applications of monotone spaces theory: peano curves...

Prologue Monotone metric spaces Urbanski Conjecture Peano curves Universal measure zero Intermezzo The plane Monotone graphs A conjecture

Applications of monotone spaces theory:Peano curves, Urbanski conjecture, differentiability. . .

Ondrej Zindulka

Czech Technical University Prague

Stara Lesna 2012

Ondrej Zindulka Applications of monotone spaces theory


Mapping a set of reals onto an interval

Theorem

Suppose C ⊆ R is compact, L(C) > 0.Then there is an onto Lipschitz mapping g : C → [0, 1].

Proof.

g(x) = L(C ∩ (−∞, x]

)g takes all values between 0 and L(C), hence g[C] contains an interval

|g(y)− g(x)| = L(C ∩ [x, y]

)6 |y − x|, i.e. g is Lipschitz



Mapping a set of reals onto an interval

Theorem

Suppose C ⊆ R is compact, Hs(C) > 0.Then there is an onto s-Holder mapping g : C → [0, 1].

Proof.

Frostman Lemma: there is a Borel measure µ on C such that µ[x, y] 6 |x− y|s.

g(x) = µ(C ∩ (−∞, x])

g[C] contains an interval

|g(y)− g(x)| 6 µ[x, y] 6 |y − x|s, i.e. g is s-Holder



Monotone metric spaces

Definition (Oz 2012)

A metric space (X, d) is

1-monotone if there is a linear order on X such that

x < y < z =⇒ d(x, y) 6 d(x, z),

monotone if it is Lipschitz equivalent to a 1-monotone space

Theorem (Keleti, Mathe, Oz 2012)

Suppose X is analytic and monotone. If Hs(X) > 0, then there is an onto s-Holder mappingg : C → [0, 1].

Corollary (Keleti, Mathe, Oz 2012)

Suppose X is analytic and monotone.If Hn(X) > 0, then there is an onto Lipschitz mapping g : X → [0, 1]n.

Theorem (Nekvinda, Oz 2011)

Every separable monotone space embeds into the line.



Lipschitz-ultrametric sets

Definition

A metric space (X, d) is

ultrametric if d(x, z) 6 max{d(x, y), d(y, z)

}.

Lipschitz-ultrametric if it is Lipschitz equivalent to an ultrametric space.

Theorem (Mendel, Naor 2012)

Let X be an analytic metric space. For every ε > 0 there is a compact Lipschitz-ultrametric setZ ⊆ X such that

dimH Y > dimH X − ε.

Theorem (Nekvinda, Oz 2011)

Every Lipschitz-ultrametric space is monotone.

Corollary (Keleti, Mathe, Oz 2012)

Suppose X is analytic. If dimH X > n, then there is an onto Lipschitz mapping g : X → [0, 1]n.



Transitive Hausdorff dimension

Definition (Transitive Hausdorff dimension — Urbanski 2009)

Let X be a separable metric space. Define

tHDX = sup{dim f [X] : f Lipschitz}

Proposition (Hurewicz–Wallman 1941)

tHDX 6 dimH X

Urbanski Conjecture

If X is a metric space with finite Hausdorff dimension, then

tHDX = bdimH Xc or tHDX = bdimH Xc − 1.

Failure

Urbanski Conjecture consistently fails: If ∃A ⊆ R2 such that |A| < c and L2(A) > 0.



Urbanski Conjecture

Corollary (Keleti–Mathe–Oz 2012)

Suppose X is analytic.If dimH X > n, then there is an onto Lipschitz mapping g : X → [0, 1]n.

Theorem (Keleti–Mathe–Oz 2012)

Suppose X is analytic.

If dimH X is finite but not an integer, then tHDX = bdimH Xc,if dimH X is an integer, then tHDX = dimH X or tHDX = dimH X − 1,

if dimH X =∞, then tHDX =∞.

Therefore Urbanski Conjecture holds for analytic spaces.



Failure of Urbanski Conjecture?

Theorem (Keleti–Mathe–Oz 2012)

There exist separable metric spaces with zero transfinite Hausdorff dimension and arbitrarily largeHausdorff dimension.

Question

Is Urbanski Conjecture consistently true?



