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The dimension of the full nonuniformly hyperbolic horseshoe Cao Yongluo Email: ylcao@suda.edu.cn Soochow University Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 1 / 30 Contents Motivation 1

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The dimension of the full nonuniformly hyperbolic horseshoe

Cao Yongluo Email: ylcao@suda.edu.cn

Soochow University

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 1 / 30

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Contents

1

Motivation

2

Hyperbolic set

3

The full non-uniformly hyperbolic horseshoe

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 2 / 30

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1

Motivation

2

Hyperbolic set

3

The full non-uniformly hyperbolic horseshoe

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 3 / 30

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Motivation

To consider the Hausdorff dimension and the dynamics of the full non-uniformly hyperbolic horseshoe. These systems was constructed by Rios for studying the bifurcation of homoclinic tangency inside horseshoe and it appears naturally in the Henon family.

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 3 / 30

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Hausdorff dimension

Definition 1.1

For any πœ€ > 0 we define ℋ𝑑

πœ€(𝐺) = inf{ ∞

βˆ‘οΈ‚

𝑗=1

|𝑉𝑗|𝑑 : {𝑉𝑗} is a πœ€ cover of 𝐺} ℋ𝑑(𝐺) = lim

πœ€β†’0 ℋ𝑑 πœ€(𝐺).

This limit exists for any subset, though the limiting value can be 0 or ∞. We call ℋ𝑑(𝐺) the 𝑑-dimensional Hausdorff measure of 𝐺. 𝐸𝑗𝑛𝐼𝐺 = inf{𝑑 : ℋ𝑑(𝐺) = 0} = sup{𝑑 : ℋ𝑑(𝐺) = ∞} .

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 4 / 30

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Topological pressure

Recall the notation of topological pressure. Let π‘ˆ : Ξ› β†’ Ξ› be a continuous transformation and πœ’ : Ξ› β†’ 𝑆 continuous function. 𝐺 is a (π‘œ, πœ—) separate set of π‘ˆ. π‘„π‘œ(𝜏, πœ’, πœ—) = sup{βˆ‘οΈ

π‘¦βˆˆπΊ 𝑓𝑇n(πœ’(𝑦))|𝐺 is(π‘œ, πœ—) separate set}

Where π‘‡π‘œ(πœ’(𝑦)) = πœ’(𝑦) + Β· Β· Β· + πœ’(πœπ‘œβˆ’1𝑦). 𝑄(𝜏, πœ’) = lim

πœ—β†’0 lim sup π‘œβ†’βˆž 1 π‘œ log π‘„π‘œ(𝜏, πœ’, πœ—)

Theorem 1.2

Bowen: 𝑄(𝜏, πœ’) topological pressure, variational principal 𝑄(𝜏, πœ’) = sup

𝜈∈M(𝜏)

{β„Žπœˆ + βˆ«οΈ‚ πœ’π‘’πœˆ}

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 5 / 30

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Topological pressure

Hausdorff dimension of Standard Cantor set.

  • 1. It is well known that the root of equation 2 Γ— ( 1

3)𝑒 = 1 is it Hausdorff

dimension. It is equivalent to 𝑄(π‘ˆ, βˆ’π‘’ log |πΈπ‘ˆ|) = 0.

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 6 / 30

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  • ne dimension repeller

2.One dimension nonlinear expanding map Hausdorff dimension of Cantor set is the root of equation 𝑄(π‘ˆ, βˆ’π‘’ log |πΈπ‘ˆ|) = 0.

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 7 / 30

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  • ne dimension repeller
  • 4. 𝑔 : 𝑁 β†’ 𝑁 be a 𝐷1 map, Ξ› is a repeller of 𝑔, Conformal.

Theorem 1.3

Let Ξ› be a 𝐷1 Conformal repeller of 𝑔. Then Hausdorff dimension of repeller is the root of equation 𝑄(π‘ˆ, βˆ’π‘’ log |πΈπ‘ˆ|) = 0. Ruelle considered the 𝐷1+𝛽 for Hausdorff. Falconer considered for Hausdorff and Box dimension Gatzouras and Peres considered the 𝐷1 case.

