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Автор: Grafiati

Опубліковано: 7 липня 2024

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1

Crane, Harry, and Peter McCullagh. "Reversible Markov structures on divisible set partitions." Journal of Applied Probability 52, no. 03 (September 2015): 622–35. http://dx.doi.org/10.1017/s0021900200113336.

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We studyk-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integerk= 1, 2, …. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, fork> 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restaurant-type seating rules for generating families of exchangeablek-divisible partitions that are consistent under random deletion. We further introduce the notion ofMarkovian partition structures, which are ensembles of exchangeable Markov chains onk-divisible partitions that are consistent under a random process ofMarkovian deletion. The Markov chains we study are reversible and refine the class of Markov chains introduced in Crane (2011).

2

Crane, Harry, and Peter McCullagh. "Reversible Markov structures on divisible set partitions." Journal of Applied Probability 52, no. 3 (September 2015): 622–35. http://dx.doi.org/10.1239/jap/1445543836.

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We study k-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integer k = 1, 2, …. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, for k > 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restaurant-type seating rules for generating families of exchangeable k-divisible partitions that are consistent under random deletion. We further introduce the notion of Markovian partition structures, which are ensembles of exchangeable Markov chains on k-divisible partitions that are consistent under a random process of Markovian deletion. The Markov chains we study are reversible and refine the class of Markov chains introduced in Crane (2011).

3

Infante, Guillermo, Anders Jonsson, and Vicenç Gómez. "Globally Optimal Hierarchical Reinforcement Learning for Linearly-Solvable Markov Decision Processes." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 6 (June 28, 2022): 6970–77. http://dx.doi.org/10.1609/aaai.v36i6.20655.

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We present a novel approach to hierarchical reinforcement learning for linearly-solvable Markov decision processes. Our approach assumes that the state space is partitioned, and defines subtasks for moving between the partitions. We represent value functions on several levels of abstraction, and use the compositionality of subtasks to estimate the optimal values of the states in each partition. The policy is implicitly defined on these optimal value estimates, rather than being decomposed among the subtasks. As a consequence, our approach can learn the globally optimal policy, and does not suffer from non-stationarities induced by high-level decisions. If several partitions have equivalent dynamics, the subtasks of those partitions can be shared. We show that our approach is significantly more sample efficient than that of a flat learner and similar hierarchical approaches when the set of boundary states is smaller than the entire state space.

4

Cawley, Elise. "Smooth Markov partitions and toral automorphisms." Ergodic Theory and Dynamical Systems 11, no. 4 (December 1991): 633–51. http://dx.doi.org/10.1017/s0143385700006404.

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AbstractWe show that the only hyperbolic toral automorphisms f for which there exist Markov partitions with piecewise smooth boundary are those for which a power fk is linearly covered by a direct product of automorphisms of the 2-torus. Only a finite number of shapes occur in a certain natural set of cross-sections of the partition boundary. The behavior of the stratified structure of a piecewise smooth boundary under the mapping forces these shapes to be self-similar. This, together with expanding properties of the mapping, means that a piecewise smooth partition is in fact piecewise linear. Orbits of affine disks in the boundary are used to construct a basis of 2-dimensional invariant toral subgroups, and then the product decomposition of a covering follows easily.

5

Jung, Ho Yub, and Kyoung Mu Lee. "Image Segmentation by Edge Partitioning over a Nonsubmodular Markov Random Field." Mathematical Problems in Engineering 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/683176.

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Edge weight-based segmentation methods, such as normalized cut or minimum cut, require a partition number specification for their energy formulation. The number of partitions plays an important role in the segmentation overall quality. However, finding a suitable partition number is a nontrivial problem, and the numbers are ordinarily manually assigned. This is an aspect of the general partition problem, where finding the partition number is an important and difficult issue. In this paper, the edge weights instead of the pixels are partitioned to segment the images. By partitioning the edge weights into two disjoints sets, that is, cut and connect, an image can be partitioned into all possible disjointed segments. The proposed energy function is independent of the number of segments. The energy is minimized by iterating the QPBO-α-expansion algorithm over the pairwise Markov random field and the mean estimation of the cut and connected edges. Experiments using the Berkeley database show that the proposed segmentation method can obtain equivalently accurate segmentation results without designating the segmentation numbers.

