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題名 樹上馬可夫系統的拓樸性質及統計量
Markov Systems on Trees : Topological Properties and Statistic Quantities作者 黃迺筑
Huang, Nai-Zhu貢獻者 班榮超
Ban, Jung-Chao
黃迺筑
Huang, Nai-Zhu關鍵詞 動態系統
馬可夫系統
樹
混合性質
熵
交換性
Dynamical systems
Markov systems
Tree
Mixing property
Entropy
Commutativity日期 2023 上傳時間 6-Jul-2023 17:06:53 (UTC+8) 摘要 本篇博士論文研究樹上馬可夫系統的拓樸性質和統計量。我們針對在多元樹上馬可夫系統的拓樸性質，包含混合性質、CPC混合性質、稠密軌道的存在性，最小系統和重現點，都提供了由馬可夫系統的鄰接矩陣判別的刻畫條件。其中幾乎所有的矩陣刻畫條件都是有限步以內可檢查的。第二部分我們討論樹上馬可夫系統的增長率的一階估計（degree）和二階估計(熵)。得到了馬可夫樹上的馬可夫系統的degree是馬可夫樹的鄰接矩陣的譜半徑取對數的結果。關於馬可夫樹上的馬可夫系統的熵，我們提供了一個由鄰接矩陣表示的遞迴計算公式。最後得到了在非週期回歸樹上的馬可夫系統都具備熵的可交換性的結果。
This dissertation presents the equivalent matrix-conditions for the topological properties of a Markov system on a $d$-tree, based on its adjacency matrix. These characterizations include various kinds of mixing properties, mixing properties in the sense of CPC (complete prefix code), the existence of dense orbits, minimal systems, and the recurrence. Notably, almost all of these equivalent matrix-conditions are finitely checkable.In the second part of this dissertation, we consider two statistical quantities, the degree and the entropy, of a Markov system on trees. It is showed that the degree of any Markov system on the Markov tree $\\mathcal{T}_D$ with adjacency matrix $D$ is the logarithm of the spectral radius of $D$. An algorithm is provided to compute the entropy of a Markov system on the Markov tree. Finally, we proved that the entropy is commutative for nonautonomous $p$-periodic Markov systems on an aperiodic recursive tree.參考文獻 [1] J. Auslander and J. A. Yorke. Interval maps, factors of maps, and chaos. Tohoku Mathematical Journal, Second Series, 32(2):177–188, 1980.[2] J.-C. Ban and C.-H. Chang. Tree-shifts: The entropy of tree-shifts of finite type. Nonlinearity, 30:2785–2804, 2017.[3] J.-C. Ban and C.-H. Chang. Characterization for entropy of shifts of finite type on cayley trees. Journal of Statistical Mechanics: Theory and Experiment, 2020(7):073412, 2020.[4] J.-C. Ban, C.-H. Chang, W.-G. Hu, G.-Y. Lai, and Y.-L. Wu. Characterization and topological behavior of homomorphism tree-shifts. Topology and its Applications, 302:107848, 2021.[5] J.-C. Ban, C.-H. Chang, and N.-Z. Huang. Entropy dimension of shift spaces on monoids. Journal of Mathematical Physics, 61(7):072702, 2020.[6] J.-C. Ban, C.-H. Chang, N.-Z. Huang, and G.-Y. Lai. On the mixing properties of the markov tree-shifts. preprint, 2023.[7] J.-C. Ban, C.-H. Chang, N.-Z. Huang, and G.-Y. Lai. Transitivity, dense orbits, minimality and recurrence of markov tree-shifts. preprint, 2023.[8] J.-C. Ban, W.-G. Hu, S.-S. Lin, and Y.-H. Lin. Verification of mixing properties in two- dimensional shifts of finite type. Journal of Mathematical Physics, 62(7):072703, 2021.[9] I. Benjamini and Y. Peres. Markov chains indexed by trees. Annals of Probability, 22(1):219–243, 1994.[10]R.Berger. The undecidability of the domino problem. Number66. American Mathematical Society, 1966.[11] T. Berger and Z.-X. Ye. Entropic aspects of random fields on trees. IEEE Transactions on Information Theory, 36(5):1006–1018, 1990.[12]R.Bowen.EquilibriumstatesandtheergodictheoryofAnosovdiffeomorphisms.Springer- Verlag, Berlin-New York, 1975.[13]M.Boyle,R.Pavlov,andM.Schraudner.Multidimensionalsoficshiftswithoutseparation and their factors. Transactions of the American Mathematical Society, 362(9):4617–4653, 2010.[14] N. Chandgotia and B. Marcus. Mixing properties for hom-shifts and the distance between walks on associated graphs. Pacific Journal of Mathematics, 294:41–69, 2018.[15] S. Friedland. On the entropy of zd subshifts of finite type. Linear Algebra and its Applications, 252(1-3):199–220, 1997.[16] H.-O. Georgii. Gibbs measures and phase transitions. In Gibbs Measures and Phase Transitions. de Gruyter, 2011.[17] M. Hochman and T. Meyerovitch. A characterization of the entropies of multidimensional shifts of finite type. Annals of Mathematics, 171(3):2011–2038, 2010.[18] S. Kolyada and L. Snoha. Topological entropy of nonautonomous dynamical systems. Random and computational dynamics, 4(2):205, 1996.