Entropy and kneading theory, Barcelona, 06-2013

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Entropy on Hubbard trees

Tan Lei

谭蕾

, Université d’Angers

(most figures are produced by W. Thurston and G. Tiozzo)

Barcelona, June 2013

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Organization of the mini course

– Definition of the topological entropy – Forf :z 7→z²+c,c ∈[−2,1

4], study various aspects of the entropy on[−β_c, βc]

1. Misiurewitcz-Szlenk’s lap numbers 2. Milnor-Thurston’s kneading theory 3. Douady’s external angle model 4. Thurston’s angle-pair model

5. Tiozzo’s parameter section model· · ·

– Case of complex polynomials : entropy on the Hubbard tree and other variants

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§1 Definition of the topological entropy

LetX be a compact topological space. Letf :X →X be continuous. Let U be anopen coverofX.

•Letn be any integer.(prescribing the itinerary) Pick a list U₀,· · · ,Un−1 of (not necessarily distinct) open sets ofU. Set

W_U₀_,···_,U_n−1 ={x ∈X |x ∈U₀,f(x)∈U₁,· · · ,fⁿ⁻¹(x)∈U_n−1}. Each suchW is open (might be empty). But they together form an open cover denoted byWnU.

•For any open cover V of X, letN(V)<∞ denote the minimal cardinal of a sub cover ofV.

•Therelative entropyh(X,U,f) :=limn→∞

logN(WnU)

n .

•Thetopological entropyh(X,f) := sup

U,open cover ofX

h(X,U,f).

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§1 Definition of the topological entropy

W_U₀_,···_,U_n−1 ={x ∈X |x ∈U₀,f(x)∈U₁,· · · ,fⁿ⁻¹(x)∈U_n−1}.

Each suchW is open (might be empty). But they together form an open cover denoted byWnU.

logN(WnU)

n .

U,open cover ofX

h(X,U,f).

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§1 Definition of the topological entropy

logN(WnU)

n .

U,open cover ofX

h(X,U,f).

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§1 Definition of the topological entropy

logN(WnU)

n .

U,open cover ofX

h(X,U,f).

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§1 Definition of the topological entropy

logN(WnU)

n .

U,open cover ofX

h(X,U,f).

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Growth and log-growth of a sequence a

_n

> 0

a_n∼λⁿ=⇒growth(a_n) =λ growth(a_n) :=lima^1/n_n = 1

convergence radius of P a_nzⁿ. log-growth(a_n) :=limloga_n

n .

Ifa_n satisfies the sub-multiplicativeproperty :a_n+m ≤a_na_m, then the actual limit exists and is the infimum of the sequencea^1/nn . Therefore therelative entropyh(X,U,f) := lim

n→∞

logN(WnU)

n is

the log-growth of the sequenceN(WnU), the minimal cardinal of a sub cover ofWnU. And the entropy h(X,f)is the supremum of h(X,U,f)for all open covers U.

This sounds difficult to compute. But it is equal to the growth of many interesting dynamical quantities, some of them are

(sometimes) computable. The variationf 7→h(f)is also interesting.

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Growth and log-growth of a sequence a

_n

> 0

a_n∼λⁿ=⇒growth(a_n) =λ

growth(a_n) :=lima^1/n_n = 1

n .

n→∞

logN(WnU)

n is

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Growth and log-growth of a sequence a

_n

> 0

a_n∼λⁿ=⇒growth(a_n) =λ growth(a_n) :=lima^1/n_n =

1

n .

n→∞

logN(WnU)

n is

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Growth and log-growth of a sequence a

_n

> 0

convergence radius of P a_nzⁿ.

log-growth(a_n) :=limloga_n n .

n→∞

logN(WnU)

n is

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Growth and log-growth of a sequence a

_n

> 0

n .

n→∞

logN(WnU)

n is

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Growth and log-growth of a sequence a

_n

> 0

n .

Ifa_n satisfies thesub-multiplicativeproperty : a_n+m ≤a_na_m, then the actual limit exists and is the infimum of the sequencea^1/nn .

Therefore therelative entropyh(X,U,f) := lim

n→∞

logN(WnU)

n is

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Growth and log-growth of a sequence a

_n

> 0

n .

Ifa_n satisfies thesub-multiplicativeproperty : a_n+m ≤a_na_m, then the actual limit exists and is the infimum of the sequencea^1/nn . Therefore therelative entropyh(X,U,f) := lim

n→∞

logN(WnU)

n is

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Growth and log-growth of a sequence a

_n

> 0

n .