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 8 / 30

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1

Motivation

2

Hyperbolic set

3

The full non-uniformly hyperbolic horseshoe

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 9 / 30

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Smale Hoesrshoes

Next we consider two dimension case. βˆ€π‘¦ ∈ Ξ›, π‘ˆπ‘¦π‘ = 𝐹𝑑 + 𝐹𝑣 ‖𝐸𝑔 π‘œ(𝑀1)β€– ≀ π‘“πœ‡π‘œβ€–π‘€1β€–, 𝑀 ∈ 𝐹𝑑 ‖𝐸𝑔 π‘œ(𝑀2)β€– β‰₯ π‘“βˆ’πœ‡π‘œβ€–π‘€2β€–, 𝑀 ∈ 𝐹𝑣 (πœ‡ < 0).

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 9 / 30

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The Hausdorff dimension of hyperbolic horseshoe

MaCluskey, H. and Manning, A., 1983(ETDS) prove that for 𝐷1+𝛽, the Hausdorff dimension of Ξ› 𝐸𝑗𝑛𝐼Λ = 𝑒𝑑 + 𝑒𝑣 Where 𝑒𝑣 is the root of 𝑄(βˆ’π‘’ log |𝑒 𝑔|𝐹u|) = 0 and 𝑒𝑑 is the root of 𝑄(𝑒 log |𝐸𝑔|𝐹s|) = 0. J.Palis and Viana prove that the formula as above holds for 𝐷1 diffeomeorphism.

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 10 / 30

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Lyapunove Exponents

Lyapunov exponents: 𝑔 : 𝑁 β†’ 𝑁 M is a Riemann manifold . βˆ€π‘¦ ∈ 𝑁, 𝑀 ∈ π‘ˆπ‘¦π‘ if lim

π‘œβ†’βˆž

1 π‘œ log ‖𝐸𝑔 π‘œ(𝑦)𝑀‖ ‖𝑀‖ exists, it is called Lyapunov exponent, and denote it by πœ‡(𝑦, 𝑀). 𝜈 is 𝑔 invariant measure, β„³(𝑔). 𝜈(𝐡) = 𝜈(𝑔 βˆ’1(𝐡)) for every measurable set. Ergodic invariant measure β„°(𝑔). Ergodic means that for every invariant set 𝐡, 𝜈(𝐡) = 0, 𝑝𝑠1.

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 11 / 30

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Oseledec Theorem

𝐡 βŠ‚ 𝑁 𝜈(𝐡) = 1 for every 𝜈 ∈ β„³(𝑔) 𝑦 ∈ 𝐡 1.πœ‡1(𝑦) ≀ Β· Β· Β· ≀ πœ‡π‘‘(𝑦) 2.π‘Š0(𝑦) βŠ‚ π‘Š1(𝑦) βŠ‚ Β· Β· Β· βŠ‚ π‘Šπ‘‘(𝑦) = π‘ˆπ‘¦π‘ lim

π‘œβ†’βˆž

1 π‘œ log |𝐸𝑔 π‘œ

𝑦 (𝑀)| = πœ‡π‘—(𝑦), 𝑀 ∈ π‘Šπ‘— βˆ– π‘Šπ‘—βˆ’1

πœ‡π‘—(𝑦) is defined on 𝐡 𝜈 is ergodic, πœ‡π‘—(𝑦) constant 𝑏.π‘“πœˆ and denote it by πœ‡π‘—(𝜈).

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 12 / 30

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1

Motivation

2

Hyperbolic set

3

The full non-uniformly hyperbolic horseshoe

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 13 / 30

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The dynamics of a = a*

Henon map 𝐼𝑏,𝑐(𝑦, 𝑧) = (1 βˆ’ 𝑏𝑦2 + 𝑧, 𝑐𝑦)

Theorem 3.1

If 𝑐 is small, then there is an 𝑏 = π‘βˆ—, the corresponding map πΌπ‘βˆ—,𝑐, βˆƒπΆ βŠ‚ Ξ› with 𝜈(𝐢) = 1 for βˆ€πœˆ ∈ β„³(𝐼, Ξ›) and βˆ€π‘¦ ∈ 𝐢 πœ‡1(𝑦) < 𝑑1 < 0 < 𝑑2 < πœ‡2(𝑦) 𝜈 is hyperbolic measure. What is Hausdorff dimension ?

