# American Institute of Mathematical Sciences

October  2012, 6(4): 427-449. doi: 10.3934/jmd.2012.6.427

## Weak mixing suspension flows over shifts of finite type are universal

 1 Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, B.C., V8W 3R4 2 Department of Mathematics and Statistics, University of Victoria, PO BOX 3060 STN CSC, Victoria, BC V8W 3R4, Canada

Received  October 2011 Revised  July 2012 Published  January 2013

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.
Citation: Anthony Quas, Terry Soo. Weak mixing suspension flows over shifts of finite type are universal. Journal of Modern Dynamics, 2012, 6 (4) : 427-449. doi: 10.3934/jmd.2012.6.427
##### References:
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##### References:
 [1] S. Alpern, Generic properties of measure preserving homeomorphisms,, In, 729 (1979), 16. Google Scholar [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. doi: 10.1007/BF02762163. Google Scholar [3] R. Bowen, The equidistribution of closed geodesics,, Amer. J. Math., 94 (1972), 413. Google Scholar [4] R. Bowen, One-dimensional hyperbolic sets for flows,, J. Differential Equations, 12 (1972), 173. Google Scholar [5] R. Bowen, Symbolic dynamics for hyperbolic flows,, Amer. J. Math., 95 (1973), 429. Google Scholar [6] R. Bowen and B. Marcus, Unique ergodicity for horocycle foliations,, Israel J. Math., 26 (1977), 43. Google Scholar [7] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows,, Invent. Math., 29 (1975), 181. Google Scholar [8] R. Bowen and P. Walters, Expansive one-parameter flows,, J. Differential Equations, 12 (1972), 180. Google Scholar [9] M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Spaces,", Lecture Notes in Mathematics, (1976). Google Scholar [10] S. J. Eigen and V. S. Prasad, Multiple Rokhlin tower theorem: A simple proof,, New York J. Math., (1997), 9. Google Scholar [11] N. A. Friedman, "Introduction to Ergodic Theory,", Van Nostrand Reinhold Mathematical Studies, (1970). Google Scholar [12] T. N. T. Goodman, Relating topological entropy and measure entropy,, Bull. London Math. Soc., 3 (1971), 176. Google Scholar [13] R. I. Jewett, The prevalence of uniquely ergodic systems,, J. Math. Mech., 19 (): 717. Google Scholar [14] M. Keane and M. Smorodinsky, A class of finitary codes,, Israel J. Math., 26 (1977), 352. Google Scholar [15] M. Keane and M. Smorodinsky, Bernoulli schemes of the same entropy are finitarily isomorphic,, Ann. of Math. (2), 109 (1979), 397. doi: 10.2307/1971117. Google Scholar [16] H. B. Keynes and J. B. Robertson, Eigenvalue theorems in topological transformation groups,, Trans. Amer. Math. Soc., 139 (1969), 359. Google Scholar [17] W. Krieger, On entropy and generators of measure-preserving transformations,, Trans. Amer. Math. Soc., 149 (1970), 453. Google Scholar [18] W. Krieger, On entropy and generators of measure-preserving transformations,, Trans. Amer. Math. Soc., 149 (1970), 453. Google Scholar [19] W. Krieger, Erratum to: "On entropy and generators of measure-preserving transformations,", Trans. Amer. Math. Soc., 168 (1972). Google Scholar [20] W. Krieger, On unique ergodicity,, in, (1972), 327. Google Scholar [21] W. Krieger, On generators in ergodic theory,, in, (1975), 303. Google Scholar [22] , F. Ledrappier, F. Rodriguez Hertz and J. Rodriguez Hertz,, personal communication., (). Google Scholar [23] D. Lind, Ergodic group automorphisms and specification,, in, 729 (1979), 93. doi: 10.1007/BFb0063287. Google Scholar [24] D. Lind, Dynamical properties of quasihyperbolic toral automorphisms,, Ergodic Theory Dynam. Systems, 2 (1982), 49. Google Scholar [25] D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Coding,", Cambridge University Press, (1995). doi: 10.1017/CBO9780511626302. Google Scholar [26] D. A. Lind, Perturbations of shifts of finite type,, SIAM J. Discrete Math., 2 (1989), 350. doi: 10.1137/0402031. Google Scholar [27] D. A. Lind and J.-P. Thouvenot, Measure-preserving homeomorphisms of the torus represent all finite entropy ergodic transformations,, Math. Systems Theory, 11 (): 275. Google Scholar [28] D. S. Ornstein, A $K$ automorphism with no square root and Pinsker's conjecture,, Advances in Math., 10 (1973), 89. Google Scholar [29] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics,, Astérisque, 187-188 (1990), 187. Google Scholar [30] A. Quas and T. Soo, Ergodic universality of some topological dynamical systems,, \arXiv{1208.3501}, (2012). Google Scholar [31] M. Ratner, Markov partitions for Anosov flows on $n$-dimensional manifolds,, Israel J. Math., 15 (1973), 92. Google Scholar [32] J. Serafin, Finitary codes, a short survey,, in, 48 (2006), 262. doi: 10.1214/lnms/1196285827. Google Scholar [33] , J.-P. Thouvenot,, personal communication., (). Google Scholar [34] H. Totoki, On a class of special flows,, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 15 (1970), 157. Google Scholar
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