uniform entropy in the realm of topological groupslucia/stw2013/slides/sanchis.pdf · groups m....

Uniform Entropy in the Realm of TopologicalGroups

M. Sanchis1

Institut Universitari de Matematiques i Aplicacions de Castello (IMAC), UJI

Brazilian Conference on General Topology and Set TheoryIn honor of O. T. Alas on the occasion of her 70 birthday

August 12th to 16th, 2013 (Sao Sebastiao, Brazil)

1Joint research with D. Alcaraz, Universidad Politecnica de Cartagenaand D. Dikranjan, Universita d’Udine

1 Introduction

2 Bowen’s Entropy and Dimension

3 h-jump endomorphisms

4 Compact abelian groups without endomorphisms of infiniteentropy

5 Open Questions

1 Introduction

5 Open Questions

1 Introduction

5 Open Questions

1 Introduction

5 Open Questions

1 Introduction

5 Open Questions

Uniform Entropy and Topological Groups

The outcomes on h-jumps endomorphisms are joint results withD. Alcaraz (Universidad de Cartagena, Spain) and D. Dikranjan(Universita di Udine, Italy). The remainder of the results is ajoint work with D. Dikranjan (Universita di Udine, Italy)

M. Sanchis Uniform Entropy and Topological Groups

The outcomes on h-jumps endomorphisms are joint results withD. Alcaraz (Universidad de Cartagena, Spain) and D. Dikranjan(Universita di Udine, Italy). The remainder of the results is ajoint work with D. Dikranjan (Universita di Udine, Italy)

1 Introduction

5 Open Questions

Introduction

Definition

Let X be a compact space. If α : X → X is a continuousfunction, then the topological entropy of α with respect to acover U is defined as

hT (α,U) = limn→∞

(n−1∧i=0

α−i(U)

where, as usual,n−1∧i=0

α−i(U) stands for the cover

U ∩ α−1(U) ∩ α−2(U) ∩ · · · · · ∩ α−(n−1)(U) .

(n−1∧i=0

α−i(U)

)≡ the smallest cardinality of a finite subcover

ofn−1∧i=0

α−i(U).

Introduction

Definition

(n−1∧i=0

α−i(U)

U ∩ α−1(U) ∩ α−2(U) ∩ · · · · · ∩ α−(n−1)(U) .

(n−1∧i=0

α−i(U)

ofn−1∧i=0

α−i(U).

Introduction

Definition

Let X be a compact space.

If α : X → X is a continuousfunction, then the topological entropy of α with respect to acover U is defined as

(n−1∧i=0

α−i(U)

U ∩ α−1(U) ∩ α−2(U) ∩ · · · · · ∩ α−(n−1)(U) .

(n−1∧i=0

α−i(U)

ofn−1∧i=0

α−i(U).

Introduction

Definition

(n−1∧i=0

α−i(U)

U ∩ α−1(U) ∩ α−2(U) ∩ · · · · · ∩ α−(n−1)(U) .

(n−1∧i=0

α−i(U)

ofn−1∧i=0

α−i(U).

Introduction

Definition

(n−1∧i=0

α−i(U)

U ∩ α−1(U) ∩ α−2(U) ∩ · · · · · ∩ α−(n−1)(U) .

(n−1∧i=0

α−i(U)

ofn−1∧i=0

α−i(U).

Introduction

Definition (Adler, Konheim and McAndrew (1965))

Let X be a compact space. If α : X → X is a continuousfunction, then the topological entropy of α is defined as

hT (α) = sup{hT (α,U) | U an open cover of X } .

Introduction

Bowen’s Entropy (Bowen (1971))

Let (X,U) be a uniform space. Let α : X → X be a uniformlycontinuous function. Consider U ∈ U

A subset F (n,U)-span a compact subset C if for every x ∈ Cthere is y ∈ F such that

(αj(x), αj(y)) ∈ U 0 ≤ j < n

Introduction

Let (X,U) be a uniform space. Let α : X → X be a uniformlycontinuous function. Consider U ∈ U

(αj(x), αj(y)) ∈ U 0 ≤ j < n

Introduction

Let (X,U) be a uniform space.

Let α : X → X be a uniformlycontinuous function. Consider U ∈ U

(αj(x), αj(y)) ∈ U 0 ≤ j < n

Introduction

Let (X,U) be a uniform space. Let α : X → X be a uniformlycontinuous function. Consider U ∈ U.

(αj(x), αj(y)) ∈ U 0 ≤ j < n

Introduction

Let (X,U) be a uniform space. Let α : X → X be a uniformlycontinuous function. Consider U ∈ U.A subset F (n,U)-span a compact subset C

if for every x ∈ Cthere is y ∈ F such that

(αj(x), αj(y)) ∈ U 0 ≤ j < n

Introduction

Let (X,U) be a uniform space. Let α : X → X be a uniformlycontinuous function. Consider U ∈ U.A subset F (n,U)-span a compact subset C if for every x ∈ Cthere is y ∈ F such that

(αj(x), αj(y)) ∈ U 0 ≤ j < n

Introduction

For a compact subset C ⊆ X, we consider the numbers

• rn(U,C) ≡ the smallest cardinality of any set F which(n,U)-spans C.