6

Ashley, Jonathan, Bruce Kitchens, and Matthew Stafford. "Boundaries of Markov partitions." Transactions of the American Mathematical Society 333, no. 1 (January 1, 1992): 177–201. http://dx.doi.org/10.1090/s0002-9947-1992-1073772-3.

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7

Wagoner, J. B. "Markov partitions and K2." Publications mathématiques de l'IHÉS 65, no. 1 (December 1987): 91–129. http://dx.doi.org/10.1007/bf02698936.

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8

Borodin, Alexei, and Grigori Olshanski. "Markov processes on partitions." Probability Theory and Related Fields 135, no. 1 (August 17, 2005): 84–152. http://dx.doi.org/10.1007/s00440-005-0458-z.

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9

NEKRASHEVYCH, VOLODYMYR. "SELF-SIMILAR INVERSE SEMIGROUPS AND SMALE SPACES." International Journal of Algebra and Computation 16, no. 05 (October 2006): 849–74. http://dx.doi.org/10.1142/s0218196706003153.

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Self-similar inverse semigroups are defined using automata theory. Adjacency semigroups of s-resolved Markov partitions of Smale spaces are introduced. It is proved that a Smale space can be reconstructed from the adjacency semigroup of its Markov partition, using the notion of the limit solenoid of a contracting self-similar semigroup. The notions of the limit solenoid and a contracting semigroup is described.

10

Friston, Karl, Conor Heins, Kai Ueltzhöffer, Lancelot Da Costa, and Thomas Parr. "Stochastic Chaos and Markov Blankets." Entropy 23, no. 9 (September 17, 2021): 1220. http://dx.doi.org/10.3390/e23091220.

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In this treatment of random dynamical systems, we consider the existence—and identification—of conditional independencies at nonequilibrium steady-state. These independencies underwrite a particular partition of states, in which internal states are statistically secluded from external states by blanket states. The existence of such partitions has interesting implications for the information geometry of internal states. In brief, this geometry can be read as a physics of sentience, where internal states look as if they are inferring external states. However, the existence of such partitions—and the functional form of the underlying densities—have yet to be established. Here, using the Lorenz system as the basis of stochastic chaos, we leverage the Helmholtz decomposition—and polynomial expansions—to parameterise the steady-state density in terms of surprisal or self-information. We then show how Markov blankets can be identified—using the accompanying Hessian—to characterise the coupling between internal and external states in terms of a generalised synchrony or synchronisation of chaos. We conclude by suggesting that this kind of synchronisation may provide a mathematical basis for an elemental form of (autonomous or active) sentience in biology.

11

HIRAIDE, Koichi. "On Homeomorphisms with Markov Partitions." Tokyo Journal of Mathematics 08, no. 1 (June 1985): 219–29. http://dx.doi.org/10.3836/tjm/1270151581.

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12

Tseng, Jimmy. "Schmidt games and Markov partitions." Nonlinearity 22, no. 3 (January 28, 2009): 525–43. http://dx.doi.org/10.1088/0951-7715/22/3/001.

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13

Adler, Roy L. "Symbolic dynamics and Markov partitions." Bulletin of the American Mathematical Society 35, no. 01 (January 1, 1998): 1–57. http://dx.doi.org/10.1090/s0273-0979-98-00737-x.

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14

Teresa Gallegos, María, and Gunter Ritter. "Balanced partitions for Markov chains." Results in Mathematics 37, no. 3-4 (May 2000): 246–73. http://dx.doi.org/10.1007/bf03321996.

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15

Fisher, Todd, and Himal Rathnakumara. "Markov partitions for hyperbolic sets." Involve, a Journal of Mathematics 2, no. 5 (March 10, 2010): 549–57. http://dx.doi.org/10.2140/involve.2009.2.549.

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16

Denker, Manfred, and Hajo Holzmann. "Markov partitions for fibre expanding systems." Colloquium Mathematicum 110, no. 2 (2008): 485–92. http://dx.doi.org/10.4064/cm110-2-11.

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17

BURKE, NIGEL D., and IAN F. PUTNAM. "Markov partitions and homology for -solenoids." Ergodic Theory and Dynamical Systems 37, no. 3 (November 27, 2015): 716–38. http://dx.doi.org/10.1017/etds.2015.71.