[19] D. Lind and B. Marcus. An introduction to symbolic dynamics and coding. Cambridge university press, 2021.[20] D. J. C. MacKay. Information theory, inference and learning algorithms. Cambridge university press, 2003.[21] T. Meyerovitch and R. Pavlov. On independence and entropy for high-dimensional isotropic subshifts. Proceedings of the London Mathematical Society, 109(4):921–945, 2014.[22] K. Petersen and I. Salama. Tree shift topological entropy. Theoretical Computer Science, 743:64–71, 2018.[23] S. Silverman. On maps with dense orbits and the definition of chaos. The Rocky Mountain Journal of Mathematics, 22(1):353–375, 1992.[24] F. Spitzer. Markov random fields on an infinite tree. Annals of Probability, 3(3):387–398, 1975.[25] W.-G. Yang and Z.-X. Ye. The asymptotic equipartition property for nonhomogeneous markov chains indexed by a homogeneous tree. IEEE Transactions on Information Theory, 53(9):3275–3280, 2007.[26] L. Zadeh and C. Desoer. Linear system theory: the state space approach. Courier Dover Publications, 2008. 描述 博士
國立政治大學
應用數學系
108751502資料來源 http://thesis.lib.nccu.edu.tw/record/#G0108751502 資料類型 thesis dc.contributor.advisor 班榮超 zh_TW dc.contributor.advisor Ban, Jung-Chao en_US dc.contributor.author (Authors) 黃迺筑 zh_TW dc.contributor.author (Authors) Huang, Nai-Zhu en_US dc.creator (作者) 黃迺筑 zh_TW dc.creator (作者) Huang, Nai-Zhu en_US dc.date (日期) 2023 en_US dc.date.accessioned 6-Jul-2023 17:06:53 (UTC+8) - dc.date.available 6-Jul-2023 17:06:53 (UTC+8) - dc.date.issued (上傳時間) 6-Jul-2023 17:06:53 (UTC+8) - dc.identifier (Other Identifiers) G0108751502 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/145949 - dc.description (描述) 博士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 應用數學系 zh_TW dc.description (描述) 108751502 zh_TW dc.description.abstract (摘要) 本篇博士論文研究樹上馬可夫系統的拓樸性質和統計量。我們針對在多元樹上馬可夫系統的拓樸性質，包含混合性質、CPC混合性質、稠密軌道的存在性，最小系統和重現點，都提供了由馬可夫系統的鄰接矩陣判別的刻畫條件。其中幾乎所有的矩陣刻畫條件都是有限步以內可檢查的。第二部分我們討論樹上馬可夫系統的增長率的一階估計（degree）和二階估計(熵)。得到了馬可夫樹上的馬可夫系統的degree是馬可夫樹的鄰接矩陣的譜半徑取對數的結果。關於馬可夫樹上的馬可夫系統的熵，我們提供了一個由鄰接矩陣表示的遞迴計算公式。最後得到了在非週期回歸樹上的馬可夫系統都具備熵的可交換性的結果。 zh_TW dc.description.abstract (摘要) This dissertation presents the equivalent matrix-conditions for the topological properties of a Markov system on a $d$-tree, based on its adjacency matrix. These characterizations include various kinds of mixing properties, mixing properties in the sense of CPC (complete prefix code), the existence of dense orbits, minimal systems, and the recurrence. Notably, almost all of these equivalent matrix-conditions are finitely checkable.In the second part of this dissertation, we consider two statistical quantities, the degree and the entropy, of a Markov system on trees. It is showed that the degree of any Markov system on the Markov tree $\\mathcal{T}_D$ with adjacency matrix $D$ is the logarithm of the spectral radius of $D$. An algorithm is provided to compute the entropy of a Markov system on the Markov tree. Finally, we proved that the entropy is commutative for nonautonomous $p$-periodic Markov systems on an aperiodic recursive tree. en_US dc.description.tableofcontents 致謝 i中文摘要 iiAbstract iiiContents iv1 Introductions p.12 Preliminaries p.52.1 Non-NegativeMatrices p.52.2 Trees p.112.2.1 d-Trees, Markov Trees and Recursive Trees p.112.2.2 Complete Recursive Markov Trees p.252.3 Markov Systems on Trees p.343 Topological Properties p.403.1 Mixing Properties on d-Trees p.403.1.1 TraditionalMixingProperties p.403.1.2 CPC-Mixing Properties p.473.2 Dense Orbit on d-Trees p.583.2.1 Dense Orbit p.583.2.2 MinimalSystems p.753.2.3 Recurrences p.794 Statistic Quantities p.834.1 Degree and Entropy on Markov Trees p.834.1.1 Degree p.834.1.2 Entropy p.914.2 The Commutativity of Entropy on Aperiodic Recursive Trees p.97References p.112 zh_TW dc.format.extent 1478976 bytes - dc.format.mimetype application/pdf - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0108751502 en_US dc.subject (關鍵詞) 動態系統 zh_TW dc.subject (關鍵詞) 馬可夫系統 zh_TW dc.subject (關鍵詞) 樹 zh_TW dc.subject (關鍵詞) 混合性質 zh_TW dc.subject (關鍵詞) 熵 zh_TW dc.subject (關鍵詞) 交換性 zh_TW dc.subject (關鍵詞) Dynamical systems en_US dc.subject (關鍵詞) Markov systems en_US dc.subject (關鍵詞) Tree en_US dc.subject (關鍵詞) Mixing property en_US dc.subject (關鍵詞) Entropy en_US dc.subject (關鍵詞) Commutativity en_US dc.title (題名) 樹上馬可夫系統的拓樸性質及統計量 zh_TW dc.title (題名) Markov Systems on Trees : Topological Properties and Statistic Quantities en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) [1] J. Auslander and J. A. Yorke. Interval maps, factors of maps, and chaos. Tohoku Mathematical Journal, Second Series, 32(2):177–188, 1980.