Transfinite Hausdorff dimension revisited

Proposition

tHDX > 1 if and only if there is an onto Lipschitz mapping f : X → [0, 1].

Proof.

Enough to show: If dimX > 0, then there is an onto Lipschitz mapping f : X → [0, 1].

If dimX > 0, there is x0 ∈ X such that all small circles centered at x0 are nonempty.

Define f(x) = d(x0, x).

Question

Is it true that tHDX > n if and only if there is an onto Lipschitz mapping f : X → [0, 1]n?

Theorem (Oz)

Yes if X is σ-compact.



Laczkovich Conjecture

Laczkovich Conjecture

Let X ⊆ Rn be compact. If Ln(X) > 0, then X maps by a Lipschitz map onto a ball.

Status of Laczkovich Conjecture

trivially holds for n = 1

[Preiss ∼1995] very nontrivially holds for n = 2

[Csornyei, Jones] hopefully holds for n > 3

Theorem (Oz)

Suppose Laczkovich Conjecture holds. If X is analytic, thentHDX > n if and only if there is an onto Lipschitz mapping f : X → [0, 1]n.



Peano curves

Peano curves

Peano curve is an onto continuous mapping p : [0, 1]m → [0, 1]n.

Holder mapping

A mapping is β-Holder if d(fx, fy) 6 Cd(x, y)β for some constant C

Holder Peano curves

There exist 12

-Holder Peano curves [0, 1]→ [0, 1]2

There exist 1n

-Holder Peano curves [0, 1]→ [0, 1]n

Question

Is there a 23

-Holder Peano curve [0, 1]2 → [0, 1]3?

Find S23 = sup{β : there is a β-Holder Peano curve [0, 1]2 → [0, 1]3}



Peano curves: partial answer

Nearly Holder

A mapping is

nearly s-Holder if it is β-Holder for all β < s,

nearly Lipschitz if it is nearly 1-Holder.

Theorem

Let X ⊆ Rn be analytic, S ⊆ Rm self-similar and s = dimH S.If Hs(X) > 0, then there is a nearly Lipschitz mapping f : Rn → Rm such that S ⊆ f(X).

Corollary

Let X ⊆ Rm be analytic.If Hs(X) > 0, then there is an onto, nearly s

n-Holder mapping f : X → [0, 1]n.

Corollary

There is a nearly mn

-Holder Peano curve [0, 1]m → [0, 1]n.

In particular, S23 =2

3.



Universal measure zero sets with large Hausdorff dimension

Universal measure zero

A separable metric space X has universal measure zero (UMZ) if there is no diffused Borel measureµ, 0 < µ(X) <∞.

Theorem (Oz 2005)

There is a UMZ set X ⊆ [0, 1] such that dimH X = 1.

Theorem (Oz)

Every analytic metric space contains a UMZ set of the same Hausdorff dimension.



Industry of monotone spaces: papers and topics

Papers

4 published

1 accepted

2 submitted

4 in preparation

1 PhD thesis

Topics

Urbanski Conjecture

Universal measure zero

Peano curves

cardinal invariants of porous sets

fractal properties

geometry of curves

differentiability of functions



Industry of monotone spaces: people involved

People involved

Pieter Allaart

Michael Hrusak

Tamas Keleti

Andras Mathe

Tamas Matrai

Ales Nekvinda

Dusan Pokorny

Arturo Rodrigues

Vasek Vlasak

Oz

Nations involved

Americans, Czechs, Hungarians, Mexicans. But no Polish and no Slovaks!



Monotone sets in the plane

c-monotone sets

X ⊆ R2 is c-monotone if there is a linear order on X such that

x < y < z =⇒ |y − x| 6 c · |z − x|.

X is monotone ⇐⇒ X is c-monotone for some c.

Theorem (Hrusak, Oz 2011)

If X ⊆ R2 is monotone, then it is porous and dimH X < 2.

Example (Nekvinda, Oz 2011)

There is a set Cantor set C ⊆ R2 with the following properties:

H1(C) = 1,

if Y ⊆ C is monotone, then H1(Y ) = 0 and Y is nowhere dense in C,

there is Y ⊆ C closed such that dimH Y = 0 and Y is not a countable union of monotone sets.



