The dimension of the full nonuniformly hyperbolic horseshoe Cao - PowerPoint PPT Presentation

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 Hyperbolic set 2 The full non-uniformly hyperbolic horseshoe 3 Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 2 / 30

Motivation 1 Hyperbolic set 2 The full non-uniformly hyperbolic horseshoe 3 Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 3 / 30

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

Hausdorff dimension Definition 1.1 For any 𝜀 > 0 we define ∞ | 𝑉 𝑗 | 𝑡 : { 𝑉 𝑗 } is a ℋ 𝑡 ∑︂ 𝜀 ( 𝐺 ) = inf { 𝜀 cover of 𝐺 } 𝑗 =1 ℋ 𝑡 ( 𝐺 ) = lim 𝜀 → 0 ℋ 𝑡 𝜀 ( 𝐺 ) . This limit exists for any subset, though the limiting value can be 0 or ∞ . We call ℋ 𝑡 ( 𝐺 ) the 𝑡 -dimensional Hausdorff measure of 𝐺 . 𝐸𝑗𝑛 𝐼 𝐺 = inf { 𝑡 : ℋ 𝑡 ( 𝐺 ) = 0 } = sup { 𝑡 : ℋ 𝑡 ( 𝐺 ) = ∞} . Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 4 / 30

Topological pressure Recall the notation of topological pressure. Let 𝑈 : Λ → Λ be a continuous transformation and 𝜒 : Λ → 𝑆 continuous function. 𝐺 is a ( 𝑜, 𝜗 ) separate set of 𝑈 . 𝑦 ∈ 𝐺 𝑓 𝑇 n ( 𝜒 ( 𝑦 )) | 𝐺 is( 𝑜, 𝜗 ) separate set } 𝑄 𝑜 ( 𝜏, 𝜒, 𝜗 ) = sup { ∑︁ Where 𝑇 𝑜 ( 𝜒 ( 𝑦 )) = 𝜒 ( 𝑦 ) + · · · + 𝜒 ( 𝜏 𝑜 − 1 𝑦 ). 1 𝑄 ( 𝜏, 𝜒 ) = lim 𝜗 → 0 lim sup 𝑜 log 𝑄 𝑜 ( 𝜏, 𝜒, 𝜗 ) 𝑜 →∞ Theorem 1.2 Bowen: 𝑄 ( 𝜏, 𝜒 ) topological pressure, variational principal ∫︂ 𝑄 ( 𝜏, 𝜒 ) = sup { ℎ 𝜈 + 𝜒𝑒𝜈 } 𝜈 ∈M ( 𝜏 ) Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 5 / 30

Topological pressure Hausdorff dimension of Standard Cantor set. 3 ) 𝑢 = 1 is it Hausdorff 1. It is well known that the root of equation 2 × ( 1 dimension. It is equivalent to 𝑄 ( 𝑈, − 𝑢 log | 𝐸𝑈 | ) = 0. Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 6 / 30

one dimension repeller 2.One dimension nonlinear expanding map Hausdorff dimension of Cantor set is the root of equation 𝑄 ( 𝑈, − 𝑢 log | 𝐸𝑈 | ) = 0. Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 7 / 30

one dimension repeller 4. 𝑔 : 𝑁 → 𝑁 be a 𝐷 1 map, Λ is a repeller of 𝑔 , Conformal. Theorem 1.3 Let Λ be a 𝐷 1 Conformal repeller of 𝑔 . Then Hausdorff dimension of repeller is the root of equation 𝑄 ( 𝑈, − 𝑢 log | 𝐸𝑈 | ) = 0 . Ruelle considered the 𝐷 1+ 𝛽 for Hausdorff. Falconer considered for Hausdorff and Box dimension Gatzouras and Peres considered the 𝐷 1 case. Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 8 / 30

Motivation 1 Hyperbolic set 2 The full non-uniformly hyperbolic horseshoe 3 Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 9 / 30

Smale Hoesrshoes Next we consider two dimension case. ∀ 𝑦 ∈ Λ , 𝑈 𝑦 𝑁 = 𝐹 𝑡 + 𝐹 𝑣 ‖ 𝐸𝑔 𝑜 ( 𝑤 1 ) ‖ ≤ 𝑓 𝜇𝑜 ‖ 𝑤 1 ‖ , 𝑤 ∈ 𝐹 𝑡 ‖ 𝐸𝑔 𝑜 ( 𝑤 2 ) ‖ ≥ 𝑓 − 𝜇𝑜 ‖ 𝑤 2 ‖ , 𝑤 ∈ 𝐹 𝑣 ( 𝜇 < 0). Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 9 / 30

The Hausdorff dimension of hyperbolic horseshoe MaCluskey, H. and Manning, A., 1983(ETDS) prove that for 𝐷 1+ 𝛽 , the Hausdorff dimension of Λ 𝐸𝑗𝑛 𝐼 Λ = 𝑢 𝑡 + 𝑢 𝑣 Where 𝑢 𝑣 is the root of 𝑄 ( − 𝑢 log | 𝑒 𝑔 | 𝐹 u | ) = 0 and 𝑢 𝑡 is the root of 𝑄 ( 𝑢 log | 𝐸𝑔 | 𝐹 s | ) = 0. J.Palis and Viana prove that the formula as above holds for 𝐷 1 diffeomeorphism. Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 10 / 30

