New PDF release: Dimension and Recurrence in Hyperbolic Dynamics

Read or Download Dimension and Recurrence in Hyperbolic Dynamics PDF

Similar fluid dynamics books

• Modern Developments in Fluid Dynamics. An Account of Theory and Experiment Relating to Boundary Layers, Turbulent Motion and Wakes. Vol. II. (The Oxford Engenieering Science Series).
• On Superconductivity and Superfluidity: A Scientific Autobiography
• Perspectives in Membrane Biology
• Advances in microfluidics
• Asymptotic modelling of fluid flow phenomena

Additional info for Dimension and Recurrence in Hyperbolic Dynamics

Example text

9) k=0 We note that these sets are pairwise disjoint for each given r. Since f is of class C 1+ε the function x → log dx f is Holder ¨ continuous with H¨ older ¨ exponent ε. 1. 10) ε 1 + C1 f x − f y 1 + C2 f n x − f n y ε β ε(k−n) , for some constants C1 , C2 > 0. 11) ε −εj 1 + C2 d β j=1 =: D < ∞. Now let h be the local inverse of f n restricted to the set Δ(ω, r), where n = n(ω, r). We obtain diam Δ(ω, r) = sup x,y∈Δ(ω,r) ≤ sup x,y∈Δ(ω,r) x−y f n x − f n y · sup z∈Rin dz h . Since (dx f n )−1 = df n x h and f is conformal on J, we have (dx f n )−1 = dx f n −1 diam Δ(ω, r) ≤ dD dx f n −1 .

U (V Vn )) for each V ∈ Wn (V), and we observe that if Γ ⊂ {U(V) : V ∈ Γ} ⊂ k∈N Wk (V) is a cover of Z, then Wk (U) k∈N is also a cover of Z. Set γ(U) = lim sup n→∞ γn (Φ, U) . n Given ε > 0, we have γn (Φ, U)/n ≤ γ(U) + ε for all sufficiently large n. Hence, ϕ(U(V)) ≤ ϕ(V) + n(γ(U) + ε) for each V ∈ Wn (V), and M (Z, α, Φ, U) ≤ M (Z, α − γ(U) − ε, Φ, V). Therefore, PZ (Φ, U) ≤ PZ (Φ, V) + γ(U) + ε, and PZ (Φ, U) − γ(U) − ε ≤ lim inf PZ (Φ, V). diam V→0 34 Chapter 3. 39) we have that γ(U) → 0 as diam U → 0.

Un ) with U0 , . , Un ∈ U, and we write m(U) = n. For each U ∈ Wn (U) we define the open set n f −k Uk . X(U) = k=0 Now we consider a sequence of continuous functions Φ = {ϕn : X → R}n∈N . For each n ∈ N we define γn (Φ, U) = sup |ϕn (x) − ϕn (y)| : x, y ∈ X(U) for some U ∈ Wn (U) . 39) (we note that since X is compact it has finite covers with diameter as small as desired). 1. 39) is automatically satisfied. More precisely, given a continuous function ϕ : X → R n−1 we consider the sequence of functions ϕn = k=0 ϕ ◦ f k .

Download PDF sample