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題名 擴張樹與非擴張樹子平移熵與混合性質之研究
Entropy and Mixing Properties of Shifts on Expandable and Non-expandable Trees
作者 蔡承育
Tsai, Cheng-Yu
貢獻者 班榮超
Ban, Jung-Chao
蔡承育
Tsai, Cheng-Yu
關鍵詞 拓樸熵;條型熵近似;收斂速率;Li-Yorke 混沌;馬可夫樹移位;具單位元半群作用
Topological entropy;strip entropy approximation;rate of convergence;Li-Yorke chaos;Markov tree shifts;monoid actions
日期 2026
上傳時間 3-Aug-2026 16:45:17 (UTC+8)
摘要 本文研究了擴張樹與非擴張樹子平移（tree shifts）熵與混合性質。 對於擴張樹子平移，我們證明了條型熵（strip entropy）近似對於黃金平均樹移位的所有路徑，以及某些類別的馬可夫樹移位的部分路徑成立。我們進一步證明，對於與本原矩陣相關聯的d-元樹移位，其收斂速率是線性的。此外，我們表明條型熵近似的收斂速率與對應馬可夫樹移位的擴張常數密切相關。 對於非擴張樹子平移，我們在具單位元半群作用（monoid actions）下推導了熵公式與混合性質，並引入了多種Li-Yorke 混沌形式。特別地，我們證明正熵等價於局部Li-Yorke 混沌，而這比傳統的Li-Yorke 混沌更強。
In this article, we investigate the entropy and mixing behaviors of both expandable and non-expandable tree shifts. For expandable tree shifts, we establish that the strip entropy approximation holds for every path in a golden-mean tree shift and for certain paths in a class of Markov tree shifts. We further show that, for d-tree shifts associated with primitive matrices, the rates of convergence are linear. Moreover, the expanding constant of the corresponding Markov tree shift plays a key role in determining the convergence rate of the strip entropy approximation. For non-expandable tree shifts, we derive entropy formulas and mixing properties under monoid actions, and we introduce various forms of Li-Yorke chaos. In particular, we prove that positive entropy is equivalent to locally Li-Yorke chaos, which is a stronger form than the classical Li-Yorke chaos.
參考文獻 [1] L. A. Pierce II. Computing entropy for Z2-actions. PhD thesis, Oregon State University, 2008. [2] E. H. Lieb. Exact solution of the problem of the entropy of two-dimensional ice. Physical Review Letters, 18(17):692, 1967. [3] J.-C. Ban and C.-H. Chang. Tree-shifts: The entropy of tree-shifts of finite type. Nonlinearity, 30(7):2785–2804, 2017. [4] K. Petersen and I. Salama. Tree shift topological entropy. Theoretical Computer Science, 743(26):64–71, 2018. [5] T.-Y. Li and J. A. Yorke. Period three implies chaos. The American Mathematical Monthly, 82(10):985–992, 1975. [6] J.-C. Ban, C.-H. Chang, Y.-L. Wu, and Y.-Y. Wu. Stem and topological entropy on Cayley trees. Mathematical Physics, Analysis and Geometry, 25(1):1–40, 2022. [7] J.-C. Ban, C.-H. Chang, W.-G. Hu, G.-Y. Lai, and Y.-L. Wu. Topological entropy and sequence entropy for hom tree-shifts on unexpandable trees. Qualitative Theory of Dynamical Systems, 23(3):108, 2024. [8] J.-C. Ban and C.-H. Chang. Tree-shifts: Irreducibility, mixing, and chaos of tree-shifts. Transactions of the American Mathematical Society, 369(12):8389–8407, 2017. [9] K. Petersen and I. Salama. Entropy on regular trees. Discrete & Continuous Dynamical Systems, 40(7):4453, 2020. [10] R. Pavlov. Approximating the hard square entropy constant with probabilistic methods. The Annals of Probability, 40(6):2362–2399, 2012. [11] J.-C. Ban and S.-S. Lin. Patterns generation and transition matrices in multi-dimensional lattice models. Discrete & Continuous Dynamical Systems, 13(3):637, 2005. [12] D. Lind and B. Marcus. An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 1995. [13] R. J. Baxter, I. G. Enting, and S. K. Tsang. Hard-square lattice gas. Journal of Statistical Physics, 22(4):465–489, 1980. [14] N. Aubrun and M. P. Béal. Tree-shifts of finite type. Theoretical Computer Science, 459(9):16–25, 2012. [15] J.-C. Ban, C.-H. Chang, W.-G. Hu, G.-Y. Lai, and Y.-L. Wu. An analogue of topological sequence entropy for Markov hom tree-shifts. Studia Mathematica, 270(3):263–283, 2022. [16] J.-C. Ban, C.-H. Chang, W.-G. Hu, and Y.-L. Wu. On structure of topological entropy for tree-shift of finite type. Journal of Differential Equations, 292(15):325–353, 2021. [17] J.-C. Ban, G.-Y. Lai, and C.-Y. Tsai. The strip entropy approximation of Markov tree-shifts on Markov-Cayley trees. Monatshefte für Mathematik, 207(4):497–519, 2025. [18] J.