Rotation, entropy, and equilibrium states

Transactions of the American Mathematical Society

Volumn 353, Issue 9, 2001, Pages 3713-3739

  Jenkinson, Oliver ^a,b  

^a AIX MARSEILLE UNIVERSITÉ   (France)

^b QUEEN MARY UNIVERSITY OF LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 23044527390     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-01-02706-4     Document Type: Article

References (59)

1. S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions 1. Exact results for the ground states, Physica 8D (1983), 381-422. MR 85f:58032
2. M. Babillot and F. Ledrappier, Lalley's theorem on periodic orbits of hyperbolic flows, Ergod. Th. Dyn. Sys., 18 (1998), 17-39. MR 99a:58128
3. V. Bangert, Mather sets for twist maps and geodesies on tori, Dynamics Reported 1 (1988), 1-45. MR 90a:58145
4. V. Bangert, Minimal geodesies, Ergod. Th. Dyn. Sys. 10 (1990), 263-286. MR 91j:58126
5. L. Barreira, Ya. Pesin, and J. Schmeling, On a general concept of multifractality: multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity, Chaos, 7:1 (1997), 27-38. MR 98g:58097
6. L. Barreira and J. Schmeling, Invariant sets with zero measure and full Hausdorff dimension, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 114-118 (electronic). MR 98f:58123
7. L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension, IST Preprint 14/97, 1997.
8. A. Blokh, Functional rotation numbers for one dimensional maps, Thins. Amer. Math. Soc., 347 (1995), 499-513. MR 95d:58044
9. T. Bousch, Le poisson n'a pas d'arêtes, Ann. I. H. P. (Prob.), 36 (2000), 489-508.
10. M. Bramson and S. Kalikow, Nonuniqueness in p-functions, Israel J. Math., 84 (1993), 153-160. MR94h:28011
11. Z. Coclho and A. Quas, Criteria for d-continuity, Trans. Amer. Math. Soc., 350 (1998), 3257-3268. MR 99d:28028
12. M. Denker, C. Grillenberger, and K. Sigmund, Ergodic Theory on Compact Spaces, Springer Lecture Notes in Mathematics, 527, 1976. MR 56:15879
13. T. Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel J. Math., 74 (1991), 241-256. MR 93e:54029
14. A. H. Fan and D. J. Feng, Analyse multifractale de la récurrence sur l'espace symbolique, C. R. Acad. Sei. Paris Sér. I Math., 327 (1998), 629-632. MR 2000f:28018
15. W. Geller and M. Misiurewicz, Rotation and entropy, Trans. Amer. Math. Soc., 351 (1999), 2927-2948. MR 99j:58125
16. E. Glasner and B. Weiss, Kazhdan's property T and the geometry of the collection of invariant measures, Geom. Fund. Anal., 7 (1997), 917-935. MR 99f:28029
17. C. Grillenberger, Construction of strictly ergodic systems I. Given entropy, Z. Wahr. venu. Geb., 25 (1973), 323-334. MR 49:5296
18. B. Grünbaum, Convex Polytopes, Pure and Applied Mathematics vol. XVI, Interscience, 1967. MR 37:2085
19. F. Hofbauer, Examples for the nonuniqueness of the equilibrium state, Trans. Amer. Math. Soc., 228 (1977) 223-241. MR 55:8312
20. O. Jenkinson, Conjugacy rigidity, cohomological triviality, and barycentres of invariant measures, Ph.D. thesis, Warwick University, 1996, http://www.maths.qau.ac.uk/~omj
21. O. Jenkinson, Geometric barycentres of invariant measures for circle maps, Ergod. Th. Dyn. Sys., 21 (2000), 511-532.
22. O. Jenkinson, Frequency-locking on the boundary of the barycentre set, Experimental Mathematics 9 (2000), 309-317. CMP 2000:17
23. G. Keller, Equilibrium states in ergodic theory, Cambridge University Press, 1998. MR 99e:28022
24. S. Kim, R. S. MacKay, J. Guckenheimer, Resonance regions for families of torus maps, Nonlinearity, 2 (1989), 391-404. MR 91d:58184
25. J. Kwapisz, Every convex polygon with rational vertices is a rotation set, Ergod. Th. Dyn. Sys. 12 (1992), 333-339. MR 93g:58082
26. J. Kwapisz, A toral diffeomorphism with a nonpolygonal rotation set, Nonlinearity 8 (1995), 461-476. MR 96j:58099
