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Weak mixing suspension flows over shifts of finite type are universal

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  • Let $S$ be an ergodic measure-preserving automorphism on a nonatomic probability space, and let $T$ be the time-one map of a topologically weak mixing suspension flow over an irreducible subshift of finite type under a Hölder ceiling function. We show that if the measure-theoretic entropy of $S$ is strictly less than the topological entropy of $T$, then there exists an embedding of the measure-preserving automorphism into the suspension flow. As a corollary of this result and the symbolic dynamics for geodesic flows on compact surfaces of negative curvature developed by Bowen [5] and Ratner [31], we also obtain an embedding of the measure-preserving automorphism into a geodesic flow whenever the measure-theoretic entropy of $S$ is strictly less than the topological entropy of the time-one map of the geodesic flow.
    Mathematics Subject Classification: Primary: 37A35; Secondary: 37D40.

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  • [1]

    S. Alpern, Generic properties of measure preserving homeomorphisms, In "Ergodic Theory" (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978), Lecture Notes in Math., 729, Springer, Berlin, (1979), 16-27.

    [2]

    A. Bellow and H. Furstenberg, An application of number theory to ergodic theory and the construction of uniquely ergodic models. A collection of invited papers on ergodic theory, Israel J. Math., 33 (1979), 231-240 (1980).doi: 10.1007/BF02762163.

    [3]

    R. Bowen, The equidistribution of closed geodesics, Amer. J. Math., 94 (1972), 413-423.

    [4]

    R. Bowen, One-dimensional hyperbolic sets for flows, J. Differential Equations, 12 (1972), 173-179.

    [5]

    R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460.

    [6]

    R. Bowen and B. Marcus, Unique ergodicity for horocycle foliations, Israel J. Math., 26 (1977), 43-67.

    [7]

    R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.

    [8]

    R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.

    [9]

    M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Spaces," Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976.

    [10]

    S. J. Eigen and V. S. Prasad, Multiple Rokhlin tower theorem: A simple proof, New York J. Math., 3A (1997/98), Proceedings of the New York Journal of Mathematics Conference, June 9-13, (1997), 11-14 (electronic).

    [11]

    N. A. Friedman, "Introduction to Ergodic Theory," Van Nostrand Reinhold Mathematical Studies, No. 29, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1970.

    [12]

    T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc., 3 (1971), 176-180.

    [13]

    R. I. JewettThe prevalence of uniquely ergodic systems, J. Math. Mech., 19 (1969/1970), 717-729.

    [14]

    M. Keane and M. Smorodinsky, A class of finitary codes, Israel J. Math., 26 (1977), 352-371.

    [15]

    M. Keane and M. Smorodinsky, Bernoulli schemes of the same entropy are finitarily isomorphic, Ann. of Math. (2), 109 (1979), 397-406.doi: 10.2307/1971117.

    [16]

    H. B. Keynes and J. B. Robertson, Eigenvalue theorems in topological transformation groups, Trans. Amer. Math. Soc., 139 (1969), 359-369.

    [17]

    W. Krieger, On entropy and generators of measure-preserving transformations, Trans. Amer. Math. Soc., 149 (1970), 453-464.

    [18]

    W. Krieger, On entropy and generators of measure-preserving transformations, Trans. Amer. Math. Soc., 149 (1970), 453-464.

    [19]

    W. Krieger, Erratum to: "On entropy and generators of measure-preserving transformations," Trans. Amer. Math. Soc., 168 (1972), 519.

    [20]

    W. Krieger, On unique ergodicity, in "Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability" (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability Theory, Univ. California Press, Berkeley, Calif., (1972), 327-346.

    [21]

    W. Krieger, On generators in ergodic theory, in "Proceedings of the International Congress of Mathematicians" (Vancouver, B. C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., (1975), 303-308.

    [22]
    [23]

    D. Lind, Ergodic group automorphisms and specification, in "Ergodic Theory" (eds. M. Denker and K. Jacobs) (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978), Lecture Notes in Mathematics, 729, Springer, Berlin, (1979), 93-104.doi: 10.1007/BFb0063287.

    [24]

    D. Lind, Dynamical properties of quasihyperbolic toral automorphisms, Ergodic Theory Dynam. Systems, 2 (1982), 49-68.

    [25]

    D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Coding," Cambridge University Press, Cambridge, 1995.doi: 10.1017/CBO9780511626302.

    [26]

    D. A. Lind, Perturbations of shifts of finite type, SIAM J. Discrete Math., 2 (1989), 350-365.doi: 10.1137/0402031.

    [27]

    D. A. Lind and J.-P. ThouvenotMeasure-preserving homeomorphisms of the torus represent all finite entropy ergodic transformations, Math. Systems Theory, 11 (1977/78), 275-282.

    [28]

    D. S. Ornstein, A $K$ automorphism with no square root and Pinsker's conjecture, Advances in Math., 10 (1973), 89-102.

    [29]

    W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188 (1990), 268 pp.

    [30]

    A. Quas and T. Soo, Ergodic universality of some topological dynamical systems, arXiv:1208.3501, 2012.

    [31]

    M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds, Israel J. Math., 15 (1973), 92-114.

    [32]

    J. Serafin, Finitary codes, a short survey, in "Dynamics & Stochastics," IMS Lecture Notes Monogr. Ser., 48, Inst. Math. Statist., Beachwood, OH, (2006), 262-273.doi: 10.1214/lnms/1196285827.

    [33]
    [34]

    H. Totoki, On a class of special flows, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 15 (1970), 157-167.

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