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Given a relatively prime pair of integers, $n\geq m>1$, there is associated a topological dynamical system which we refer to as an $n/m$-solenoid. It is also a Smale space, as defined by David Ruelle, meaning that it has local coordinates of contracting and expanding directions. In this case, these are locally products of the real and various $p$-adic numbers. In the special case, $m=2,n=3$ and for $n>3m$, we construct Markov partitions for such systems. The second author has developed a homology theory for Smale spaces and we compute this in these examples, using the given Markov partitions, for all values of $n\geq m>1$ and relatively prime.

18

Jiang, Yunping. "Markov partitions and Feigenbaum-like mappings." Communications in Mathematical Physics 171, no. 2 (August 1995): 351–63. http://dx.doi.org/10.1007/bf02099274.

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19

Franceschini, Valter, and Fernando Zironi. "On constructing Markov partitions by computer." Journal of Statistical Physics 40, no. 1-2 (July 1985): 69–91. http://dx.doi.org/10.1007/bf01010527.

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20

James, Lancelot. "Single-Block Recursive Poisson–Dirichlet Fragmentations of Normalized Generalized Gamma Processes." Mathematics 10, no. 4 (February 11, 2022): 561. http://dx.doi.org/10.3390/math10040561.

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Dong, Goldschmidt and Martin (2006) (DGM) showed that, for 0<α<1, and θ>−α, the repeated application of independent single-block fragmentation operators based on mass partitions following a two-parameter Poisson–Dirichlet distribution with parameters (α,1−α) to a mass partition having a Poisson–Dirichlet distribution with parameters (α,θ) leads to a remarkable nested family of Poisson—Dirichlet distributed mass partitions with parameters (α,θ+r) for r=0,1,2,⋯. Furthermore, these generate a Markovian sequence of α-diversities following Mittag-Leffler distributions, whose ratios lead to independent Beta-distributed variables. These Markov chains are referred to as Mittag-Leffler Markov chains and arise in the broader literature involving Pólya urn and random tree/graph growth models. Here we obtain explicit descriptions of properties of these processes when conditioned on a mixed Poisson process when it equates to an integer n, which has interpretations in a species sampling context. This is equivalent to obtaining properties of the fragmentation operations of (DGM) when applied to mass partitions formed by the normalized jumps of a generalized gamma subordinator and its generalizations. We focus primarily on the case where n=0,1.

21

Guo, Wenyuan, and Tze-Yun Leong. "An Analytic Characterization of Model Minimization in Factored Markov Decision Processes." Proceedings of the AAAI Conference on Artificial Intelligence 24, no. 1 (July 4, 2010): 1077–82. http://dx.doi.org/10.1609/aaai.v24i1.7743.

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Model minimization in Factored Markov Decision Processes (FMDPs) is concerned with finding the most compact partition of the state space such that all states in the same block are action-equivalent. This is an important problem because it can potentially transform a large FMDP into an equivalent but much smaller one, whose solution can be readily used to solve the original model. Previous model minimization algorithms are iterative in nature, making opaque the relationship between the input model and the output partition. We demonstrate that given a set of well-defined concepts and operations on partitions, we can express the model minimization problem in an analytic fashion. The theoretical results developed can be readily applied to solving problems such as estimating the size of the minimum partition, refining existing algorithms, and so on.

22

Van Cutsem, Bernard, and Bernard Ycart. "Renewal-type behavior of absorption times in Markov chains." Advances in Applied Probability 26, no. 4 (December 1994): 988–1005. http://dx.doi.org/10.2307/1427901.

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This paper studies the absorption time of an integer-valued Markov chain with a lower-triangular transition matrix. The main results concern the asymptotic behavior of the absorption time when the starting point tends to infinity (asymptotics of moments and central limit theorem). They are obtained using stochastic comparison for Markov chains and the classical theorems of renewal theory. Applications to the description of large random chains of partitions and large random ordered partitions are given.

23

Van Cutsem, Bernard, and Bernard Ycart. "Renewal-type behavior of absorption times in Markov chains." Advances in Applied Probability 26, no. 04 (December 1994): 988–1005. http://dx.doi.org/10.1017/s0001867800026720.