27. J. Lindenstrauss, G. H. Olsen, Y. Sternfeld, The Poulsen simplex, Ann. Inst. Fourier (Grenoble), 28 (1978), 91-114. MR 80b:46019a
28. D. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. MR 97a:58050
29. A. Livsic, Homology properties of Y-systems, Math. Zametki, 10 (1971), 758-763. MR 45:2746
30. G. McShane and I. Rivin, Simple curves on hyperbolic tori, C. R. Acad. Sei. Paris Sér. I Math., 320 (1995), 1523-1528. MR 96g:57018
31. G. McShane and I. Rivin, A norm on homology of surfaces and counting simple geodesies, Internat. Math. Res. Notices, (1995), 61-69 (electronic). MR 96b:57014
32. R. Mañé, On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, 5 (1992), 623-638. MR 93h:58059
33. R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, 9 (1996), 273-310. MR 97d:58118
34. B. Marcus and S. Tuncel, The weight-per-symbol polytope and scaffolds of invariants associated with Markov chains, Ergod. Th. Dyn. Sys., 11 (1991), 129-180. MR 92g:28038
35. D. Massart, Normes stables des surfaces, Ph.D. Thesis, ENS Lyon, 1996.
36. D. Massart, Stable norms of surfaces: local structure of the unit ball of rational directions, Geom. Fund. Anal. 7 (1997), 996-1010. MR 99b:53061
37. J. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology 21 (1982), 457-467. MR 84g:58084
38. J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. MR 92m:58048
39. S. New-house, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235.
40. S. Newhouse, J. Palis, F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Haut. Études Sei. Publ. Math. 57 (1983), 5-71.
41. W. Parry, Intrinsic Markov Chains, Trans. Amer. Math. Soc., 112 (1964), 55-65. MR 28:4579
42. W. Parry and M. Pollicott, Zeta functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Astérisque, 187-188, 1990. MR 92f:58141
43. W. Parry and S. Tuncel, On the classification of Markov chains by finite equivalence, Ergod. Th. Dyn. Sys., 1 (1981), 303-335. MR 83j:28020
44. Ya. Pesin, Dimension Theory in Dynamical Systems, University of Chicago Press, 1997. MR 99b:58003
45. Ya. Pesin and B. Pitskel, Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl. 18 (1984), 307-318. MR 86i:28031
46. H. Poincaré, Sur les courbes définies par les équations différentielles, œuvres Complètes, tome 1, Gauthier-Villars, Paris, 1952, 137-158.
47. R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970. MR 97m:49001
48. D. Ruelle, Statistical mechanics on a compact set with Z" action satisfying expansiveness and spécification, Trans. Amer. Math. Soc., 187 (1973), 237-253. MR 54:5441
49. D. Ruelle, Thermodynamic Formalism, Encyclopaedia of Mathematics and its Applications, vol. 5, Addison-Wesley, 1978. MR 80g:82017
50. S. Schwartzman, Asymptotic cycles, Ann. of Math. (2) 66 (1957), 270-284. MR 19:5681
51. Y. G. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surv., 27 (1972), 21-70. MR 85f:58071
52. F. Takens and E. Verbitski, On the variational principle for the topological entropy of certain non-compact sets, preprint, 1999, University of Groningen.
53. P. Walters, Ruelle's operator theorem and p-measures, Trans. Amer. Math. Soc., 214 (1975), 375-387. MR 54:515
54. P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1976), 937-971. MR 52:11006
55. P. Walters, An Introduction to Ergodic Theory, Springer-Verlag 1982. MR 84c:28017
56. P. Walters, Differentiability properties of the pressure of a continuous transformation on a compact metric space, J. London Math. Soc. (2), 46 (1992), 471-481. MR 94a:28038
57. F. Warner, Foundations of differentiate manifolds and Lie groups, Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983. MR 84k:58001
58. S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Z. Wahr. verw. Geb., 67 (1984), 95-107. MR 86k:54062
59. K. Ziemian, Rotation sets for subshifts of finite type, Fund. Math., 146 (1995), 189-201. MR 96b:58072