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The mathematical research of William Parry FRS

Published online by Cambridge University Press:  01 April 2008

M. POLLICOTT ,
R. SHARP ,
S. TUNCEL  and
P. WALTERS
M. POLLICOTT
Affiliation:
Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK (email: [email protected])
R. SHARP
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
S. TUNCEL
Affiliation:
Department of Mathematics, University of Washington, Box 354350 Seattle, WA 98195-4350, USA
P. WALTERS
Affiliation:
Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK (email: [email protected])
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Abstract

In this article we survey the mathematical research of the late William (Bill) Parry, FRS.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008 , pp. 321 - 337
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Parry, W.. On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.CrossRefGoogle Scholar
[2]Parry, W.. Ergodic properties of some permutation processes. Biometrika 49 (1962), 151–154.CrossRefGoogle Scholar
[3]Kakutani, S. and Parry, W.. Infinite measure preserving transformations with mixing. Bull. Amer. Math. Soc. 69 (1963), 752–756.Google Scholar
[4]Parry, W.. An ergodic theorem of information theory without invariant measure. Proc. London Math. Soc. 13(3) (1963), 605–612.CrossRefGoogle Scholar
[5]Parry, W.. Intrinsic Markov chains. Trans. Amer. Math. Soc. 112 (1964), 55–66.CrossRefGoogle Scholar
[6]Parry, W.. On Rohlin’s formula for entropy. Acta Math. Acad. Sci. Hungar. 15 (1964), 107–113.Google Scholar
[7]Parry, W.. Note on the ergodic theorem of Hurewicz. J. London Math. Soc. 39 (1964), 202–210.CrossRefGoogle Scholar
[8]Parry, W.. Representations for real numbers. Acta Math. Acad. Sci. Hungar. 15 (1964), 95–105.CrossRefGoogle Scholar
[9]Hahn, F. and Parry, W.. Minimal dynamical systems with quasi-discrete spectrum. J. London Math. Soc. 40 (1965), 309–323.CrossRefGoogle Scholar
[10]Parry, W.. Ergodic and spectral analysis of certain infinite measure preserving transformations. Proc. Amer. Math. Soc. 16 (1965), 960–966.Google Scholar
[11]Hoare, H. and Parry, W.. Affine transformations with quasi-discrete spectrum. I. J. London Math. Soc. 41 (1966), 88–96.CrossRefGoogle Scholar
[12]Parry, W.. Generators and strong generators in ergodic theory. Bull. Amer. Math. Soc. 72 (1966), 294–296.Google Scholar
[13]Hoare, H. and Parry, W.. Affine transformations with quasi-discrete spectrum. II. J. London Math. Soc. 41 (1966), 529–530.CrossRefGoogle Scholar
[14]Parry, W.. Symbolic dynamics and transformations of the unit interval. Trans. Amer. Math. Soc. 122 (1966), 368–378.CrossRefGoogle Scholar
[15]Parry, W.. On the coincidence of three invariant σ-algebras associated with an affine transformation. Proc. Amer. Math. Soc. 17 (1966), 1297–1302.Google Scholar
[16]Hoare, H. and Parry, W.. Semi-groups of affine transformations. Quart. J. Math. Oxford Ser. (2) 17 (1966), 106–111.CrossRefGoogle Scholar
[17]Newton, D. and Parry, W.. On a factor automorphism of a normal dynamical system. Ann. Math. Statist. 37 (1966), 1528–1533.Google Scholar
[18]Laxton, R. and Parry, W.. On the periodic points of certain automorphisms and a system of polynomial identities. J. Algebra 6 (1967), 388–393.Google Scholar
[19]Parry, W.. Principal partitions and generators. Bull. Amer. Math. Soc. 73 (1967), 307–309.CrossRefGoogle Scholar