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This paper studies the absorption time of an integer-valued Markov chain with a lower-triangular transition matrix. The main results concern the asymptotic behavior of the absorption time when the starting point tends to infinity (asymptotics of moments and central limit theorem). They are obtained using stochastic comparison for Markov chains and the classical theorems of renewal theory. Applications to the description of large random chains of partitions and large random ordered partitions are given.

24

Jakobson, Michael, and Lucia D. Simonelli. "Countable Markov partitions suitable for thermodynamic formalism." Journal of Modern Dynamics 13, no. 1 (2018): 199–219. http://dx.doi.org/10.3934/jmd.2018018.

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25

Reiter, Clifford A. "Random Markov matrices and partitions of integers." ACM SIGAPL APL Quote Quad 22, no. 3 (March 1992): 7–9. http://dx.doi.org/10.1145/142263.142265.

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26

Snavely, Mark R. "Markov partitions for the two-dimensional torus." Proceedings of the American Mathematical Society 113, no. 2 (February 1, 1991): 517. http://dx.doi.org/10.1090/s0002-9939-1991-1076579-0.

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27

Bunimovich, L. A., Yakov G. Sinai, and N. I. Chernov. "Markov partitions for two-dimensional hyperbolic billiards." Russian Mathematical Surveys 45, no. 3 (June 30, 1990): 105–52. http://dx.doi.org/10.1070/rm1990v045n03abeh002355.

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28

Stojanovski, Toni Draganov, and Ljupco Kocarev. "Construction of Markov Partitions in PL1D Maps." IEEE Transactions on Circuits and Systems II: Express Briefs 60, no. 10 (October 2013): 702–6. http://dx.doi.org/10.1109/tcsii.2013.2278106.

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29

Jakobson, M. "Uniformly scaled Markov partitions for unimodal maps." Journal of Mathematical Sciences 95, no. 5 (July 1999): 2583–608. http://dx.doi.org/10.1007/bf02169058.

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30

Nicolis, G., C. Nicolis, and John S. Nicolis. "Chaotic dynamics, Markov partitions, and Zipf's law." Journal of Statistical Physics 54, no. 3-4 (February 1989): 915–24. http://dx.doi.org/10.1007/bf01019781.

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31

Ward, Thomas, and Yuki Yayama. "Markov partitions reflecting the geometry of $\times2$, $\times3$." Discrete & Continuous Dynamical Systems - A 24, no. 2 (2009): 613–24. http://dx.doi.org/10.3934/dcds.2009.24.613.

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32

Madritsch, Manfred G. "Non-normal numbers with respect to Markov partitions." Discrete and Continuous Dynamical Systems 34, no. 2 (August 2013): 663–76. http://dx.doi.org/10.3934/dcds.2014.34.663.

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33

Rubido, Nicolás, Celso Grebogi, and Murilo S. Baptista. "Entropy-based generating Markov partitions for complex systems." Chaos: An Interdisciplinary Journal of Nonlinear Science 28, no. 3 (March 2018): 033611. http://dx.doi.org/10.1063/1.5002097.

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34

Stafford, Matthew. "Markov partitions for expanding maps of the circle." Transactions of the American Mathematical Society 324, no. 1 (January 1, 1991): 385–403. http://dx.doi.org/10.1090/s0002-9947-1991-1049617-3.

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35

Coulson, Richard M. R., Nathalie Touboul, and Christos A. Ouzounis. "Lineage-specific partitions in archaeal transcription." Archaea 2, no. 2 (2006): 117–25. http://dx.doi.org/10.1155/2006/629868.

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The phylogenetic distribution of the components comprising the transcriptional machinery in the crenarchaeal and euryarchaeal lineages of the Archaea was analyzed in a systematic manner by genome-wide profiling of transcription complements in fifteen complete archaeal genome sequences. Initially, a reference set of transcription-associated proteins (TAPs) consisting of sequences functioning in all aspects of the transcriptional process, and originating from the three domains of life, was used to query the genomes. TAP-families were detected by sequence clustering of the TAPs and their archaeal homologues, and through extensive database searching, these families were assigned a function. The phylogenetic origins of archaeal genes matching hidden Markov model profiles of protein domains associated with transcription, and those encoding the TAP-homologues, showed there is extensive lineage-specificity of proteins that function as regulators of transcription: most of these sequences are present solely in the Euryarchaeota, with nearly all of them homologous to bacterial DNA-binding proteins. Strikingly, the hidden Markov model profile searches revealed that archaeal chromatin and histone-modifying enzymes also display extensive taxon-restrictedness, both across and within the two phyla.