Ifa_n satisfies thesub-multiplicativeproperty : a_n+m ≤a_na_m, then the actual limit exists and is the infimum of the sequencea^1/nn . Therefore therelative entropyh(X,U,f) := lim

n→∞

logN(WnU)

n is

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§2. f (x ) = x

²

+ c , -2≤ c ≤ 1

4 , I = [-β

_c

, β

_c

], f (β

_c

) = β

_c

> 0

Γ0 ={0},Γi ={x,fⁱ(x) =0, i is minimal}. Setγi = #Γi. The intervals ofI r∪ⁱ_j₌₀Γj are maximal injective intervals offⁱ⁺¹. The number of them is 2+ ( X

j=0,···,i

γ_j−1) =1+ X

j=0,···,i

γ_j =:`_i+1. We havegrowth(γ_i) =growth(`_i)=:s.

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§2. f (x ) = x

²

+ c , -2≤ c ≤ 1

4 , I = [-β

_c

, β

_c

], f (β

_c

) = β

_c

> 0

j=0,···,i

γ_j−1) =1+ X

j=0,···,i

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§2. f (x ) = x

²

+ c , -2≤ c ≤ 1

4 , I = [-β

_c

, β

_c

], f (β

_c

) = β

_c

> 0

j=0,···,i

γ_j−1) =1+ X

j=0,···,i

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§2. f (x ) = x

²

+ c , -2≤ c ≤ 1

4 , I = [-β

_c

, β

_c

], f (β

_c

) = β

_c

> 0

j=0,···,i

γ_j−1) =1+ X

j=0,···,i

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§2. f (x ) = x

²

+ c , -2≤ c ≤ 1

4 , I = [-β

_c

, β

_c

], f (β

_c

) = β

_c

> 0

j=0,···,i

γ_j−1) =1+ X

j=0,···,i

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§2. f (x ) = x

²

+ c , -2≤ c ≤ 1

4 , I = [-β

_c

, β

_c

], f (β

_c

) = β

_c

> 0

j=0,···,i

γ_j−1) =1+ X

j=0,···,i

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§2. f (x ) = x

²

+ c , -2≤ c ≤ 1

4 , I = [-β

_c

, β

_c

], f (β

_c

) = β

_c

> 0

j=0,···,i

γ_j−1) =1+ X

j=0,···,i

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§2. f (x ) = x

²

+ c , -2≤ c ≤ 1

4 , I = [-β

_c

, β

_c

], f (β

_c

) = β

_c

> 0

j=0,···,i

γ_j−1) =1+ X

j=0,···,i

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§2. f (x ) = x

²

+ c , -2≤ c ≤ 1

4 , I = [-β

_c

, β

_c

], f (β

_c

) = β

_c

> 0

j=0,···,i

γ_j−1) =1+ X

j=0,···,i

Proof. Forγ(t) =X

i≥0

γ_itⁱ,, `(t) =X

i≥0

`_i+1tⁱ :`(t)(1−t) =1+γ(t).

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Theorem 1.(Misiurewitz-Szlenk) h(I,f) =logs.

Proof ofh(I,f)≥logs. Or ∃ U_k,h(I,U_k,f)≥logs−εk (→0). Fix anyk>0. LetU_k =U ={[−β,0[,]ε, ε[,]0, β[} with

]−ε, ε[∩Sk−1

j=0 Γ_j ={0}. Then f^k is injective on]−ε,0[ and]0, ε[ andf^k|_]−ε,ε[ is at most 2:1.

LetW ∈WnU, i.e.∃ V₀,· · ·,V_n−1 elements of U such that W =V₀∩f⁻¹V₁∩ · · · ∩f⁻⁽ⁿ⁻¹⁾V_n−1.

In particularfⁱ(W)⊂V_i. Let i(<n−1) such that f|_fi(W) is not injective. ThenV_i =]−ε, ε[andf^k|_fi(W) is at most 2:1. We want to estimate#W ∩Γ_n. Setm_i = #fⁱ(W)∩Γn−i. Then

mn−1≤#Vn−1∩Γ1≤1 m_i ≤

m_i₊₁ iff|_fi(W) is injective 2m_i_+k iff|_fi(W) is not injective

Som₀ ≤2^1+n/k. ForW a sub cover ofWnU, we assignΓn→ W, x 7→Wx 3x. Such a map exists sinceW is a cover. And the fiber over aW ∈ W is at most 2^1+n/k.

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Proof ofh(I,f)≥logs.