[20]Parry, W.. Generators for perfect partitions. Dokl. Akad. Nauk SSSR 173 (1967), 264–266.Google Scholar
[21]Parry, W.. Aperiodic transformations and generators. J. London Math. Soc. 43 (1968), 191–194.CrossRefGoogle Scholar
[22]Hahn, F. and Parry, W.. Some characteristic properties of dynamical systems with quasi-discrete spectra. Math. Systems Theory 2 (1968), 179–190.Google Scholar
[23]Parry, W.. Zero entropy of distal and related transformations. Topological Dynamics (Symposium, Colorado State University, Fort Collins, CO, 1967). Eds. J. Auslander and W. H. Gottschalk. Benjamin, New York, 1968, pp. 383–389.Google Scholar
[24]Parry, W.. Ergodic properties of affine transformations and flows on nilmanifolds. Amer. J. Math. 91 (1969), 757–771.CrossRefGoogle Scholar
[25]Parry, W.. Compact abelian group extensions of discrete dynamical systems. Z. Wahrsch. Verw. Gebiete 13 (1969), 95–113.Google Scholar
[26]Parry, W.. Entropy and Generators in Ergodic Theory. W. A. Benjamin, New York, 1969.Google Scholar
[27]Parry, W.. Spectral analysis of G-extensions of dynamical systems. Topology 9 (1970), 217–224.Google Scholar
[28]Parry, W.. Dynamical systems on nilmanifolds. Bull. London Math. Soc. 2 (1970), 37–40.Google Scholar
[29]Parry, W. and Walters, P.. Minimal skew-product homeomorphisms and coalescence. Compos. Math. 22 (1970), 283–288.Google Scholar
[30]Parry, W.. Metric classification of ergodic nilflows and unipotent affines. Amer. J. Math. 93 (1971), 819–828.Google Scholar
[31]Parry, W.. Ergodic theory of G-spaces. Actes du Congrès International des Mathématiciens (Nice, 1970, Tome 2). Gauthier-Villars, Paris, 1971, pp. 921–924.Google Scholar
[32]Parry, W. and Walters, P.. Endomorphisms of a Lebesgue space. Bull. Amer. Math. Soc. 78 (1972), 272–276. Also: Erratum to ‘Endomorphisms of a Lebesgue space’. Bull. Amer. Math. Soc. 78 (1972), 628.CrossRefGoogle Scholar
[33]Azencott, R. and Parry, W.. Stability of group representations and Haar spectrum. Trans. Amer. Math. Soc. 172 (1972), 317–327.CrossRefGoogle Scholar
[34]Parry, W.. Cocycles and velocity changes. J. London Math. Soc. 5(2) (1972), 511–516.Google Scholar
[35]Jones, R. and Parry, W.. Compact abelian group extensions of dynamical systems. II. Compos. Math. 25 (1972), 135–147.Google Scholar
[36]Parry, W.. Dynamical representations in nilmanifolds. Compos. Math. 26 (1973), 159–174.Google Scholar
[37]Parry, W.. Notes on a posthumous paper by F. Hahn. Israel J. Math. 16 (1973), 38–45.Google Scholar
[38]Parry, W.. Class properties of dynamical systems. Recent Advances in Topological Dynamics (Proc. Conf., Yale University, New Haven, CT, 1972; in honor of Gustav Arnold Hedlund) (Lecture Notes in Mathematics, 318). Springer, Berlin, 1973, pp. 218–225.Google Scholar
[39]Parry, W.. A note on cocycles in ergodic theory. Compos. Math. 28 (1974), 343–350.Google Scholar
[40]Fellgett, R. and Parry, W.. Endomorphisms of a Lebesgue space. II. Bull. London Math. Soc. 7 (1975), 151–158.CrossRefGoogle Scholar
[41]Parry, W.. Endomorphisms of a Lebesgue space. III. Conference on Ergodic Theory and Topological Dynamics (Kibbutz Lavi, 1974). Israel J. Math. 21 (1975), 167–172.Google Scholar
[42]Parry, W. and Sullivan, D.. A topological invariant of flows on 1-dimensional spaces. Topology 14 (1975), 297–299.CrossRefGoogle Scholar
[43]Parry, W. and Schmidt, K.. A note on cocycles of unitary representations. Proc. Amer. Math. Soc. 55 (1976), 185–190.CrossRefGoogle Scholar
