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, 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

[1]

Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643

[2]

Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365

[3]

Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581

[4]

Zhenqi Jenny Wang. The twisted cohomological equation over the geodesic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3923-3940. doi: 10.3934/dcds.2019158

[5]

Anke D. Pohl. Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2173-2241. doi: 10.3934/dcds.2014.34.2173

[6]

Vladimir S. Matveev and Petar J. Topalov. Metric with ergodic geodesic flow is completely determined by unparameterized geodesics. Electronic Research Announcements, 2000, 6: 98-104.

[7]

Bendong Lou. Spiral rotating waves of a geodesic curvature flow on the unit sphere. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 933-942. doi: 10.3934/dcdsb.2012.17.933

[8]

Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148

[9]

César J. Niche. Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 577-580. doi: 10.3934/dcds.2004.11.577

[10]

Vincenzo Ambrosio, Giovanni Molica Bisci, Dušan Repovš. Nonlinear equations involving the square root of the Laplacian. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 151-170. doi: 10.3934/dcdss.2019011

[11]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115

[12]

R.L. Sheu, M.J. Ting, I.L. Wang. Maximum flow problem in the distribution network. Journal of Industrial & Management Optimization, 2006, 2 (3) : 237-254. doi: 10.3934/jimo.2006.2.237

[13]

Daniel J. Thompson. A criterion for topological entropy to decrease under normalised Ricci flow. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1243-1248. doi: 10.3934/dcds.2011.30.1243

[14]

Partha Sharathi Dutta, Soumitro Banerjee. Period increment cascades in a discontinuous map with square-root singularity. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 961-976. doi: 10.3934/dcdsb.2010.14.961

[15]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547

[16]

Laurence Guillot, Maïtine Bergounioux. Existence and uniqueness results for the gradient vector flow and geodesic active contours mixed model. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1333-1349. doi: 10.3934/cpaa.2009.8.1333

[17]

Xian-Gao Liu, Jie Qing. Globally weak solutions to the flow of compressible liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 757-788. doi: 10.3934/dcds.2013.33.757

[18]

Roman M. Taranets, Jeffrey T. Wong. Existence of weak solutions for particle-laden flow with surface tension. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4979-4996. doi: 10.3934/dcds.2018217

[19]

Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357

[20]

Yinnian He, Yi Li. Asymptotic behavior of linearized viscoelastic flow problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 843-856. doi: 10.3934/dcdsb.2008.10.843

2018 Impact Factor: 0.295

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]