Walters’ regularity condition. (La condition de Walters.) (French) Zbl 0988.37036
In the study of invariant measures and equilibrium states for some mappings which expand distances [Trans. Am. Math. Soc. 236, 127-153 (1978; Zbl 0375.28009)] P. Walters introduced a regularity condition for the needs of the thermodynamic formalism. In particular that refers to the topological conjugacy of dynamical systems and links with Bernoulli shifts.
By generalizing observations due to R. Mañé [Nonlinearity 9, 273-310 (1996; Zbl 0886.58037)] on minimizing measures of Lagrangian systems to the hyperbolic case, the Walter condition is shown to be necessary in the study of maximizing measures, and in the search of normal forms modulo coboundaries of continuous functions.
By generalizing observations due to R. Mañé [Nonlinearity 9, 273-310 (1996; Zbl 0886.58037)] on minimizing measures of Lagrangian systems to the hyperbolic case, the Walter condition is shown to be necessary in the study of maximizing measures, and in the search of normal forms modulo coboundaries of continuous functions.
Reviewer: Piotr Garbaczewski (Zielona Gora)
MSC:
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
| 37A60 | Dynamical aspects of statistical mechanics |
Keywords:
invariant measures; equilibrium states; thermodynamic formalism; Walter condition; maximizing measures; normal formsReferences:
| [1] | Aliprantis C.D , Border K.C , Infinite-Dimensional Analysis: A Hitchhiker’s Guide , Springer-Verlag , 1999 . MR 1717083 | Zbl 0839.46001 · Zbl 0839.46001 |
| [2] | Bousch T , Le poisson n’a pas d’arêtes , Preprint, Université d’Orsay , 1999 , [à paraître dans Ann. Inst. Henri-Poincaré, Probab.-Statis.]. Numdam |
| [3] | Carlson D.A , Haurie A.B , Leizarowitz A , Infinite Horizon Optimal Control: Deterministic and Stochastic Systems , Springer-Verlag , 1991 . Zbl 0758.49001 · Zbl 0758.49001 |
| [4] | Coelho Z , Entropy and ergodicity of skew-products over subshifts of finite type and central limit asymptotics , Thèse, Université de Warwick , 1990 . |
| [5] | Coelho Z , Quas A.N , Criteria for \(\overline d\)-continuity , Trans. Amer. Math. Soc. 350 ( 1998 ). Zbl 0907.28013 · Zbl 0907.28013 · doi:10.1090/S0002-9947-98-01923-0 |
| [6] | Contreras G , Lopes A , Thieullen P , Lyapunov minimizing measures for expanding maps of the circle , Manuscrit , 1999 . · Zbl 0997.37016 |
| [7] | Fathi A , Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens , C. R. Acad. Sci. Paris, Série I 324 ( 1997 ). MR 1451248 | Zbl 0885.58022 · Zbl 0885.58022 · doi:10.1016/S0764-4442(97)87883-4 |
| [8] | Hunt B.R , Ott E , Optimal periodic orbits of chaotic systems occur at low period , Phys. Rev. E 54 ( 1996 ) 328 . |
| [9] | Jenkinson O.M , Conjugacy rigidity, cohomological triviality, and barycentres of invariant measures , Thèse, Université de Warwick , 1996 . |
| [10] | Jenkinson O.M , Frequency-locking on the boundary of the barycentre set , Experimental Mathematics 9 ( 2000 ). Article | MR 1780215 | Zbl 1106.37303 · Zbl 1106.37303 · doi:10.1080/10586458.2000.10504655 |
| [11] | Kondah A , Maume V , Schmitt B , Vitesse de convergence vers l’état d’équilibre pour des dynamiques markoviennes non höldériennes , Ann. Inst. Henri-Poincaré, Probab.-Statis. 33 ( 1997 ). Numdam | MR 1484537 | Zbl 0913.60046 · Zbl 0913.60046 · doi:10.1016/S0246-0203(97)80109-4 |
| [12] | Livšic A.N , Homology properties of Y-systems , Math. Zametki 10 ( 1971 ), [Traduction anglaise: Math. Notes 10 (1971)]. MR 293669 | Zbl 0235.58010 · Zbl 0235.58010 · doi:10.1007/BF01109040 |
| [13] | Mañé R , Generic properties and problems of minimizing measures of Lagrangian systems , Nonlinearity 9 ( 1996 ). MR 1384478 | Zbl 0886.58037 · Zbl 0886.58037 · doi:10.1088/0951-7715/9/2/002 |
| [14] | Rudin W , Real and Complex Analysis , McGraw-Hill , 1987 . MR 924157 | Zbl 0925.00005 · Zbl 0925.00005 |
| [15] | Sinaĭ Y.G , Gibbs measures in ergodic theory , Uspekhi Math. Nauk 27 ( 1972 ), [Traduction anglaise: Russian Math. Surveys 27 (1972)]. MR 399421 | Zbl 0255.28016 · Zbl 0255.28016 · doi:10.1070/rm1972v027n04ABEH001383 |
| [16] | Walters P , Invariant measures and equilibrium states for some mappings which expand distances , Trans. Amer. Math. Soc. 236 ( 1978 ). MR 466493 | Zbl 0375.28009 · Zbl 0375.28009 · doi:10.2307/1997777 |
| [17] | Yuan G , Hunt B.R , Optimal orbits of hyperbolic systems , Nonlinearity 12 ( 1999 ). MR 1709845 | Zbl 0951.37006 · Zbl 0951.37006 · doi:10.1088/0951-7715/12/4/325 |
| [18] | Ziemian K , Rotation sets for subshifts of finite type , Fund. Math. 146 ( 1995 ). MR 1314983 | Zbl 0821.58017 · Zbl 0821.58017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