[44]Parry, W.. Some classification problems in ergodic theory. Sankhyā Ser. A 38 (1976), 38–43.Google Scholar
[45]Parry, W. and Williams, R.. Block coding and a zeta function for finite Markov chains. Proc. London Math. Soc. (3) 35 (1977), 483–495.Google Scholar
[46]Parry, W.. A finitary classification of topological Markov chains and sofic systems. Bull. London Math. Soc. 9 (1977), 86–92.CrossRefGoogle Scholar
[47]Parry, W.. The information cocycle and ε-bounded codes. Israel J. Math. 29 (1978), 205–220.CrossRefGoogle Scholar
[48]Palmer, M. R., Parry, W. and Walters, P.. Large sets of endomorphisms and of g-measures. The Structure of Attractors in Dynamical Systems (Proc. Conf., North Dakota State University, Fargo, ND, 1977) (Lecture Notes in Mathematics, 668). Eds. N. G. Markley, J. C. Martin and W. Perrizo. Springer, Berlin, 1978, pp. 191–210.Google Scholar
[49]Helson, H. and Parry, W.. Cocycles and spectra. Ark. Mat. 16(2) (1978), 195–206.CrossRefGoogle Scholar
[50]Parry, W.. An information obstruction to finite expected coding length. Ergodic Theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978) (Lecture Notes in Mathematics, 729). Springer, Berlin, 1979, pp. 163–168.Google Scholar
[51]Parry, W.. Finitary isomorphisms with finite expected code lengths. Bull. London Math. Soc. 11 (1979), 170–176.CrossRefGoogle Scholar
[52]Parry, W.. The Lorenz attractor and a related population model. Ergodic Theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978) (Lecture Notes in Mathematics, 729). Eds. M. Denker and K. Jacobs. Springer, Berlin, 1979, pp. 169–187.Google Scholar
[53]Parry, W.. Topics in Ergodic Theory (Cambridge Tracts in Mathematics, 75). Cambridge University Press, Cambridge, 1981.Google Scholar
[54]Parry, W.. Finitary isomorphisms with finite expected code-lengths. II. J. London Math. Soc. (2) 24 (1981), 569–576.Google Scholar
[55]Parry, W.. Self-generation of self-replicating maps of an interval. Ergod. Th. & Dynam. Sys. 1 (1981), 197–208.Google Scholar
[56]Parry, W. and Tuncel, S.. On the classification of Markov chains by finite equivalence. Ergod. Th. & Dynam. Sys. 1 (1981), 303–335.Google Scholar
[57]Parry, W.. The classification of topological Markov chains: adapted shift equivalence. Israel J. Math. 38 (1981), 335–344.CrossRefGoogle Scholar
[58]Parry, W. and Tuncel, S.. Classification Problems in Ergodic Theory (London Mathematical Society Lecture Note Series, 67). Cambridge University Press, Cambridge, 1982.Google Scholar
[59]Parry, W. and Tuncel, S.. On the stochastic and topological structure of Markov chains. Bull. London Math. Soc. 14 (1982), 16–27.Google Scholar
[60]Parry, W. and Tuncel, S.. Two classification problems for finite state Markov chains. Ergodic Theory and Related Topics (Proc. Conf., Vitte, 1981) (Mathematical Research, 12). Ed. H. Michel. Akademie-Verlag, Berlin, 1982, pp. 153–159.Google Scholar
[61]Parry, W.. An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions. Israel J. Math. 45 (1983), 41–52.Google Scholar
[62]Parry, W. and Pollicott, M.. An analogue of the prime number theorem for closed orbits of Axiom A flows. Ann. of Math. (2) 118 (1983), 573–591.Google Scholar
[63]Parry, W. and Schmidt, K.. Invariants of finitary isomorphisms with finite expected code-lengths. Conference in Modern Analysis and Probability (New Haven, CT, 1982) (Contemporary Mathematics, 26). Eds. R. Beals, A. Beck, A. Bellow and A. Hajian. American Mathematical Society, Providence, RI, 1984, pp. 301–307.Google Scholar
