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題名 Measure of the full Hausdorff dimension for sofic affine-invariant sets on T^2 作者 班榮超
Ban, Jung-Chao
Chang, Chih-Hung
Chen, Ting-Ju貢獻者 應數系 關鍵詞 sofic measure ; Sierpiński carpet ; matrix-valued potential ; Gibbs measure ; a-weighted thermodynamic 日期 2018-05 上傳時間 22-Jun-2020 13:41:43 (UTC+8) 摘要 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 . 關聯 AMS/IP Studies in Advanced Mathematics, Vol.2018 資料類型 article DOI http://dx.doi.org/10.13140/RG.2.2.16505.21600 dc.contributor 應數系 - dc.creator (作者) 班榮超 - dc.creator (作者) Ban, Jung-Chao - dc.creator (作者) Chang, Chih-Hung - dc.creator (作者) Chen, Ting-Ju - dc.date (日期) 2018-05 - dc.date.accessioned 22-Jun-2020 13:41:43 (UTC+8) - dc.date.available 22-Jun-2020 13:41:43 (UTC+8) - dc.date.issued (上傳時間) 22-Jun-2020 13:41:43 (UTC+8) - dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/130195 - dc.description.abstract (摘要) 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 . - dc.format.extent 214 bytes - dc.format.mimetype text/html - dc.relation (關聯) AMS/IP Studies in Advanced Mathematics, Vol.2018 - dc.subject (關鍵詞) sofic measure ; Sierpiński carpet ; matrix-valued potential ; Gibbs measure ; a-weighted thermodynamic - dc.title (題名) Measure of the full Hausdorff dimension for sofic affine-invariant sets on T^2 - dc.type (資料類型) article - dc.identifier.doi (DOI) 10.13140/RG.2.2.16505.21600 - dc.doi.uri (DOI) http://dx.doi.org/10.13140/RG.2.2.16505.21600 -
