Please use this identifier to cite or link to this item:
https://ah.lib.nccu.edu.tw/handle/140.119/130195
DC Field | Value | Language |
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dc.contributor | 應數系 | - |
dc.creator | 班榮超 | - |
dc.creator | Ban, Jung-Chao | - |
dc.creator | Chang, Chih-Hung | - |
dc.creator | Chen, Ting-Ju | - |
dc.date | 2018-05 | - |
dc.date.accessioned | 2020-06-22T05:41:43Z | - |
dc.date.available | 2020-06-22T05:41:43Z | - |
dc.date.issued | 2020-06-22T05:41:43Z | - |
dc.identifier.uri | http://nccur.lib.nccu.edu.tw/handle/140.119/130195 | - |
dc.description.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 . | - |
dc.format.extent | 214 bytes | - |
dc.format.mimetype | text/html | - |
dc.relation | AMS/IP Studies in Advanced Mathematics, Vol.2018 | - |
dc.subject | sofic measure ; Sierpiński carpet ; matrix-valued potential ; Gibbs measure ; a-weighted thermodynamic | - |
dc.title | Measure of the full Hausdorff dimension for sofic affine-invariant sets on T^2 | - |
dc.type | article | - |
dc.identifier.doi | 10.13140/RG.2.2.16505.21600 | - |
dc.doi.uri | http://dx.doi.org/10.13140/RG.2.2.16505.21600 | - |
item.grantfulltext | restricted | - |
item.openairecristype | http://purl.org/coar/resource_type/c_18cf | - |
item.fulltext | With Fulltext | - |
item.cerifentitytype | Publications | - |
item.openairetype | article | - |
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