Measure of the full Hausdorff dimension for sofic affine-invariant sets on T^2

Abstract

Measures of the full Hausdorff dimension for sofic affine-invariant sets on T 2. (English) £ ¢ ¡ Zbl 07128819 Ji, Lizhen (ed.) et al., In this paper, the authors have studied the measure of the full dimension for a general Sierpiński carpet. They have firstly given a criterion for the measure of the full Hausdorff dimension of a Sierpiński carpet, that is, \"the conditional equilibrium measure of zero potential with respect to some Gibbs measure ν α of matrix-valued potential αN\". In the second part of the paper, they have given a criterion for the Markov projection measure and estimated its number of steps by means of the induced matrix-valued potential. The main results of this paper are the following: Theorem 1. Let Z be a Markov Sierpiński carpet and N = (N ij) n i,j=1 be the induced potential from A. Assume N is irreducible, then (i) The following statements are equivalent. (a) µ is the unique measure of the full Hausdorff dimension. (b) µ is the unique conditional equilibrium measure of the zero potential function on Z with respect to ν α , where ν α is the unique equilibrium measure of the matrix-valued potential αN = (∥N J ∥ α) J∈Y *. (ii) The following Hausdorff dimension formula holds: dim H Z = h top (Z) log m + P (σ Y , αN) log n Theorem 2. Let Z be a Markov Sierpiński carpet and N = (N ij) n i,j=1 be the induced matrix-valued potential from A. Then, ν is a k-step Markov measure on Y if and only if N satisfies the Markov condition of order k. Furthermore, if ν is a k-step Markov measure, then k ≤ m − n. Theorem 3. If N satisfies the Markov condition of order k, then ν is the unique maximal measure of the subshift of finite type X M with adjacency matrix M = [m (J, J ′)] J,J ′ ∈Y k. Theorem 4. Let Z = Z (m,n) (A) be a Markov Sierpiński carpet with A, assume that N the induced potential from A is irreducible. Then, dim H Z = 1 log n lim n→∞ 1 n log ∑ J∈Yn ∥N J ∥ α where α = log n/ log m. Theorem 5. Let N = (N i) i∈S be a family of d × d matrices with entries in R. If N = (N i) i∈S is irreducible. Then for each α > 0, P (σ Y , αN) has a unique α-equilibrium measure µ α which satisfies the Gibbs property: ∀n ∈ N and J ∈ Y n , there exists c > 0 such that c −1 exp(−nP (σ Y , αN)) ∥N J ∥ α ≤ µ α ([J]) ≤ c exp(−nP (σ Y , αN)) ∥N J ∥ α .