Localized asymptotic behavior for almost additive potentials

Abstract

We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost additive continuous potentials (φn) ∞ n=1 on a topologically mixing subshift of finite type X endowed itself with a metric associated with such a potential. We work without additional regularity assumption other than continuity. Our approach differs from those used previously to deal with this question under stronger assumptions on the potentials. As a consequence, it provides a new description of the structure of the spectrum in terms of weak concavity. Also, the lower bound for the spectrum is obtained as a consequence of the study sets of points at which the asymptotic behavior of φn(x) is localized, i.e. depends on the point x rather than being equal to a constant. Specifically, we compute the Hausdorff dimension of sets of the form {x ∈ X : limn→∞ φn(x)/n = ξ(x)}, where ξ is a given continuous function. This has interesting geometric applications to fixed points in the asymptotic average for dynamical systems in R d , as well as the fine local behavior of the harmonic measure on conformal planar Cantor sets.it is ready to show that χ is also injective. Next we show that T (Σ, which is a contradiction. Next we show that for any Gibbs measure µ we have µ(Z ∞ ) = 0. Define Z n := w∈Σ A, * , |w|≤n f w (∂V ) and Z n = χ −1 ( Z n ). The sequence (Z n ) n≥1 is non decreasing and Z ∞ = n≥1 Z n . Since the IFS is conformal we can easily get T (Z n ) ⊂ Z n−1 for n ≥ 1 and T (Z 0 ) ⊂ Z 0 . Consequently T (Z n ) ⊂ Z n . By the ergodicity we have µ(Z n ) = 0 or 1. By the SOSC, Σ A \Z n is nonempty and open, thus by the Gibbs property of µ we get µ(Σ A \ Z n ) > 0, hence µ(Z n ) = 0.Proof of Theorem 3.2. (1) At first we notice that by the property (4) assumed in the construction of J the mapping χ is Lipschitz. This is enough to get the desired upper bounds from Theorem 2.2(1). Now we deal with the lower bound for dimensions and the equality L Φ = L Φ . We notice that the inclusion L Φ ⊂ L Φ holds by construction.Suppose J is a conformal repeller. Since we have χ • T = g • χ on Σ A and χ is surjective, it is seen that χ −1 (E Φ (α)) = E Φ (α) for any α ∈ L Φ . Thus L Φ = L Φ and by Proposition 5, we have dim H E Φ (α) = dim H E Φ (α).Suppose J is the attractor of a conformal IFS with SOSC. Let α ∈ L Φ . Let Z = χ −1 (∂V ). The set Z is closed and by Lemma 3.

Multifractal Analysis of Dimension Spectrum in Non-uniformly Hyperbolic Systems

“…Their methods are very interesting, and they do not use thermodynamic formalism or large deviation method. We generalize the results to the additive vector-valued potentials in [19], and our methods combine the ideas in [1,7,17]. In one dimension, the compact and convex set L f is just a interval or a point.…”

“…Both in [19] and this paper, we combine the method of constructing nth Bernoulli measure first appeared in [17] and concatenation technique for measures appeared in [1,7] to construct a Moran subset M ⊂ X α , on which we support a suitable measure, whose dimension can be arbitrary close to what we expected. By this way we obtain a unified way to treat the lower bound.…”

“…The method is based on the partition of convex polyhedron. We remark that our method not only works in our settings of additive potentials, almost additive potentials, but also the setting considered in [1].…”

“…In the case of subshift of finite type, the multifractal analysis for the level sets of almost additive potentials or quotient of almost additive potentials has been well understood [1,2,5]. However, there is still not a complete picture for the multifractal analysis of non-uniform hyperbolic dynamic systems.…”

“…Some interesting applications for the fine local behavior of the harmonic measure on conformal planar cantor sets can be found in [1]. In [2], they generalized the results in [1] to consider the level set for the quotient of almost additive potentials.…”