Large deviations for systems with non-uniform structure

Abstract

Abstract. We use a weak Gibbs property and a weak form of specification to derive level-2 large deviations principles for symbolic systems equipped with a large class of reference measures. This has applications to a broad class of symbolic systems, including β-shifts, S-gap shifts, and their factors. A crucial step in our approach is to prove a 'horseshoe theorem' for these systems.

Distributional chaos in multifractal analysis, recurrence and transitivity

“…So we can find a k ∈ N large enough, such that ξ ′ ξ 1 ξ 2 · · · (ξ k + 1)ξ k+1 ξ k+2 · · · < i(1, β) and ξ ′ ∈ U . Then by Lemma 5.1, we conclude that η ξ 1 ξ 2 · · · ξ k ω ∈ U and σ k η = ω. Lemma 5.3 is a main application in [17]. Reader can refer to [17] for the details of the proof.…”

“…Then by Lemma 5.1, we conclude that η ξ 1 ξ 2 · · · ξ k ω ∈ U and σ k η = ω. Lemma 5.3 is a main application in [17]. Reader can refer to [17] for the details of the proof. The lemma above shows us that to figure out the irregular set for the whole space(Σ β ), it is sufficient to study the irregular set for certain asymptotic 'horseshoe-like'(Σ n β ) of the whole space. }…”

Quantitative Recurrence Properties for Systems with Non-uniform Structure

“…In [4], S-gap shifts, β shifts and their factors satisfy the non-unform structure i.e., for X := S or β there exists G ⊂ L(X) has (W )-specification and L(X) is edit approachable by G. Set R(ψ) := {x ∈ X : d(σ n x, x) < ψ(n) for infinitely many n ∈ N}. Let f be a positive continuous function defined on X, set R(f ) = {x ∈ X : d(σ n x, x) ≤ e −Snf (x) for infinitely many n ∈ N}.…”

“…In this paper, we consider a class of symbolic systems which is studied in [4]. That is, (X, σ) is a symbolic system with non-uniform structure for the symbolic systems (X, σ).…”

Positive entropy equilibrium states

“…Remark 1.4. Another class of shift spaces studied in [CT12,CTY17] are the S-gap shifts, for which there is no function g as in Theorem 1.1; the best that can be done in general is g(n) ≈ √ n, see [CTY17, §5. Given k ∈ N, let F k be the set of w ∈ L such that for every v ∈ L, there is u ∈ L with |u| ≤ k such that wuv ∈ L. (Then L has specification iff there is k such that F k = L.) The 'local specification' property from [HK82, Theorem 3] is equivalent to: for every x ∈ X and every infinite J ⊂ N, there is k ∈ N and an infinite J ′ ⊂ J such that x 1 · · · x j ∈ F k for every j ∈ J ′ .…”