• hB(α,C) = supU∈U{ lim supn→∞1n log rn (U,C) }

Definition

The Bowen entropy hB(α) of α with respect to the uniformityU is defined as

hB(α) = sup{hB(α,C) | C ⊆ X, compact }

Introduction

Definition

Introduction

Definition

Introduction

Definition

Introduction

Definition

Introduction

Definition

Introduction

Definition

Introduction

Definition

Introduction

Metric Entropy

Let (X, d) be a metric space. If α : X → X is a uniformlycontinuous function, then

hd(α) = hB(α)

where hB(α) is the Bowen’s Entropy with respect to theuniformity induced by the metric d .

Introduction

Metric Entropy

hd(α) = hB(α)

Introduction

Metric Entropy

hd(α) = hB(α)

Introduction

Metric Entropy

hd(α) = hB(α)

Introduction

Metric Entropy

hd(α) = hB(α)

Introduction

Theorem (Bowen, 1971)

Let (X, d) a compact metric space. If α : X → X is acontinuous function, then

hd(α) = hT (α) .

Introduction

hd(α) = hT (α) .

Introduction

Let (X, d) a compact metric space.

If α : X → X is acontinuous function, then

hd(α) = hT (α) .

Introduction

Let (X, d) a compact metric space. If α : X → X is acontinuous function,

hd(α) = hT (α) .

Introduction

hd(α) = hT (α) .

Introduction

Theorem

Let X be a compact space. If α : X → X is a continuousfunction, then

hB(α) = hT (α)

where hB(α) is the Bowen Entropy with respect to the uniqueuniformity on X.

Introduction

Theorem

hB(α) = hT (α)

Introduction

Theorem

Let X be a compact space.

If α : X → X is a continuousfunction, then

hB(α) = hT (α)

Introduction

Theorem

Let X be a compact space. If α : X → X is a continuousfunction,

hB(α) = hT (α)

Introduction

Theorem

hB(α) = hT (α)

Introduction

Let (G, τ) be a topological group.

• UR ≡ the right uniformity .

• UL ≡ the left uniformity .

Theorem

Let (G, τ) be a topological group. If α : G→ G is a continuousendomorphism, then the uniform entropy of α with respect tothe right uniformity on G coincides with the uniform entropy ofα with respect to the left uniformity on G.

Introduction

Theorem

Introduction

Theorem

Introduction

Theorem

Introduction

Theorem

Introduction

Theorem

Introduction

Theorem

Let (G, τ) be a topological group. If α : G→ G is a continuousendomorphism,

then the uniform entropy of α with respect tothe right uniformity on G coincides with the uniform entropy ofα with respect to the left uniformity on G.

Introduction

Theorem

1 Introduction

5 Open Questions

Bowen’s Entropy and Dimension

Given an integer k, let mGk denote the continuous endomorphism

mGk (x) = kx x ∈ G .

Theorem

Let G be a topological abelian group that is either locallycompact or ω-bounded. Then

hB(mGk ) = dimG · log k for every k > 1 .

Moreover, the following are equivalent:

G is totally disconnected.

hB(mGk ) = 0 for every integer k.

hB(mGk ) = 0 for some integer k > 1.

Theorem

Let G be a topological abelian group that is either locallycompact or ω-bounded.

Theorem

Let G be a compact connected group G such that thecommutator of G is finite dimensional. Then G is finitedimensional if and only if every continuous endomorphism hasfinite entropy.

Theorem

Example An infinite-dimensional compact connected group Gmay have no continuous automorphisms of infinite entropy

• Consider, for each n ≥ 1, a connected compact simple Liegroup Ln which are pairwise non-isomorphic.• Every automorphism α on Ln is continuous and, moreoversome finite power of α is an inner automorphism (van derWaerden’s theorem).• Then α has zero topological entropy.• G =

∏∞n=1 Ln. (Notice that every continuous automorphism

on G is a product of continuous automorphisms of the singlecomponents Ln).

Example

An infinite-dimensional compact connected group Gmay have no continuous automorphisms of infinite entropy

• Consider, for each n ≥ 1, a connected compact simple Liegroup Ln which are pairwise non-isomorphic.

• Every automorphism α on Ln is continuous and, moreoversome finite power of α is an inner automorphism (van derWaerden’s theorem).• Then α has zero topological entropy.• G =

• Consider, for each n ≥ 1, a connected compact simple Liegroup Ln which are pairwise non-isomorphic.• Every automorphism α on Ln is continuous and, moreoversome finite power of α is an inner automorphism (van derWaerden’s theorem).

• Then α has zero topological entropy.• G =

• Consider, for each n ≥ 1, a connected compact simple Liegroup Ln which are pairwise non-isomorphic.• Every automorphism α on Ln is continuous and, moreoversome finite power of α is an inner automorphism (van derWaerden’s theorem).• Then α has zero topological entropy.

• G =∏∞n=1 Ln. (Notice that every continuous automorphism

∏∞n=1 Ln.