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 13 / 30

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Horseshoe with infinite branches

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 14 / 30

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The example of Rios’s for homoclinic tangency inside

I.Rios gave an example of systems with homoclinic tangeny inside of invariant set in 2001, Nonlinearity. Luzzatto, Rios and Cao prove that all the Lyapunov exponents of all invariant measures are uniformly bounded away from 0(2006, DCDS). Now it is called the full nonuniformly hyperbolic

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 15 / 30

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Horseshoe with infinite branches

Furthermore, we will consider the ergodicity of this map. We will construct a inducing map ( Horseshoe with infinite branches.) One dimension 𝑔4(𝑦) = 4𝑦(1 βˆ’ 𝑦).

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 16 / 30

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Horseshoe with infinite branches

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 17 / 30

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Horseshoe with infinite branches

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 18 / 30

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˜ Ξ› = βˆͺ∞

𝑗=2Λ𝑗 and 𝐺(𝑦) = 𝑔 𝜐(𝑦)(𝑦) and 𝜐(𝑦) = 𝑗 for 𝑦 ∈ Λ𝑗.

(˜ Ξ›, 𝐺), the first return map to ˜ Ξ›. There hyperbolic product structure. Stable foliation 𝛿𝑑 and unstable foliation 𝛿𝑣. For 𝑧 ∈ 𝛿𝑑(𝑦), log Π∞

𝑗=π‘œ

𝐸𝐺 𝑣(𝐺 𝑗(𝑦) 𝐸𝐺 𝑣(𝐺 𝑗(𝑧)) ≀ π‘‘πœ‡π‘œ. For 𝑧 ∈ 𝛿𝑣(𝑦), and they are in the same π‘œ cylinder, log Ξ π‘œ

𝑗=0

𝐸𝐺 𝑣(𝐺 𝑗(𝑦) 𝐸𝐺 𝑣(𝐺 𝑗(𝑧)) ≀ π‘‘πœ‡π‘œ.

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 19 / 30

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Consider the one-side full shift of countable type (𝑇N, 𝜏), where 𝑇 countable set. Let Ξ¦ : 𝑇N β†’ 𝑆 be a some real function. The variations of Ξ¦ are defined as π‘Š π‘π‘ π‘œ(Ξ¦) = sup{|Ξ¦(𝑦) βˆ’ Ξ¦(𝑧)| : 𝑦, 𝑧 in the same n-cylinder} If there are constants 𝐷 > 0 and πœ„ ∈ (0, 1) such that π‘Š π‘π‘ π‘œ(Ξ¦) < π·πœ„π‘œ for all π‘œ β‰₯ 2, then we call Ξ¦ the weakly Holder continuous. A Gibbs measure 1 𝐢 ≀ 𝑛[𝑏0, Β· Β· Β· , π‘π‘œβˆ’1] 𝑓Φn(𝑦)βˆ’π‘œπ‘„ ≀ 𝐢 for all 𝑦 ∈ [𝑏0, Β· Β· Β· , π‘π‘œβˆ’1]. Ξ¦π‘œ = βˆ‘οΈπ‘œβˆ’1

𝑗=0 Ξ¦(πœπ‘—(𝑦)).

The Gurevich pressure of Ξ¦ 𝑄𝐻(Ξ¦) = lim

π‘œβ†’βˆž

1 π‘œ log βˆ‘οΈ‚

𝜏n(𝑦)=𝑦

𝑓Φn(𝑦)1[𝑏](𝑦)