-C. Ban, G.-Y. Lai, and C.-Y. Tsai. The strip entropy approximation of Markov tree-shifts. Dynamical Systems, 40(4):586–601, 2025. [19] J.-C. Ban, G.-Y. Lai, and C.-Y. Tsai. Entropy, mixing properties and Li-Yorke chaos of monoid actions. Journal of Australian Mathematical Society, pages 1–15, 2026. [20] F. Blanchard. Topological chaos: what may this mean? Journal of Difference Equations and Applications, 15(1):23–46, 2009. [21] M. Foryś, W. Huang, J. Li, and P. Oprocha. Invariant scrambled sets, uniform rigidity and weak mixing. Israel Journal of Mathematics, 211(1):447–472, 2016. [22] W. Huang, J. Li, X. Ye, and X. Zhou. Positive topological entropy and Δ-weakly mixing sets. Advances in Mathematics, 360(14):653–683, 2017. [23] W. Huang and L. Jin. Stable sets and mean Li-Yorke chaos in positive entropy actions of bi-orderable amenable groups. Ergodic Theory and Dynamical Systems, 36(8):2482–2497, 2016. [24] W. Huang and X. Ye. Positive entropy implies chaos along any infinite sequence. Transactions of the Moscow Mathematical Society, 82:1–14, 2021. [25] D. Lind. The entropies of topological Markov shifts and a related class of algebraic integers. Ergodic Theory and Dynamical Systems, 4(2):283–300, 1984. [26] R. Pavlov. Perturbations of multidimensional shifts of finite type. Ergodic Theory and Dynamical Systems, 31(2):483–526, 2011. [27] A. Quas and P. Trow. Subshifts of multi-dimensional shifts of finite type. Ergodic Theory and Dynamical Systems, 20(3):859–874, 2000. [28] R. J. Baxter. Solvable models in statistical mechanics: From Ising to chiral Potts. In B. K. Chung, Q.-Han Park, and C. Rim, editors, Yang–Baxter Systems, Nonlinear Models and Their Applications, pages 1–206. World Scientific, Singapore, 1999. [29] R. J. Baxter. Exactly solved models in statistical mechanics. Academic Press, London, 1982. [30] H. O. Georgii. Gibbs measures and phase transitions, volume 9 of De Gruyter Studies in Mathematics. Walter de Gruyter, Berlin/Boston, 2 edition, 2011. [31] B. Marcus and R. Pavlov. Approximating entropy for a class of Z2 markov random fields and pressure for a class of functions on Z2 shifts of finite type. Ergodic Theory and Dynamical Systems, 33(1):186–220, 2013. [32] R. Grigorchuk and A. Stepin. Gibbs states on countable groups. Theory of Probability & Its Applications, 29(2):359–362, 1985. [33] C. J. Preston. Gibbs states on countable sets: Gibbs states and Markov random fields. Cambridge University Press, London/New York, 1974. [34] S. Zachary. Countable state space Markov random fields and Markov chains on trees. The Annals of Probability, 11(4):894–903, 1983. [35] F. Blanchard, E. Glasner, S. Kolyada, and A. Maass. On Li-Yorke pairs. Journal fur die Reine und Angewandte Mathematik, 2002(547):51–68, 2002. [36] D. Kerr and H. Li. Independence in topological and C∗-dynamics. Mathematische Annalen, 338(4):869–926, 2007. [37] D. Kerr and H. Li. Combinatorial independence and sofic entropy. Communications in Mathematics and Statistics, 1(2):213–257, 2013. [38] W. Huang, J. Li, and X. Ye. Stable sets and mean Li-Yorke chaos in positive entropy systems. Journal of Functional Analysis, 266(6):3377–3394, 2014. [39] J. Li and X. Ye. Recent development of chaos theory in topological dynamics. Acta Mathematica Sinica, English Series, 32(1):83–114, 2016. [40] K. Petersen and I. Salama. Asymptotic pressure on some self-similar trees. Stochastics and Dynamics, 23(02):2350009, 2023. [41] T. Ceccherini-Silberstein and M. Coornaert. Cellular automata and groups. Springer Science & Business Media, 2010. [42] J.-C. Ban and N.-Z. Huang. Commutativity of entropy for nonautonomous systems on trees. Journal of Mathematical Analysis and Applications, 517(2):126621, 2022. [43] H.-Y. Wang and J.-C. Xiong. Chaos for subshifts of finite type. Acta Mathematica Sinica, English Series, 21(6):1407–1414, 2005.