[2] J.-C. Ban and C.-H. Chang. Tree-shifts: The entropy of tree-shifts of finite type. Nonlinearity, 30:2785–2804, 2017.[3] J.-C. Ban and C.-H. Chang. Characterization for entropy of shifts of finite type on cayley trees. Journal of Statistical Mechanics: Theory and Experiment, 2020(7):073412, 2020.[4] J.-C. Ban, C.-H. Chang, W.-G. Hu, G.-Y. Lai, and Y.-L. Wu. Characterization and topological behavior of homomorphism tree-shifts. Topology and its Applications, 302:107848, 2021.[5] J.-C. Ban, C.-H. Chang, and N.-Z. Huang. Entropy dimension of shift spaces on monoids. Journal of Mathematical Physics, 61(7):072702, 2020.[6] J.-C. Ban, C.-H. Chang, N.-Z. Huang, and G.-Y. Lai. On the mixing properties of the markov tree-shifts. preprint, 2023.[7] J.-C. Ban, C.-H. Chang, N.-Z. Huang, and G.-Y. Lai. Transitivity, dense orbits, minimality and recurrence of markov tree-shifts. preprint, 2023.[8] J.-C. Ban, W.-G. Hu, S.-S. Lin, and Y.-H. Lin. Verification of mixing properties in two- dimensional shifts of finite type. Journal of Mathematical Physics, 62(7):072703, 2021.[9] I. Benjamini and Y. Peres. Markov chains indexed by trees. Annals of Probability, 22(1):219–243, 1994.[10]R.Berger. The undecidability of the domino problem. Number66. American Mathematical Society, 1966.[11] T. Berger and Z.-X. Ye. Entropic aspects of random fields on trees. IEEE Transactions on Information Theory, 36(5):1006–1018, 1990.[12]R.Bowen.EquilibriumstatesandtheergodictheoryofAnosovdiffeomorphisms.Springer- Verlag, Berlin-New York, 1975.[13]M.Boyle,R.Pavlov,andM.Schraudner.Multidimensionalsoficshiftswithoutseparation and their factors. Transactions of the American Mathematical Society, 362(9):4617–4653, 2010.[14] N. Chandgotia and B. Marcus. Mixing properties for hom-shifts and the distance between walks on associated graphs. Pacific Journal of Mathematics, 294:41–69, 2018.[15] S. Friedland. On the entropy of zd subshifts of finite type. Linear Algebra and its Applications, 252(1-3):199–220, 1997.[16] H.-O. Georgii. Gibbs measures and phase transitions. In Gibbs Measures and Phase Transitions. de Gruyter, 2011.[17] M. Hochman and T. Meyerovitch. A characterization of the entropies of multidimensional shifts of finite type. Annals of Mathematics, 171(3):2011–2038, 2010.[18] S. Kolyada and L. Snoha. Topological entropy of nonautonomous dynamical systems. Random and computational dynamics, 4(2):205, 1996.[19] D. Lind and B. Marcus. An introduction to symbolic dynamics and coding. Cambridge university press, 2021.[20] D. J. C. MacKay. Information theory, inference and learning algorithms. Cambridge university press, 2003.[21] T. Meyerovitch and R. Pavlov. On independence and entropy for high-dimensional isotropic subshifts. Proceedings of the London Mathematical Society, 109(4):921–945, 2014.[22] K. Petersen and I. Salama. Tree shift topological entropy. Theoretical Computer Science, 743:64–71, 2018.[23] S. Silverman. On maps with dense orbits and the definition of chaos. The Rocky Mountain Journal of Mathematics, 22(1):353–375, 1992.[24] F. Spitzer. Markov random fields on an infinite tree. Annals of Probability, 3(3):387–398, 1975.[25] W.-G. Yang and Z.-X. Ye. The asymptotic equipartition property for nonhomogeneous markov chains indexed by a homogeneous tree. IEEE Transactions on Information Theory, 53(9):3275–3280, 2007.[26] L. Zadeh and C. Desoer. Linear system theory: the state space approach. Courier Dover Publications, 2008. zh_TW
