Non-uniform Specification and Large Deviations for Weak Gibbs Measures

Abstract

Let (X, T) be a dynamical system, where X is a compact metric space and T : X → X a continuous onto map. For weak Gibbs measures we prove large deviations estimates.

Multifractal analysis of ergodic averages in some non-uniformly hyperbolic systems

“…However, in order to obtain the lower bound on P (K(ϕ, α), ψ), the dynamical system should be endowed with some mixing property such as specification by Takens & Verbitskiy [30], Tomphson [32], g almost product property by Pfister & Sullivan [28,29], Pei & Chen [26], Yamamoto [36]. Here, we will use the weak specification introduced by Varandas [34]. The proof will be divided into the following two subsection.…”

“…In this section, by the work of Paulo Varandas [34], Theorem 1.1 can be applied (i) to Maneville-Pomeau map, (ii) to multidimensional local diffeomorphisms, and (iii) Viana maps. …”

“…Paulo Varandas [34] proved that when m is SRB measure or the maximal entropy measure, ([0, 1], T, m) satisfies non-uniform specification. Example 2 multidimensional local diffeomorphisms Let T 0 be an expanding map in T n and take a periodic point p for T 0 .…”

“…(1) every point x ∈ T n has some preimage outside U; (2) DT (x) −1 ≤ σ −1 for every x ∈ T n \ U, and DT (x) −1 ≤ L for every x ∈ T n where σ > 1 is large enough or L > 0 is sufficiently close to1; (3) T is topologically exact: for every open set U there is N ≥ 1 for which T N (U) = T n Paulo Varandas [34] proved that if m be the unique ergodic equilibrium state for Holder continuous potential − log |detDT |, then (T, m) satisfies non-uniform specification. where d ≥ 16 is an integer, a is a Misiurewicz parameter for the quadratic family, and α is small.…”

“…Definition 1.1. [34] We say that (T, m) satisfy the non-uniform specification property if there exists δ > 0 such that for m-almost every x and every 0 < ǫ < δ there exists an integer p(x, n, ǫ) ≥ 1 satisfying lim ǫ→0 lim sup n→∞ 1 n p(x, n, ǫ) = 0 and so that the following holds: given points x 1 , · · · , x k in a full m-measure set and positive integers n 1 , · · · , n k , if p i ≥ p(x i , n i , ǫ) then there exists z that ǫ-shadows the orbits of each x i during n i iterates with a time lag of p(x i , n i , ǫ) in between T n i (x i ) and…”