描述 博士
國立政治大學
應用數學系
111751501
資料來源 https://thesis.lib.nccu.edu.tw/thesis/detail/9e9be23dfb28bf2b28e0c49aea16186b/
資料類型 thesis
dc.contributor.advisor 班榮超zh_TW
dc.contributor.advisor Ban, Jung-Chaoen_US
dc.contributor.author (Authors) 蔡承育zh_TW
dc.contributor.author (Authors) Tsai, Cheng-Yuen_US
dc.creator (作者) 蔡承育zh_TW
dc.creator (作者) Tsai, Cheng-Yuen_US
dc.date (日期) 2026
dc.date.accessioned 3-Aug-2026 16:45:17 (UTC+8)-
dc.date.available 3-Aug-2026 16:45:17 (UTC+8)-
dc.date.issued (上傳時間) 3-Aug-2026 16:45:17 (UTC+8)-
dc.identifier.uri (URI) https://ah.lib.nccu.edu.tw/item?item_id=183782-
dc.description (描述) 博士
dc.description (描述) 國立政治大學
dc.description (描述) 應用數學系
dc.description (描述) 111751501
dc.description.abstract (摘要) 本文研究了擴張樹與非擴張樹子平移（tree shifts）熵與混合性質。 對於擴張樹子平移，我們證明了條型熵（strip entropy）近似對於黃金平均樹移位的所有路徑，以及某些類別的馬可夫樹移位的部分路徑成立。我們進一步證明，對於與本原矩陣相關聯的d-元樹移位，其收斂速率是線性的。此外，我們表明條型熵近似的收斂速率與對應馬可夫樹移位的擴張常數密切相關。 對於非擴張樹子平移，我們在具單位元半群作用（monoid actions）下推導了熵公式與混合性質，並引入了多種Li-Yorke 混沌形式。特別地，我們證明正熵等價於局部Li-Yorke 混沌，而這比傳統的Li-Yorke 混沌更強。zh_TW
dc.description.abstract (摘要) In this article, we investigate the entropy and mixing behaviors of both expandable and non-expandable tree shifts. For expandable tree shifts, we establish that the strip entropy approximation holds for every path in a golden-mean tree shift and for certain paths in a class of Markov tree shifts. We further show that, for d-tree shifts associated with primitive matrices, the rates of convergence are linear. Moreover, the expanding constant of the corresponding Markov tree shift plays a key role in determining the convergence rate of the strip entropy approximation. For non-expandable tree shifts, we derive entropy formulas and mixing properties under monoid actions, and we introduce various forms of Li-Yorke chaos. In particular, we prove that positive entropy is equivalent to locally Li-Yorke chaos, which is a stronger form than the classical Li-Yorke chaos.en_US
dc.description.tableofcontents 中文摘要 i Abstract ii Contents iii 1 Introduction 1 1.1 Monoid actions on shifts 1 1.2 Zd shift of finite type 4 2 Fundamental Definitions and Results 7 2.1 1-dimensional shift spaces 7 2.1.1 Higher block shifts 9 2.1.2 Sliding block codes 11 2.1.3 Shifts of finite type 13 2.2 Topological structure of 1-dimensional shift spaces 17 2.2.1 Cylinder sets in a shift space 18 2.3 Topological entropy of 1-dimensional shift spaces 20 2.3.1 Computing entropy 23 3 Main Results and Examples 25 3.1 Non-expandable Markov tree shifts 26 3.1.1 Mixing 26 3.1.2 Topological entropy 28 3.1.3 Li-Yorke chaos 35 3.2 Expandable Markov tree shifts 38 3.2.1 d-tree shifts 38 3.2.2 Golden mean tree shifts 44 3.2.3 Complete recursive tree shifts 45 4 Proof of Main Results 47 4.1 Proof of Theorem 3.1.1 47 4.2 Proof of Theorem 3.1.3 49 4.3 Proof of Theorem 3.1.8 57 4.4 Proof of Theorem 3.2.1 58 4.5 Proof of Theorem 3.2.3 63 4.6 Proof of Theorem 3.2.5 68 4.7 Proof of Theorem 3.2.6 74 4.8 Proof of Theorem 3.2.7 80 5 Conclusion 84 References 86
dc.format.extent 142 bytes-
dc.format.mimetype text/html-
dc.source.uri (資料來源) https://thesis.lib.nccu.edu.tw/thesis/detail/9e9be23dfb28bf2b28e0c49aea16186b/
dc.subject (關鍵詞) 拓樸熵;條型熵近似;收斂速率;Li-Yorke 混沌;馬可夫樹移位;具單位元半群作用zh_TW