[64]Parry, W. and Schmidt, K.. Natural coefficients and invariants for Markov-shifts. Invent. Math. 76 (1984), 15–32.CrossRefGoogle Scholar
[65]Parry, W.. Bowen’s equidistribution theory and the Dirichlet density theorem. Ergod. Th. & Dynam. Sys. 4 (1984), 117–134.CrossRefGoogle Scholar
[67]Parry, W. and Pollicott, M.. The Chebotarov theorem for Galois coverings of Axiom A flows. Ergod. Th. & Dynam. Sys. 6 (1986), 133–148.CrossRefGoogle Scholar
[68]Parry, W.. Synchronisation of canonical measures for hyperbolic attractors. Comm. Math. Phys. 106 (1986), 267–275.CrossRefGoogle Scholar
[69]Alexander, J. C. and Parry, W.. Discerning fat baker’s transformations. Dynamical Systems (College Park, MD, 1986–87) (Lecture Notes in Mathematics, 1342). Springer, Berlin, 1988, pp. 1–6.Google Scholar
[70]Parry, W.. Equilibrium states and weighted uniform distribution of closed orbits. Dynamical Systems (College Park, MD, 1986–87) (Lecture Notes in Mathematics, 1342). Springer, Berlin, 1988,pp. 617–625.Google Scholar
[71]Parry, W.. Problems and perspectives in the theory of Markov shifts. Dynamical Systems (College Park, MD, 1986–87) (Lecture Notes in Mathematics, 1342). Springer, Berlin, 1988, pp. 626–637.Google Scholar
[72]Parry, W.. Decoding with two independent processes. Measure and Measurable Dynamics (Rochester, NY, 1987) (Contemporary Mathematics, 94). Eds. R. D. Mauldin, R. M. Shortt and C. E. Silva. American Mathematical Society, Providence, RI, 1989, pp. 207–209.CrossRefGoogle Scholar
[73]Parry, W.. Temporal and spatial distribution of closed orbits of hyperbolic dynamical systems. Measure and Measurable Dynamics (Rochester, NY, 1987) (Contemporary Mathematics, 94). Eds. R. D. Mauldin, R. M. Shortt and C. E. Silva. American Mathematical Society, Providence, RI, 1989, pp. 211–216.Google Scholar
[74]Coelho, Z. and Parry, W.. Central limit asymptotics for shifts of finite type. Israel J. Math. 69 (1990), 235–249.CrossRefGoogle Scholar
[75]Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990), 1–268.Google Scholar
[76]Parry, W.. Notes on coding problems for finite state processes. Bull. London Math. Soc. 23 (1991), 1–33.Google Scholar
[77]Parry, W.. A cocycle equation for shifts. Symbolic Dynamics and its Applications (New Haven, CT, 1991) (Contemporary Mathematics, 135). Ed. P. Walters. American Mathematical Society, Providence, RI, 1992, pp. 327–333.Google Scholar
[78]Parry, W.. In general a degree two map is an automorphism. Symbolic Dynamics and its Applications (New Haven, CT, 1991) (Contemporary Mathematics, 135). Ed. P. Walters. American Mathematical Society, Providence, RI, 1992, pp. 335–338.Google Scholar
[79]Noorani, M. S. M. and Parry, W.. A Chebotarev theorem for finite homogeneous extensions of shifts. Bol. Soc. Brasil. Mat. (N.S.) 23 (1992), 137–151.CrossRefGoogle Scholar
[80]Parry, W.. Remarks on Williams’ problem. Differential Equations, Dynamical Systems, and Control Science (Lecture Notes in Pure and Applied Mathematics, 152). Eds. K. D. Elworthy, W. N. Everitt and E. B. Lee. Dekker, New York, 1994, pp. 235–246.Google Scholar
[82]Parry, W.. Instances of cohomological triviality and rigidity. Ergod. Th. & Dynam. Sys. 15 (1995), 685–696.CrossRefGoogle Scholar
[83]Parry, W.. Ergodic properties of a one-parameter family of skew-products. Nonlinearity 8 (1995), 821–825.CrossRefGoogle Scholar
[84]Parry, W.. Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys. 16 (1996), 519–529.Google Scholar