36

Bedford, Tim. "Generating special Markov partitions for hyperbolic toral automorphisms using fractals." Ergodic Theory and Dynamical Systems 6, no. 3 (September 1986): 325–33. http://dx.doi.org/10.1017/s0143385700003527.

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AbstractWe show that given some natural conditions on a 3 × 3 hyperbolic matrix of integers A(det A = 1) there exists a Markov partition for the induced map A(x + ℤ3) = A(x)+ℤ3 on T3 whose transition matrix is (A−1)t. For expanding endomorphisms of T2 we construct a Markov partition so that there is a semiconjugacy from a full (one-sided) shift.

37

Krüger, Tyll, and Serge Troubetzkoy. "Markov partitions and shadowing for non-uniformly hyperbolic systems with singularities." Ergodic Theory and Dynamical Systems 12, no. 3 (September 1992): 487–508. http://dx.doi.org/10.1017/s014338570000691x.

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38

Crane, Harry. "A Consistent Markov Partition Process Generated from the Paintbox Process." Journal of Applied Probability 48, no. 03 (September 2011): 778–91. http://dx.doi.org/10.1017/s0021900200008317.

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We study a family of Markov processes onP(k), the space of partitions of the natural numbers with at mostkblocks. The process can be constructed from a Poisson point process onR+x ∏i=1kP(k)with intensity dt⊗ ϱν(k), where ϱνis the distribution of the paintbox based on the probability measure ν onPm, the set of ranked-mass partitions of 1, and ϱν(k)is the product measure on ∏i=1kP(k). We show that these processes possess a unique stationary measure, and we discuss a particular set of reversible processes for which transition probabilities can be written down explicitly.

39

Crane, Harry. "A Consistent Markov Partition Process Generated from the Paintbox Process." Journal of Applied Probability 48, no. 3 (September 2011): 778–91. http://dx.doi.org/10.1239/jap/1316796914.

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We study a family of Markov processes on P(k), the space of partitions of the natural numbers with at most k blocks. The process can be constructed from a Poisson point process on R+ x ∏i=1kP(k) with intensity dt ⊗ ϱν(k), where ϱν is the distribution of the paintbox based on the probability measure ν on Pm, the set of ranked-mass partitions of 1, and ϱν(k) is the product measure on ∏i=1kP(k). We show that these processes possess a unique stationary measure, and we discuss a particular set of reversible processes for which transition probabilities can be written down explicitly.

40

Praggastis, Brenda. "Numeration systems and Markov partitions from self similar tilings." Transactions of the American Mathematical Society 351, no. 8 (April 8, 1999): 3315–49. http://dx.doi.org/10.1090/s0002-9947-99-02360-0.

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41

Bollt, Erik, Paweł Góra, Andrzej Ostruszka, and Karol Życzkowski. "Basis Markov Partitions and Transition Matrices for Stochastic Systems." SIAM Journal on Applied Dynamical Systems 7, no. 2 (January 2008): 341–60. http://dx.doi.org/10.1137/070686111.

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42

Cerqueti, Roy, Paolo Falbo, Gianfranco Guastaroba, and Cristian Pelizzari. "Approximating multivariate Markov chains for bootstrapping through contiguous partitions." OR Spectrum 37, no. 3 (April 26, 2015): 803–41. http://dx.doi.org/10.1007/s00291-015-0397-8.

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43

Rykken, E. "Markov Partitions for Hyperbolic Toral Automorphisms of $\t^2$." Rocky Mountain Journal of Mathematics 28, no. 3 (September 1998): 1103–24. http://dx.doi.org/10.1216/rmjm/1181071758.

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44

Hu, Yunchun. "Markov partitions, Martingale and symmetric conjugacy of circle endomorphisms." Proceedings of the American Mathematical Society 145, no. 6 (December 9, 2016): 2557–66. http://dx.doi.org/10.1090/proc/13400.