(Notice that every continuous automorphismon G is a product of continuous automorphisms of the singlecomponents Ln).

1 Introduction

5 Open Questions

h-jump endomorphisms

Let G be an abstract abelian group.

• G# ≡ G endowed with the weak topology associated to thefamily of all homomorphisms from G into the circle T (the Bohrtopology)

• bG ≡ its Weil completion (the Bohr compactification of G)

Theorem

For every infinite abelian group G there exists an h-jumpendomorphism α : G# → G#, i.e., such that hB(b(α)) =∞ forthe extension b(α) : bG→ bG of α.

Case 1. G is not bounded torsion.

(The functionmGk (k > 1).)

Case 2. G =⊕

κ Z(n), where κ is an infinite cardinal andn > 1.

(The left Bernoulli shift.)

Case 3. There exists a natural n > 1 such that nG = 0.

Theorem

Case 2. G =⊕

Theorem

For every infinite abelian group G there exists an h-jumpendomorphism α : G# → G#,

i.e., such that hB(b(α)) =∞ forthe extension b(α) : bG→ bG of α.

Case 2. G =⊕

Theorem

Case 2. G =⊕

Theorem

Case 2. G =⊕

Theorem

Case 2. G =⊕

Theorem

Case 2. G =⊕

Theorem

Case 2. G =⊕

Theorem

Case 1. G is not bounded torsion. (The functionmGk (k > 1).)

Case 2. G =⊕

Theorem

Case 2. G =⊕

κ Z(n), where κ is an infinite cardinal andn > 1. (The left Bernoulli shift.)

Theorem

Case 2. G =⊕

κ Z(n), where κ is an infinite cardinal andn > 1. (The left Bernoulli shift.)

Case 3. There exists a natural n > 1 such that nG = 0.(The left Bernoulli shift.)

1 Introduction

5 Open Questions

Compact abelian groups without endomorphisms of infinite entropy

Definition

A group of the form G =∏p Z

npp × Fp, where np ≥ 0 is an

integer and Fp is a finite p-group for every prime p, is called anOrsatti group.

Definition

Theorem (Orsatti, 1967)

An abelian group G is an Orsatti group if and only if theZ-topology on G is compact.

• The Z-topology of G is the group topology of G having asbasic neighborhoods of 0 all subgroups of G of the form nG,n > 0.

Theorem

Every endomorphism on an Orsatti group has zero entropy.

Theorem

For every compact finite-dimensional abelian group G thefollowing statements are equivalent:

(1) G/c(G) is an Orsatti group.

(2) G/pG is finite for every prime p.

(3) For every prime p, every endomorphism on G/pG has finiteentropy.

(4) Every endomorphism on a quotient of G has finite entropy.

(5) There exists a closed subgroup G1 of G containing c(G),such that G = G1 ×

∏p Z

npp and G1/c(G) ∼=

∏p Fp, where

np is a non-negative integer and Fp is a finite p-group forevery prime p.

In case these conditions hold, G is metrizable.

Theorem

∏p Z

∏p Fp, where

Theorem

∏p Z

∏p Fp, where

Theorem

∏p Z

∏p Fp, where

Theorem

∏p Z

∏p Fp, where

Theorem

∏p Z

∏p Fp, where

Theorem

∏p Z

∏p Fp, where

Theorem

∏p Z

∏p Fp, where

Theorem

∏p Z

∏p Fp, where

Theorem

∏p Z

∏p Fp, where

Theorem

For every totally disconnected compact abelian group G thefollowing statements are equivalent:

(1) G is an Orsatti group.

(5) Every endomorphism on a quotient of G has zero entropy.

Theorem

1 Introduction

5 Open Questions

Open Questions

E ≡ the class of topological groups without endomorphisms ofinfinite entropy.

E0 ≡ the class of topological groups with the property thatevery endomorphism has zero entropy.

(P1) Are the classes (of compact abelian groups in) E and E0

closed with respect to: (i) taking finite products, and (ii)taking extensions?

(P2) Is there a metrizable abelian group which hash-endomorphisms?

(P3) (J. Peters, 1981) Let G be a locally compact abelian group.Is the function

α ∈ Aut(G)→ hB(α) ∈ [0,∞]

continuous?

Open Questions

α ∈ Aut(G)→ hB(α) ∈ [0,∞]

continuous?

Open Questions

α ∈ Aut(G)→ hB(α) ∈ [0,∞]

continuous?

Open Questions

α ∈ Aut(G)→ hB(α) ∈ [0,∞]

continuous?

Open Questions

α ∈ Aut(G)→ hB(α) ∈ [0,∞]

continuous?M. Sanchis Uniform Entropy and Topological Groups

Open Questions

α ∈ Aut(G)→ hB(α) ∈ [0,∞]

Open Questions

α ∈ Aut(G)→ hB(α) ∈ [0,∞]

Open Questions

α ∈ Aut(G)→ hB(α) ∈ [0,∞]

Open Questions

Muito obrigado pela sua atencao !!

Open Questions

Muito obrigado pela sua atencao !!

uniform entropy in the realm of topological groupslucia/stw2013/slides/sanchis.pdf · groups m....

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