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 20 / 30

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Consider the full shift of countable type (𝑇Z, 𝜏), where 𝑇 countable set. Let πœ” : 𝑇Z β†’ 𝑆 be a some real function. The variations of Ξ¦ are defined as π‘Š π‘π‘ π‘œ(πœ”) = sup{|πœ”(𝑦) βˆ’ πœ”(𝑧)| : 𝑦𝑗 = 𝑧𝑗(𝑗 = βˆ’(π‘œ βˆ’ 1), Β· Β· Β· , π‘œ βˆ’ 1)} A function πœ” : 𝑇Z β†’ R is called one-side, if πœ”(𝑦) = πœ”(𝑧) for every 𝑦, 𝑧 ∈ 𝑇Z with 𝑦𝑗 = 𝑧𝑗 for all 𝑗 β‰₯ 0. It has

Proposition 1

If πœ” : 𝑇Z β†’ 𝑆 is weakly Holder continuous and 𝑀𝑏𝑠1πœ’ < ∞, then there exists a bound Holder continuous function 𝜚 such that πœ’ = πœ” + 𝜚 βˆ’ 𝜚 ∘ 𝜏 is weakly Holder continuous and one-side.

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 21 / 30

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Sarig, Mauldin & Urbanski

Theorem 1

For full shift map as above, if βˆ‘οΈ

𝑙β‰₯1 π‘Š 𝑏𝑠𝑙(Ξ¦) < ∞, then Ξ¦ has an invariant

Gibbs measure iff Ξ¦ has finite Gurevich pressure.

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 22 / 30

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If πœ’ : Ξ› β†’ 𝑆 is Holder continuous function, then we can define an induced potential on ˜ Ξ› as follow: Ξ¦ = βˆ‘οΈπœ(𝑦)βˆ’1

𝑗=0

πœ’(𝑔 𝑗(𝑦)). We can prove that βˆ‘οΈ

𝑙β‰₯1 π‘Š 𝑏𝑠𝑙(Ξ¦) < ∞.

Lemma 3.2

If 𝜈𝐺 is an ergodic measure on (˜ Ξ›, 𝐺) with ∫︁ πœπ‘’πœˆπΊ < ∞, and 𝜈 is the projected measure on (πœ‡, 𝑔) then β„ŽπœˆF (𝐺) = ( βˆ«οΈ‚ πœπ‘’πœˆπΊ )β„Žπœˆ(𝑔) βˆ«οΈ‚ Ξ¦π‘’πœˆπΊ = ( βˆ«οΈ‚ πœπ‘’πœˆπΊ ) βˆ«οΈ‚ πœ’π‘’πœˆ. πœ”π‘‘ = πœ’ βˆ’ 𝑑, then induced potential Ψ𝑑 = Ξ¦ βˆ’ 𝑒𝑑.

Lemma 3.3

If 𝑄𝐻(Ξ¨π‘‘βˆ—) < ∞, then 𝑄𝐻(Ψ𝑑) is decrease and continuous on [π‘‘βˆ—, ∞).

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 23 / 30

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Equilibrium state

Furthermore if πœ’ = βˆ’π‘’ log |𝐸𝑔 𝑣(𝑦)|, 𝑒 > 0, then we can prove βˆ‘οΈ

𝑙β‰₯1 π‘Š 𝑏𝑠𝑙(Ξ¦) < ∞. There exists an unique Gibbs measure πœˆπ‘’ which is

equilibrium state of induced systems (˜ Ξ›, 𝐺).

Theorem 3.4

There is an open set 𝑉 such that if 𝑒 ∈ 𝑉 then 𝑄𝐻(˜ Ξ›, Φ𝑒) has an unique equilibrium state of Φ𝑒 which is Gibbs measure and the project

  • f πœˆπ‘’ onto (Ξ›, 𝑔) is the equilibrium state of βˆ’π‘’ log |𝐸𝑔 𝑣(𝑦)|.

The properties of Gibbs measure πœˆπ‘’ has exponential decay of correlations and satisfies the CLT for the class of functions whose induced functions are Holder continuous.

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 24 / 30

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Upper semi-continuity

We also use the stable foliation and unstable foliation as above to prove that there is a surjective Ξ  : {0, 1}π‘Ž β†’ Ξ›, finite to one and Holder continuous. Full probability set: one to one. Measure entropy β„Žπœˆ : (𝑔) β†’ 𝑆 upper semi-continuity. In fact it is continuous. Rios and Leplaideur have related results for the nonhyperbolic horseshoe which was constructed by Rios. In fact, our method for constructing infinite branches horseshoe can be applied to the nonhyperbolic horseshoe which was constructed by Rios.