[85] W. Parry. Squaring and cubing the circle–Rudolph’s theorem. Ergodic Theory of  Actions (Warwick, 1993–1994) (London Mathematical Society Lecture Note Series, 228). Eds. M. Pollicott and K. Schmidt. Cambridge University Press, Cambridge, 1996, pp. 177–183.Google Scholar
[86]Parry, W.. Cohomology of permutative cellular automata. Israel J. Math. 99 (1997), 315–333.Google Scholar
[87]Parry, W.. Skew products of shifts with a compact Lie group. J. London Math. Soc. (2) 56 (1997), 395–404.Google Scholar
[88]Parry, W. and Pollicott, M.. The Livšic cocycle equation for compact Lie group extensions of hyperbolic systems. J. London Math. Soc. (2) 56 (1997), 405–416.CrossRefGoogle Scholar
[89]Parry, W. and Pollicott, M.. Stability of mixing for toral extensions of hyperbolic systems. Proc. Steklov Inst. Math. 216 (1997), 350–359.Google Scholar
[90]Coelho, Z., Parry, W. and Williams, R.. A note on Livšic’s periodic point theorem. Topological Dynamics and Applications (Contemporary Mathematics, 215). American Mathematical Society, Providence, RI, 1998, pp. 223–230.CrossRefGoogle Scholar
[91]Coelho, Z. and Parry, W.. Shift endomorphisms and compact Lie extensions. Bol. Soc. Brasil. Mat. (N.S.) 29 (1998), 163–179.CrossRefGoogle Scholar
[93]Field, M. and Parry, W.. Stable ergodicity of skew extensions by compact Lie groups. Topology 38 (1999), 167–187.Google Scholar
[94]Parry, W.. The Livšic periodic point theorem for non-abelian cocycles. Ergod. Th. & Dynam. Sys. 19 (1999), 687–701.CrossRefGoogle Scholar
[95]Coelho, Z. and Parry, W.. Ergodicity of p-adic multiplications and the distribution of Fibonacci numbers. Topology, Ergodic Theory, Real Algebraic Geometry (American Mathematical Society Translation Series 2, 202). Eds. V. Turaev and A. Vershik. American Mathematical Society, Providence, RI, 2001, pp. 51–70.Google Scholar
[96]Parry, W. and Pollicott, M.. Skew products and Livˇsic theory. Representation Theory, Dynamical Systems, and Asymptotic Combinatorics (American Mathematical Society Translation Series 2, 217). Eds. V. Kaimanovich and A. Lodkin. American Mathematical Society, Providence, RI, 2006, pp. 139–165.Google Scholar
[97]Parry, W. and Pollicott, M.. An analogue of Bauer’s theorem for closed orbits of skew products. Ergod. Th. & Dynam. Sys. 28 (2008), 535–546.Google Scholar
[98]Hamdan, D., Parry, W. and Thouvenot, J.-P.. Shannon entropy for stationary processes and dynamical systems. Ergod. Th. & Dynam. Sys. 28 (2008), 447–480.CrossRefGoogle Scholar
[99]Parry, W.. An elementary construction of C r renormalizing maps, volume in honour of Zeeman’s 60th Birthday, unpublished.Google Scholar
[100]Ashley, J.. Bounded-to-1 factors of an aperiodic shift of finite type are 1-to-1 almost everywhere factors also. Ergod. Th. & Dynam. Sys. 10 (1990), 615–625.CrossRefGoogle Scholar
[101]Bowen, R.. The equidistribution of closed geodesics. Amer. J. Math. 94 (1972), 413–423.Google Scholar
[102]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.Google Scholar
[103]Bowen, R. and Franks, J.. Homology for zero-dimensional nonwandering sets. Ann. of Math. (2) 106 (1977), 73–92.Google Scholar
[104]Franks, J.. Flow equivalence of subshifts of finite type. Ergod. Th. & Dynam. Sys. 4 (1984), 53–66.CrossRefGoogle Scholar
[105]Friedman, N. A. and Ornstein, D. S.. On isomorphism of weak Bernoulli transformations. Adv. Math. 5 (1970), 365–394.CrossRefGoogle Scholar