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45

Vago, Gioia M. "Conjugate unstable manifolds and their underlying geometrized Markov partitions." Topology and its Applications 104, no. 1-3 (June 2000): 255–91. http://dx.doi.org/10.1016/s0166-8641(99)00017-6.

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46

Duchamps, Jean-Jil. "Fragmentations with self-similar branching speeds." Advances in Applied Probability 53, no. 4 (November 22, 2021): 1149–89. http://dx.doi.org/10.1017/apr.2021.11.

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AbstractWe consider fragmentation processes with values in the space of marked partitions of $\mathbb{N}$, i.e. partitions where each block is decorated with a nonnegative real number. Assuming that the marks on distinct blocks evolve as independent positive self-similar Markov processes and determine the speed at which their blocks fragment, we get a natural generalization of the self-similar fragmentations of Bertoin (Ann. Inst. H. Poincaré Prob. Statist.38, 2002). Our main result is the characterization of these generalized fragmentation processes: a Lévy–Khinchin representation is obtained, using techniques from positive self-similar Markov processes and from classical fragmentation processes. We then give sufficient conditions for their absorption in finite time to a frozen state, and for the genealogical tree of the process to have finite total length.

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FULMAN, JASON. "A PROBABILISTIC PROOF OF THE ROGERS–RAMANUJAN IDENTITIES." Bulletin of the London Mathematical Society 33, no. 4 (July 2001): 397–407. http://dx.doi.org/10.1017/s0024609301008207.

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The asymptotic probability theory of conjugacy classes of the finite general groups leads to a probability measure on the set of all partitions of natural numbers. A simple method of understanding these measures in terms of Markov chains is given in this paper, leading to an elementary probabilistic proof of the Rogers–Ramanujan identities. This is compared with work on the uniform measure. The main case of Bailey's lemma is interpreted as finding eigenvectors of the transition matrix of a Markov chain. It is shown that the viewpoint of Markov chains extends to quivers.

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Aleshin-Guendel, Serge, and Rebecca C. Steorts. "Convergence Diagnostics for Entity Resolution." Annual Review of Statistics and Its Application 11, no. 1 (April 22, 2024): 419–35. http://dx.doi.org/10.1146/annurev-statistics-040522-114848.

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Entity resolution is the process of merging and removing duplicate records from multiple data sources, often in the absence of unique identifiers. Bayesian models for entity resolution allow one to include a priori information, quantify uncertainty in important applications, and directly estimate a partition of the records. Markov chain Monte Carlo (MCMC) sampling is the primary computational method for approximate posterior inference in this setting, but due to the high dimensionality of the space of partitions, there are no agreed upon standards for diagnosing nonconvergence of MCMC sampling. In this article, we review Bayesian entity resolution, with a focus on the specific challenges that it poses for the convergence of a Markov chain. We review prior methods for convergence diagnostics, discussing their weaknesses. We provide recommendations for using MCMC sampling for Bayesian entity resolution, focusing on the use of modern diagnostics that are commonplace in applied Bayesian statistics. Using simulated data, we find that a commonly used Gibbs sampler performs poorly compared with two alternatives.

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Field, Michael, and Matthew Nicol. "Ergodic theory of equivariant diffeomorphisms: Markov partitions and stable ergodicity." Memoirs of the American Mathematical Society 169, no. 803 (2004): 0. http://dx.doi.org/10.1090/memo/0803.

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YURI, MICHIKO. "Zeta functions for certain non-hyperbolic systems and topological Markov approximations." Ergodic Theory and Dynamical Systems 18, no. 6 (December 1998): 1589–612. http://dx.doi.org/10.1017/s0143385798117972.

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We study dynamical (Artin–Mazur–Ruelle) zeta functions for piecewise invertible multi-dimensional maps. In particular, we direct our attention to non-hyperbolic systems admitting countable generating definite partitions which are not necessarily Markov but satisfy the finite range structure (FRS) condition. We define a version of Gibbs measure (weak Gibbs measure) and by using it we establish an analogy with thermodynamic formalism for specific cases, i.e. a characterization of the radius of convergence in terms of pressure. The FRS condition leads us to nice countable state symbolic dynamics and allows us to realize it as towers over Markov systems. The Markov approximation method then gives a product formula of zeta functions for certain weighted functions.