Or∃ U_k,h(I,U_k,f)≥logs−εk (→0). Fix anyk>0. LetU_k =U ={[−β,0[,]ε, ε[,]0, β[} with

]−ε, ε[∩Sk−1

mn−1≤#Vn−1∩Γ1≤1 m_i ≤

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Proof ofh(I,f)≥logs. Or ∃ U_k,h(I,U_k,f)≥logs−ε_k (→0).

Fix anyk>0. LetU_k =U ={[−β,0[,]ε, ε[,]0, β[} with ]−ε, ε[∩Sk−1

mn−1≤#Vn−1∩Γ1≤1 m_i ≤

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j=0 Γ_j ={0}.

Then f^k is injective on]−ε,0[ and]0, ε[ andf^k|_]−ε,ε[ is at most 2:1.

mn−1≤#Vn−1∩Γ1≤1 m_i ≤

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j=0 Γ_j ={0}. Thenf^k is injective on ]−ε,0[ and]0, ε[

andf^k|_]−ε,ε[ is at most 2:1.

mn−1≤#Vn−1∩Γ1≤1 m_i ≤

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LetW ∈WnU, i.e.∃V₀,· · ·,V_n−1 elements of U such that W =V₀∩f⁻¹V₁∩ · · · ∩f⁻⁽ⁿ⁻¹⁾V_n−1.

mn−1≤#Vn−1∩Γ1≤1 m_i ≤

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LetW ∈WnU, i.e.∃V₀,· · ·,V_n−1 elements of U such that W =V₀∩f⁻¹V₁∩ · · · ∩f⁻⁽ⁿ⁻¹⁾V_n−1. In particularfⁱ(W)⊂V_i.

Let i(<n−1) such that f|_fi(W) is not injective. ThenV_i =]−ε, ε[andf^k|_fi(W) is at most 2:1. We want to estimate#W ∩Γ_n. Setm_i = #fⁱ(W)∩Γn−i. Then

mn−1≤#Vn−1∩Γ1≤1 m_i ≤

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In particularfⁱ(W)⊂V_i. Leti(<n−1) such that f|_fi(W) is not injective.

ThenV_i =]−ε, ε[andf^k|_fi(W) is at most 2:1. We want to estimate#W ∩Γ_n. Setm_i = #fⁱ(W)∩Γn−i. Then

mn−1≤#Vn−1∩Γ1≤1 m_i ≤

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In particularfⁱ(W)⊂V_i. Leti(<n−1) such that f|_fi(W) is not injective. ThenV_i =]−ε, ε[andf^k|_fi(W) is at most 2:1.

We want to estimate#W ∩Γ_n. Setm_i = #fⁱ(W)∩Γn−i. Then mn−1≤#Vn−1∩Γ1≤1

m_i ≤

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We want to estimate#W ∩Γ_n.

Setm_i = #fⁱ(W)∩Γn−i. Then mn−1≤#Vn−1∩Γ1≤1

m_i ≤

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We want to estimate#W ∩Γ_n. Setm_i = #fⁱ(W)∩Γn−i.

Then mn−1≤#Vn−1∩Γ1≤1

m_i ≤

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m_i ≤

m_i+1 iff|_fi(W) is injective 2m_i_+k iff|_fi(W) is not injective

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m_i ≤

Som₀ ≤2^1+n/k.

ForW a sub cover ofWnU, we assignΓn→ W, x 7→Wx 3x. Such a map exists sinceW is a cover. And the fiber over aW ∈ W is at most 2^1+n/k.

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m_i ≤

Som₀ ≤2^1+n/k. ForW a sub cover ofWnU, we assignΓ_n→ W, x 7→W_x 3x.

Such a map exists sinceW is a cover. And the fiber over aW ∈ W is at most 2^1+n/k.

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m_i ≤

Som₀ ≤2^1+n/k. ForW a sub cover ofWnU, we assignΓ_n→ W, x 7→W_x 3x. Such a map exists sinceW is a cover.

And the fiber over aW ∈ W is at most 2^1+n/k.

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m_i ≤

Som₀ ≤2^1+n/k. ForW a sub cover ofWnU, we assignΓ_n→ W, x 7→W_x 3x. Such a map exists sinceW is a cover. And the fiber over aW ∈ W is at most 2^1+n/k.

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It follows that

2^1+n/k·#W ≥2^1+n/k·#{W_x,x ∈Γ_n} ≥ |list{W_x,x ∈Γ_n}|=γ_n.

So 1

k log 2+h(I,f)≥ 1

k log 2+h(I,U,f)≥logs. Lettingk → ∞, we get h(I,f)≥logs.

