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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: 2023-07-06 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
    顯示於類別:[應用數學系] 學位論文

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