dc.subject (關鍵詞) Topological entropy;strip entropy approximation;rate of convergence;Li-Yorke chaos;Markov tree shifts;monoid actionsen_US
dc.title (題名) 擴張樹與非擴張樹子平移熵與混合性質之研究zh_TW
dc.title (題名) Entropy and Mixing Properties of Shifts on Expandable and Non-expandable Treesen_US
dc.type (資料類型) thesis
dc.relation.reference (參考文獻) [1] L. A. Pierce II. Computing entropy for Z2-actions. PhD thesis, Oregon State University, 2008. [2] E. H. Lieb. Exact solution of the problem of the entropy of two-dimensional ice. Physical Review Letters, 18(17):692, 1967. [3] J.-C. Ban and C.-H. Chang. Tree-shifts: The entropy of tree-shifts of finite type. Nonlinearity, 30(7):2785–2804, 2017. [4] K. Petersen and I. Salama. Tree shift topological entropy. Theoretical Computer Science, 743(26):64–71, 2018. [5] T.-Y. Li and J. A. Yorke. Period three implies chaos. The American Mathematical Monthly, 82(10):985–992, 1975. [6] J.-C. Ban, C.-H. Chang, Y.-L. Wu, and Y.-Y. Wu. Stem and topological entropy on Cayley trees. Mathematical Physics, Analysis and Geometry, 25(1):1–40, 2022. [7] J.-C. Ban, C.-H. Chang, W.-G. Hu, G.-Y. Lai, and Y.-L. Wu. Topological entropy and sequence entropy for hom tree-shifts on unexpandable trees. Qualitative Theory of Dynamical Systems, 23(3):108, 2024. [8] J.-C. Ban and C.-H. Chang. Tree-shifts: Irreducibility, mixing, and chaos of tree-shifts. Transactions of the American Mathematical Society, 369(12):8389–8407, 2017. [9] K. Petersen and I. Salama. Entropy on regular trees. Discrete & Continuous Dynamical Systems, 40(7):4453, 2020. [10] R. Pavlov. Approximating the hard square entropy constant with probabilistic methods. The Annals of Probability, 40(6):2362–2399, 2012. [11] J.-C. Ban and S.-S. Lin. Patterns generation and transition matrices in multi-dimensional lattice models. Discrete & Continuous Dynamical Systems, 13(3):637, 2005. [12] D. Lind and B. Marcus. An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 1995. [13] R. J. Baxter, I. G. Enting, and S. K. Tsang. Hard-square lattice gas. Journal of Statistical Physics, 22(4):465–489, 1980. [14] N. Aubrun and M. P. Béal. Tree-shifts of finite type. Theoretical Computer Science, 459(9):16–25, 2012. [15] J.-C. Ban, C.-H. Chang, W.-G. Hu, G.-Y. Lai, and Y.-L. Wu. An analogue of topological sequence entropy for Markov hom tree-shifts. Studia Mathematica, 270(3):263–283, 2022. [16] J.-C. Ban, C.-H. Chang, W.-G. Hu, and Y.-L. Wu. On structure of topological entropy for tree-shift of finite type. Journal of Differential Equations, 292(15):325–353, 2021. [17] J.-C. Ban, G.-Y. Lai, and C.-Y. Tsai. The strip entropy approximation of Markov tree-shifts on Markov-Cayley trees. Monatshefte für Mathematik, 207(4):497–519, 2025. [18] J.-C. Ban, G.-Y. Lai, and C.-Y. Tsai. The strip entropy approximation of Markov tree-shifts. Dynamical Systems, 40(4):586–601, 2025. [19] J.-C. Ban, G.-Y. Lai, and C.