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

Weak mixing suspension flows over shifts of finite type are universal

Anthony Quas 1, and  Terry Soo 2,

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.
Keywords: square root problem, weak topological mixing., Embedding, suspension flow, universality, geodesic flow.
Mathematics Subject Classification: Primary: 37A35; Secondary: 37D4.
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, 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

show all references

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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