# 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:
 [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. Jewett, The prevalence of uniquely ergodic systems,, J. Math. Mech., 19 (): 717. [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] , F. Ledrappier, F. Rodriguez Hertz and J. Rodriguez Hertz,, personal communication., (). [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. Thouvenot, Measure-preserving homeomorphisms of the torus represent all finite entropy ergodic transformations,, Math. Systems Theory, 11 (): 275. [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] , J.-P. Thouvenot,, personal communication., (). [34] H. Totoki, On a class of special flows, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 15 (1970), 157-167.

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##### References:
 [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. Jewett, The prevalence of uniquely ergodic systems,, J. Math. Mech., 19 (): 717. [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] , F. Ledrappier, F. Rodriguez Hertz and J. Rodriguez Hertz,, personal communication., (). [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. Thouvenot, Measure-preserving homeomorphisms of the torus represent all finite entropy ergodic transformations,, Math. Systems Theory, 11 (): 275. [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] , J.-P. Thouvenot,, personal communication., (). [34] H. Totoki, On a class of special flows, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 15 (1970), 157-167.
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