Weak mixing suspension flows over shifts of finite type are universal

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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

Abstract
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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.

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 "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.

show all references

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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