Lyapunove Exponents Lyapunov exponents: 𝑔 : 𝑁 → 𝑁 M is a Riemann manifold . ∀ 𝑦 ∈ 𝑁, 𝑤 ∈ 𝑈 𝑦 𝑁 if 𝑜 log ‖ 𝐸𝑔 𝑜 ( 𝑦 ) 𝑤 ‖ 1 lim ‖ 𝑤 ‖ 𝑜 →∞ exists, it is called Lyapunov exponent, and denote it by 𝜇 ( 𝑦, 𝑤 ). 𝜈 is 𝑔 invariant measure, ℳ ( 𝑔 ). 𝜈 ( 𝐵 ) = 𝜈 ( 𝑔 − 1 ( 𝐵 )) for every measurable set. Ergodic invariant measure ℰ ( 𝑔 ). Ergodic means that for every invariant set 𝐵 , 𝜈 ( 𝐵 ) = 0 , 𝑝𝑠 1. Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 11 / 30

Oseledec Theorem 𝐵 ⊂ 𝑁 𝜈 ( 𝐵 ) = 1 for every 𝜈 ∈ ℳ ( 𝑔 ) 𝑦 ∈ 𝐵 1 .𝜇 1 ( 𝑦 ) ≤ · · · ≤ 𝜇 𝑡 ( 𝑦 ) 2 .𝑊 0 ( 𝑦 ) ⊂ 𝑊 1 ( 𝑦 ) ⊂ · · · ⊂ 𝑊 𝑡 ( 𝑦 ) = 𝑈 𝑦 𝑁 1 𝑜 log | 𝐸𝑔 𝑜 lim 𝑦 ( 𝑤 ) | = 𝜇 𝑗 ( 𝑦 ) , 𝑤 ∈ 𝑊 𝑗 ∖ 𝑊 𝑗 − 1 𝑜 →∞ 𝜇 𝑗 ( 𝑦 ) is defined on 𝐵 𝜈 is ergodic, 𝜇 𝑗 ( 𝑦 ) constant 𝑏.𝑓𝜈 and denote it by 𝜇 𝑗 ( 𝜈 ). Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 12 / 30

Motivation 1 Hyperbolic set 2 The full non-uniformly hyperbolic horseshoe 3 Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 13 / 30

The dynamics of a = a * Henon map 𝐼 𝑏,𝑐 ( 𝑦, 𝑧 ) = (1 − 𝑏𝑦 2 + 𝑧, 𝑐𝑦 ) Theorem 3.1 If 𝑐 is small, then there is an 𝑏 = 𝑏 ∗ , the corresponding map 𝐼 𝑏 ∗ ,𝑐 , ∃ 𝐶 ⊂ Λ with 𝜈 ( 𝐶 ) = 1 for ∀ 𝜈 ∈ ℳ ( 𝐼, Λ) and ∀ 𝑦 ∈ 𝐶 𝜇 1 ( 𝑦 ) < 𝑑 1 < 0 < 𝑑 2 < 𝜇 2 ( 𝑦 ) 𝜈 is hyperbolic measure. What is Hausdorff dimension ? Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 13 / 30

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

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

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 𝑔 4 ( 𝑦 ) = 4 𝑦 (1 − 𝑦 ). Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 16 / 30

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

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

˜ Λ = ∪ ∞ 𝑗 =2 Λ 𝑗 and 𝐺 ( 𝑦 ) = 𝑔 𝜐 ( 𝑦 )( 𝑦 ) and 𝜐 ( 𝑦 ) = 𝑗 for 𝑦 ∈ Λ 𝑗 . (˜ Λ , 𝐺 ), the first return map to ˜ Λ. There hyperbolic product structure. Stable foliation 𝛿 𝑡 and unstable foliation 𝛿 𝑣 . For 𝑧 ∈ 𝛿 𝑡 ( 𝑦 ) , 𝐸𝐺 𝑣 ( 𝐺 𝑗 ( 𝑦 ) log Π ∞ 𝐸𝐺 𝑣 ( 𝐺 𝑗 ( 𝑧 )) ≤ 𝑑𝜇 𝑜 . 𝑗 = 𝑜 For 𝑧 ∈ 𝛿 𝑣 ( 𝑦 ) , and they are in the same 𝑜 cylinder, 𝐸𝐺 𝑣 ( 𝐺 𝑗 ( 𝑦 ) log Π 𝑜 𝐸𝐺 𝑣 ( 𝐺 𝑗 ( 𝑧 )) ≤ 𝑑𝜇 𝑜 . 𝑗 =0 Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 19 / 30

Consider the one-side full shift of countable type ( 𝑇 N , 𝜏 ), where 𝑇 countable set. Let Φ : 𝑇 N → 𝑆 be a some real function. The variations of Φ are defined as 𝑊 𝑏𝑠 𝑜 (Φ) = sup {| Φ( 𝑦 ) − Φ( 𝑧 ) | : 𝑦, 𝑧 in the same n-cylinder } If there are constants 𝐷 > 0 and 𝜄 ∈ (0 , 1) such that 𝑊 𝑏𝑠 𝑜 (Φ) < 𝐷𝜄 𝑜 for all 𝑜 ≥ 2, then we call Φ the weakly Holder continuous. A Gibbs measure 𝐶 ≤ 𝑛 [ 𝑏 0 , · · · , 𝑏 𝑜 − 1 ] 1 ≤ 𝐶 for all 𝑦 ∈ [ 𝑏 0 , · · · , 𝑏 𝑜 − 1 ] . 𝑓 Φ n ( 𝑦 ) − 𝑜𝑄 Φ 𝑜 = ∑︁ 𝑜 − 1 𝑗 =0 Φ( 𝜏 𝑗 ( 𝑦 )). The Gurevich pressure of Φ 1 ∑︂ 𝑓 Φ n ( 𝑦 ) 1 [ 𝑏 ] ( 𝑦 ) 𝑄 𝐻 (Φ) = lim 𝑜 log 𝑜 →∞ 𝜏 n ( 𝑦 )= 𝑦 Cao Yongluo (Suzhou University) The dimension of the full nonuniformly hyperbolic horseshoe 20 / 30