Non-σ-monotone dust in the plane



Curves

Question

Do monotone curves have low Hausdorff dimension?

Proposition (Pieter Allaart, Oz)

Koch curve is monotone. The optimal witnessing constant is c =

√490747

277636.



Fuctions with monotone graphs

Framework

f : I → R is a continuous function on an interval I

Gr(f) is the graph of f

ψ(x) = (x, f(x)) is the natural parametrization of the graph

Dini derivatives

Right upper Dini derivative: f+(x)

Derivative: f ′(x)

D-point: f ′(x) exists

D is the set of all D-points

knot point: left and right derivatives do not exist, all Dini derivatives infinite

K is the set of all knot points

Approximate Dini derivatives

f+app(x), f

′app(x), Dapp etc.



M-points

Vague question

How is differentiability of f related to monotonicity of Gr(f)?

Definition (M-points)

y ∈ I is an M-point if there are c and ε > 0 such that

x ∈ (y − ε, y), z ∈ (y, y + ε) =⇒ |ψ(y)− ψ(x)| 6 c|ψ(z)− ψ(x)|.

M is the set of all M-points.



M-points

Theorem (Oz, Hrusak, Matrai, Nekvinda, Vlasak 201?)

There is a set E ⊆ I such that H1(Gr(f |E)) = 0 and

D ⊆M ⊆ Dapp ∪Kapp ∪E.

Therefore

Every D-point is an M-point.

Almost every M-point is either an approximate D-point or a knot point.

Theorem (Oz, Hrusak, Matrai, Nekvinda, Vlasak 201?)

The set Gr(f |M) has σ-finite linear measure. In particular, dimH Gr(f |M) = 1.

Corollary

If Gr(f) is monotone, then Gr(f) has σ-finite linear measure and thus dimH Gr(f) = 1.



Differentiability when the graph is monotone

Proposition

If Gr(f) is monotone, then f+app(a) = f+(a) for all a ∈ [0, 1]. Likewise for all Dini derivatives.

Theorem

If Gr(f) is monotone, then almost every point is either a D-point or a knot point.

Theorem

If Gr(f) is monotone, then every interval contains a perfect set of D-points.In particular, f is differentiable at a dense set.

D-points vs. M-points

D ⊆M ⊆ess Dapp ∪KintM⊆ D



Some examples

Let ‖x‖ = dist(x,Z) be the triangle wave function.

Example

Let f(x) =∑∞k=0 2

−k‖2k2

x‖.f is continuous,

f has no M-points,

every monotone subset of Gr(f) is meager,

f is nowhere differentiable.

Example

Let g(x) = (x− 12) sin 1

2x−1f(x).

g is continuous,

g has exactly one M-point,

g is nowhere differentiable,

D is not dense in M.



Some examples

Example

Let f(x) =∑∞n=0 2

−n‖2nx‖ be the Takagi function.

f is continuous,

f does not possess a finite one-sided derivative at any point,

if x is a dyadic rational, then f+(x) = +∞ and f−(x) = −∞,

f ′(x) = +∞ at a dense set,

D is dense and co-dense,

M is dense and co-dense.



Nondifferentiable function with monotone graph

Theorem

There is a continuous, almost nowhere differentiable function f : [0, 1]→ R with a monotone graph.

Properties of the function

Every point is an M-point,

the function is almost nowhere approximately differentiable,

almost all points are knot points (actually approximate knot points),

the function has a derivative at a perfectly dense set.

Theorem

There is an absolutely continuous function f such that M is co-dense.

Properties of the function

f is differentiable almost everywhere,

almost all points are M-points,

all monotone subsets of Gr(f) are nowhere dense (in Gr(f)).



A conjecture

Theorem

If f : [0, 1]→ R has a 1-monotone graph, then it is of bounded variation.

Theorem (Nekvinda, Pokorny, Vlasak)

If X ⊆ Rn is 1-monotone, then dimH X = 1.

Conjecture 1

There is a function f : [1,∞)→ [1, 2] such that

limc→1+ f(c) = 1,

if X ⊆ R2 is c-monotone, then dimH X 6 f(c).

Conjecture 2

The same, but for c-monotone curves in R2.


applications of monotone spaces theory: peano curves...

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