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 25 / 30

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Hausdorff dimension

Furthermore we study the Hausdorff dimension of Ξ›. We can prove that 𝐸𝑗𝑛𝐼Λ = 𝑒𝑑 + 𝑒𝑣 Where 𝑒𝑣 is the root of 𝑄𝐻( ˜ βˆ’π‘’ log |𝐸𝑔 𝑣(𝑦)|) = 0 and 𝑒𝑑 is the root of 𝑄𝐻( ˜ 𝑒 log |𝐸𝑔 𝑑(𝑦)|) = 0. Where ˜ βˆ’π‘’ log |𝐸𝑔 𝑣(𝑦)|, and ˜ 𝑒 log |𝐸𝑔 𝑑(𝑦)| are induced potential of βˆ’π‘’ log |𝐸𝑔 𝑣(𝑦)| and 𝑒 log |𝐸𝑔 𝑑(𝑦)| respectively.

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 26 / 30

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Hausdorff dimension

In fact, we can prove that 𝑄𝐻( ˜ βˆ’π‘’π‘£ log |𝐸𝑔 𝑣(𝑦)|) = π‘ž(βˆ’π‘’π‘£ log |𝐸𝑔 𝑣(𝑦)|) = sup{β„Žπœˆ βˆ’ 𝑒𝑣 βˆ«οΈ‚ log |𝐸𝑔 𝑣(𝑦)|π‘’πœˆ} = 0, 𝑄𝐻( ˜ 𝑒𝑑 log |𝐸𝑔 𝑑(𝑦)|) = π‘ž(𝑒𝑑 log |𝐸𝑔 𝑑(𝑦)|) = sup{β„Žπœˆ + 𝑒𝑑 βˆ«οΈ‚ log |𝐸𝑔 𝑑(𝑦)|π‘’πœˆ} = 0.

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 27 / 30

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Multifractal structure of LE

Now we consider the level set 𝐿𝛽,𝛾 = {𝑦 ∈ Ξ› : lim

π‘œβ†’βˆž

1 π‘œ

π‘œβˆ’1

βˆ‘οΈ‚

𝑗=0

log |𝑒 𝑔(𝑔 𝑗(𝑦))|𝐹s

fi(x)| = 𝛽,

βˆ’ lim

π‘œβ†’βˆž

1 π‘œ

π‘œβˆ’1

βˆ‘οΈ‚

𝑗=0

log |𝑒 𝑔 βˆ’1(𝑔 βˆ’π‘—(𝑦))|𝐹u

fβˆ’i(x)| = 𝛾, }.

Let 𝒬+ = { βˆ«οΈ‚ log |𝑒 𝑔|𝐹s(𝑦)|π‘’πœˆ, 𝜈 ∈ β„³} and π’¬βˆ’ = { βˆ«οΈ‚ log |𝑒 𝑔|𝐹u(𝑦)|π‘’πœˆ, 𝜈 ∈ β„³}.

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 28 / 30

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Multifractal structure of LE

We can prove that for (𝛽, 𝛾) ∈ π‘—π‘œπ‘’π’¬+ Γ— π‘—π‘œπ‘’π’¬βˆ’, 𝑒𝑗𝑛𝐼𝐿𝛽,𝛾 = max{ β„Žπœˆ βˆ’π›½ : 𝜈 ∈ β„³ and βˆ«οΈ‚ log |𝑒 𝑔|𝐹s(𝑦)|π‘’πœˆ = 𝛽} + max{β„Žπœˆ 𝛾 : 𝜈 ∈ β„³ and βˆ«οΈ‚ log |𝑒 𝑔|𝐹u(𝑦)|π‘’πœˆ = 𝛾}. The same result as above for uniformally hyperbolic horseshoe in the surface was obtained by Barreira and Valls 2006(CMP), Barreira and Doutor 2009(Nonlinearity).

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 29 / 30

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THANK YOU! 谒谒！

Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 30 / 30