[106]Kim, K. H. and Roush, F. W.. The Williams conjecture is false for irreducible subshifts. Ann. of Math. (2) 149 (1999), 545–558.Google Scholar
[107]Marcus, B. and Tuncel, S.. Matrices of polynomials, positivity, and finite equivalence of Markov chains. J. Amer. Math. Soc. 6 (1993), 131–147.Google Scholar
[108]Margulis, G. A.. On Some Aspects of the Theory of Anosov Systems (With a Survey by Richard Sharp: Periodic Orbits of Hyperbolic Flows). Springer, Berlin, 2004.CrossRefGoogle Scholar
[109]Morris, D. W.. Ratner’s Theorems on Unipotent Flows (Chicago Lectures in Mathematics). University of Chicago Press, Chicago, IL, 2005.Google Scholar
[110]Nogueira, A.. The three-dimensional Poincaré continued fraction algorithm. Israel J. Math. 90(1–3) (1995), 373–401.Google Scholar
[111]Ornstein, D.. Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4 (1970), 337–352.Google Scholar
[112]Rohlin, V. A.. Lectures on the entropy theory of transformations with invariant measure. Uspehi Mat. Nauk 22 (1967), 3–56 (in Russian).Google Scholar
[113]Ruelle, D.. A variational formulation of equilibrium statistical mechanics and the Gibbs phase rule. Comm. Math. Phys. 5 (1967), 324–329.CrossRefGoogle Scholar
[114]Schmidt, K.. Invariants for finitary isomorphisms with finite expected code lengths. Invent. Math. 76 (1984), 33–40.Google Scholar
[115]Schmidt, K.. Remarks on Livšic’ theory for nonabelian cocycles. Ergod. Th. & Dynam. Sys. 19(3) (1999), 703–721.CrossRefGoogle Scholar
[116]Schweiger, F.. Ergodic Theory of Fibred Systems and Metric Number Theory (Oxford Science Publications). The Clarendon Press, Oxford University Press, New York, 1995.Google Scholar
[117]Shub, M. and Sullivan, D.. Expanding endomorphisms of the circle revisited. Ergod. Th. & Dynam. Sys. 5 (1985), 285–289.Google Scholar
[118]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[119]Walters, P.. A variational principle for the pressure of continuous transformations. Amer. J. Math. 97 (1975), 937–971.CrossRefGoogle Scholar
[120]Williams, R.. Classification of subshifts of finite type. Ann. of Math. (2) 98 (1973), 120–153. Also: Erratum to ‘Classification of subshifts of finite type’. Ann. of Math. (2) 99 (1974), 380–381.Google Scholar
[121]Rényi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493.Google Scholar
[122]Rohlin, V. A.. Generators in ergodic theory II. Vestnik Leningrad. Univ. 20 (1965), 68–72.Google Scholar
[123]Adler, R. and Weiss, B.. Similarity of Automorphisms of the Torus (Memoirs of the American Mathematical Society, 98). American Mathematical Society, Providence, RI, 1970.Google Scholar
[124]Adler, R. and Marcus, B.. Topological Entropy and Equivalence of Dynamical Systems (Memoirs of the American Mathematical Society, 219). American Mathematical Society, Providence, RI, 1979.Google Scholar
[125]Keane, M. and Smorodinsky, M.. Bernoulli schemes of the same entropy are finitarily isomorphic. Ann. of Math. 109 (1979), 397–406.Google Scholar
[126]Tuncel, S.. Conditional pressure and coding. Israel J. Math. 39 (1981), 101–112.Google Scholar
[127]Krieger, W.. On the finitary isomorphisms of Markov shifts that have finite expected coding time. Z. Wahrsch. Verw. Gebiete 65 (1983), 323–328.Google Scholar
[128]Ruelle, D.. Zeta-functions for expanding maps and Anosov flows. Invent. Math. 34 (1976), 231–242.Google Scholar
[129]Livšic, A. N.. Certain properties of the homology of Y-systems. Math. Notes 10 (1971), 758–763.Google Scholar