Proof of logs≥h(I,f)=suph(I,U,f). Orlogs≥h(I,U,f),∀ U. ChooseI ⊃S ⊃ {0} ∪∂I a large finite subset such that every consecutive interval[a_i,a_i+1]is contained in some open set of U. Then the setSn−1

j=0 f^−jS partitions I into closed intervals [a,b]’s s.t. points in a[a.b]have a common itinerary U₀,U₁,· · ·,Un−1 in U (see Claim 1. below), therefore is contained in aW_[a,b]∈WnU. The collection ofW_[a,b]’s forms a sub cover, with cardinality

∈[N(WnU),#

n−1

[

j=0

f^−jS]. So

log-growth#

n−1

[

j=0

f^−jS ≥log-growthN(

n

_U) =h(I,U,f).

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It follows that

2^1+n/k·#W ≥2^1+n/k·#{W_x,x ∈Γ_n} ≥ |list{W_x,x ∈Γ_n}|=γ_n. So 1

k log 2+h(I,f)≥ 1

k log 2+h(I,U,f)≥logs.

Lettingk → ∞, we get h(I,f)≥logs.

∈[N(WnU),#

n−1

[

j=0

f^−jS]. So

log-growth#

n−1

[

j=0

n

_U) =h(I,U,f).

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It follows that

k log 2+h(I,f)≥ 1

∈[N(WnU),#

n−1

[

j=0

f^−jS]. So

log-growth#

n−1

[

j=0

n

_U) =h(I,U,f).

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It follows that

k log 2+h(I,f)≥ 1

Proof of logs≥h(I,f)=suph(I,U,f).

Orlogs≥h(I,U,f),∀ U. ChooseI ⊃S ⊃ {0} ∪∂I a large finite subset such that every consecutive interval[a_i,a_i+1]is contained in some open set of U. Then the setSn−1

∈[N(WnU),#

n−1

[

j=0

f^−jS]. So

log-growth#

n−1

[

j=0

n

_U) =h(I,U,f).

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It follows that

k log 2+h(I,f)≥ 1

Proof of logs≥h(I,f)=suph(I,U,f). Orlogs≥h(I,U,f),∀ U.

ChooseI ⊃S ⊃ {0} ∪∂I a large finite subset such that every consecutive interval[a_i,a_i+1]is contained in some open set of U. Then the setSn−1

∈[N(WnU),#

n−1

[

j=0

f^−jS]. So

log-growth#

n−1

[

j=0

n

_U) =h(I,U,f).

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It follows that

k log 2+h(I,f)≥ 1

Proof of logs≥h(I,f)=suph(I,U,f). Orlogs≥h(I,U,f),∀ U. ChooseI ⊃S ⊃ {0} ∪∂I a large finite subset such that every consecutive interval[a_i,a_i+1]is contained in some open set of U.

Then the setSn−1

∈[N(WnU),#

n−1

[

j=0

f^−jS]. So

log-growth#

n−1

[

j=0

n

_U) =h(I,U,f).

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It follows that

k log 2+h(I,f)≥ 1

j=0 f^−jS partitions I into closed intervals [a,b]’s s.t. points in a[a.b]have a common itinerary U₀,U₁,· · ·,Un−1 in U (see Claim 1. below),

therefore is contained in a W_[a,b]∈WnU. The collection ofW_[a,b]’s forms a sub cover, with cardinality

∈[N(WnU),#

n−1

[

j=0

f^−jS]. So

log-growth#

n−1

[

j=0

n

_U) =h(I,U,f).

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It follows that

k log 2+h(I,f)≥ 1

j=0 f^−jS partitions I into closed intervals [a,b]’s s.t. points in a[a.b]have a common itinerary U₀,U₁,· · ·,Un−1 in U (see Claim 1. below), therefore is contained in aW_[a,b]∈WnU.

The collection ofW_[a,b]’s forms a sub cover, with cardinality

∈[N(WnU),#

n−1

[

j=0

f^−jS]. So

log-growth#

n−1

[

j=0

n

_U) =h(I,U,f).

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It follows that

k log 2+h(I,f)≥ 1

j=0 f^−jS partitions I into closed intervals [a,b]’s s.t. points in a[a.b]have a common itinerary U₀,U₁,· · ·,Un−1 in U (see Claim 1. below), therefore is contained in aW_[a,b]∈WnU. The collection ofW_[a,b]’s forms a sub cover,

with cardinality

∈[N(WnU),#

n−1

[

j=0

f^−jS]. So

log-growth#

n−1

[

j=0

n

_U) =h(I,U,f).