µΓ ).
The proof of
F (Γ )C ⊇ supp(µΓ )
follows from the fact that there is at most exponential volume-growth ofanalytic branches of Γ , and that F (Γ ) does not depend on the size of ε along as 0 < ε < 1. We then apply (∗∗).
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Theorem B
Theorem B (B., 2015)
Let X be a compact Riemann surface. Let Γ and µΓ be as in Theorem A.Suppose the postcritical set of Γ is disjoint from supp(µΓ ). Define:
F (Γ ) := the largest open subset of X consisting of points z0 ∈ X such
that most analytic branches of Γ around z0 converge.
Then, F (Γ )C = supp(µΓ ).
The proof of
F (Γ )C ⊇ supp(µΓ )
follows from the fact that there is at most exponential volume-growth ofanalytic branches of Γ , and that F (Γ ) does not depend on the size of ε along as 0 < ε < 1. We then apply (∗∗).
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Theorem B
Theorem B (B., 2015)
Let X be a compact Riemann surface. Let Γ and µΓ be as in Theorem A.Suppose the postcritical set of Γ is disjoint from supp(µΓ ). Define:
F (Γ ) := the largest open subset of X consisting of points z0 ∈ X such
that most analytic branches of Γ around z0 converge.
Then, F (Γ )C = supp(µΓ ).
The proof of
F (Γ )C ⊇ supp(µΓ )
follows from the fact that there is at most exponential volume-growth ofanalytic branches of Γ , and that F (Γ ) does not depend on the size of ε along as 0 < ε < 1. We then apply (∗∗).
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Theorem B
Theorem B (B., 2015)
Let X be a compact Riemann surface. Let Γ and µΓ be as in Theorem A.Suppose the postcritical set of Γ is disjoint from supp(µΓ ). Define:
F (Γ ) := the largest open subset of X consisting of points z0 ∈ X such
that most analytic branches of Γ around z0 converge.
Then, F (Γ )C = supp(µΓ ).
The proof of
F (Γ )C ⊇ supp(µΓ )
follows from the fact that there is at most exponential volume-growth ofanalytic branches of Γ ,
and that F (Γ ) does not depend on the size of ε along as 0 < ε < 1. We then apply (∗∗).
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Theorem B
Theorem B (B., 2015)
Let X be a compact Riemann surface. Let Γ and µΓ be as in Theorem A.Suppose the postcritical set of Γ is disjoint from supp(µΓ ). Define:
F (Γ ) := the largest open subset of X consisting of points z0 ∈ X such
that most analytic branches of Γ around z0 converge.
Then, F (Γ )C = supp(µΓ ).
The proof of
F (Γ )C ⊇ supp(µΓ )
follows from the fact that there is at most exponential volume-growth ofanalytic branches of Γ , and that F (Γ ) does not depend on the size of ε along as 0 < ε < 1. We then apply (∗∗).
Gautam Bharali Dynamics of holomorphic correspondences