-Y. Tsai. Entropy, mixing properties and Li-Yorke chaos of monoid actions. Journal of Australian Mathematical Society, pages 1–15, 2026. [20] F. Blanchard. Topological chaos: what may this mean? Journal of Difference Equations and Applications, 15(1):23–46, 2009. [21] M. Foryś, W. Huang, J. Li, and P. Oprocha. Invariant scrambled sets, uniform rigidity and weak mixing. Israel Journal of Mathematics, 211(1):447–472, 2016. [22] W. Huang, J. Li, X. Ye, and X. Zhou. Positive topological entropy and Δ-weakly mixing sets. Advances in Mathematics, 360(14):653–683, 2017. [23] W. Huang and L. Jin. Stable sets and mean Li-Yorke chaos in positive entropy actions of bi-orderable amenable groups. Ergodic Theory and Dynamical Systems, 36(8):2482–2497, 2016. [24] W. Huang and X. Ye. Positive entropy implies chaos along any infinite sequence. Transactions of the Moscow Mathematical Society, 82:1–14, 2021. [25] D. Lind. The entropies of topological Markov shifts and a related class of algebraic integers. Ergodic Theory and Dynamical Systems, 4(2):283–300, 1984. [26] R. Pavlov. Perturbations of multidimensional shifts of finite type. Ergodic Theory and Dynamical Systems, 31(2):483–526, 2011. [27] A. Quas and P. Trow. Subshifts of multi-dimensional shifts of finite type. Ergodic Theory and Dynamical Systems, 20(3):859–874, 2000. [28] R. J. Baxter. Solvable models in statistical mechanics: From Ising to chiral Potts. In B. K. Chung, Q.-Han Park, and C. Rim, editors, Yang–Baxter Systems, Nonlinear Models and Their Applications, pages 1–206. World Scientific, Singapore, 1999. [29] R. J. Baxter. Exactly solved models in statistical mechanics. Academic Press, London, 1982. [30] H. O. Georgii. Gibbs measures and phase transitions, volume 9 of De Gruyter Studies in Mathematics. Walter de Gruyter, Berlin/Boston, 2 edition, 2011. [31] B. Marcus and R. Pavlov. Approximating entropy for a class of Z2 markov random fields and pressure for a class of functions on Z2 shifts of finite type. Ergodic Theory and Dynamical Systems, 33(1):186–220, 2013. [32] R. Grigorchuk and A. Stepin. Gibbs states on countable groups. Theory of Probability & Its Applications, 29(2):359–362, 1985. [33] C. J. Preston. Gibbs states on countable sets: Gibbs states and Markov random fields. Cambridge University Press, London/New York, 1974. [34] S. Zachary. Countable state space Markov random fields and Markov chains on trees. The Annals of Probability, 11(4):894–903, 1983. [35] F. Blanchard, E. Glasner, S. Kolyada, and A. Maass. On Li-Yorke pairs. Journal fur die Reine und Angewandte Mathematik, 2002(547):51–68, 2002. [36] D. Kerr and H. Li. Independence in topological and C∗-dynamics. Mathematische Annalen, 338(4):869–926, 2007. [37] D. Kerr and H. Li. Combinatorial independence and sofic entropy. Communications in Mathematics and Statistics, 1(2):213–257, 2013. [38] W. Huang, J. Li, and X. Ye. Stable sets and mean Li-Yorke chaos in positive entropy systems. Journal of Functional Analysis, 266(6):3377–3394, 2014. [39] J. Li and X. Ye. Recent development of chaos theory in topological dynamics. Acta Mathematica Sinica, English Series, 32(1):83–114, 2016. [40] K. Petersen and I. Salama. Asymptotic pressure on some self-similar trees. Stochastics and Dynamics, 23(02):2350009, 2023. [41] T. Ceccherini-Silberstein and M. Coornaert. Cellular automata and groups. Springer Science & Business Media, 2010. [42] J.-C. Ban and N.-Z. Huang. Commutativity of entropy for nonautonomous systems on trees. Journal of Mathematical Analysis and Applications, 517(2):126621, 2022. [43] H.-Y. Wang and J.-C. Xiong. Chaos for subshifts of finite type. Acta Mathematica Sinica, English Series, 21(6):1407–1414, 2005.