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.

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.

[1]

Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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.

[8]

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

[9]

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

[10]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2891-2905. doi: 10.3934/dcds.2020390

[11]

Dubi Kelmer, Hee Oh. Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds. Journal of Modern Dynamics, 2021, 17: 401-434. doi: 10.3934/jmd.2021014

[12]

Gabriela P. Ovando. The geodesic flow on nilpotent Lie groups of steps two and three. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 327-352. doi: 10.3934/dcds.2021119

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

Theresa Lange, Wilhelm Stannat. Mean field limit of Ensemble Square Root filters - discrete and continuous time. Foundations of Data Science, 2021, 3 (3) : 563-588. doi: 10.3934/fods.2021003

2020 Impact Factor: 0.848

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]