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    政大機構典藏 > 理學院 > 應用數學系 > 學位論文 >  Item 140.119/145949
    Please use this identifier to cite or link to this item: https://nccur.lib.nccu.edu.tw/handle/140.119/145949

    Title: 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
    Authors: 黃迺筑
    Huang, Nai-Zhu
    Contributors: 班榮超
    Ban, Jung-Chao
    Huang, Nai-Zhu
    Keywords: 動態系統


    Dynamical systems
    Markov systems
    Mixing property
    Date: 2023
    Issue Date: 2023-07-06 17:06:53 (UTC+8)
    Abstract: 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。
    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.
    Reference: [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.
    Description: 博士
    Source URI: http://thesis.lib.nccu.edu.tw/record/#G0108751502
    Data Type: thesis
    Appears in Collections:[應用數學系] 學位論文

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