Ergodic measures of intermediate entropies for dynamical systems with the approximate product property.(English)Zbl 07989620
Summary: For a dynamical system satisfying the approximate product property and asymptotically entropy expansiveness, we characterize a delicate structure of the space of invariant measures: The ergodic measures of intermediate entropies and the ones of intermediate pressures are generic in certain subspaces. Consequently, the conjecture of Katok that ergodic measures of arbitrary intermediate entropy exist is verified for a broad class of systems.
MSC:
| 37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |
| 37B65 | Approximate trajectories, pseudotrajectories, shadowing and related notions for topological dynamical systems |
| 37B40 | Topological entropy |
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
| 37C50 | Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics |
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
Keywords:
intermediate entropy;entropy dense;approximate product property;asymptotically entropy expansive;specificationCite
References:
| [1] | Béguin, F.; Crovisier, S.; Le Roux, F., Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique, Ann. Sci. Éc. Norm. Supér. (4), 40, 251-308, 2007 ·Zbl 1132.37003 |
| [2] | Bessa, M.; Torres, M. J.; Varandas, P., On the periodic orbits, shadowing and strong transitivity of continuous flows, Nonlinear Anal., 175, 191-209, 2018 ·Zbl 1404.53111 |
| [3] | Bomfim, T.; Torres, M. J.; Varandas, P., Topological features of flows with the reparametrized gluing orbit property, J. Differ. Equ., 262, 8, 4292-4313, 2017 ·Zbl 1381.37024 |
| [4] | Bomfim, T.; Varandas, P., The gluing orbit property, uniform hyperbolicity and large deviations principles for semiflows, J. Differ. Equ., 267, 1, 228-266, 2019 ·Zbl 1412.37023 |
| [5] | Bowen, R., Periodic points and measures for Axiom A diffeomorphisms, Transl. Am. Math. Soc., 154, 377-397, 1971 ·Zbl 0212.29103 |
| [6] | Bowen, R., Entropy-expansive maps, Transl. Am. Math. Soc., 164, 323-331, 1972 ·Zbl 0229.28011 |
| [7] | Burguet, D., Topological and almost Borel universality for systems with the weak specification property, Ergod. Theory Dyn. Syst., 40, 8, 2098-2115, 2020 ·Zbl 1447.37018 |
| [8] | Buzzi, J., Intrinsic ergodicity of smooth interval maps, Isr. J. Math., 100, 125-161, 1997 ·Zbl 0889.28009 |
| [9] | Chandgotia, N.; Meyerovitch, T., Borel subsystems and ergodic universality for compact \(\mathbb{Z}^d\)-systems via specification and beyond, Proc. Lond. Math. Soc., 123, 231-312, 2021 ·Zbl 1506.37011 |
| [10] | Climenhaga, V.; Thompson, D. J., Unique equilibrium states for flows and homeomorphisms with non-uniform structure, Adv. Math., 303, 745-799, 2016 ·Zbl 1366.37084 |
| [11] | Constantine, D.; Lafont, J.; Thompson, D. J., The weak specification property for geodesic flows on CAT(-1) spaces, Groups Geom. Dyn., 14, 1, 297-336, 2020 ·Zbl 1457.37049 |
| [12] | Cuneo, N., Additive, almost additive and asymptotically additive potential sequences are equivalent, Commun. Math. Phys., 377, 2579-2595, 2020 ·Zbl 1447.37045 |
| [13] | Dateyama, M., The almost weak specification property for ergodic group automorphisms of abelian groups, J. Math. Soc. Jpn., 42, 2, 341-351, 1990 ·Zbl 0722.28014 |
| [14] | Denker, M.; Grillenberger, C.; Sigmund, K., Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, vol. 527, 1976, Springer-Verlag: Springer-Verlag Berlin-New York ·Zbl 0328.28008 |
| [15] | Feng, D.; Huang, W., Lyapunov spectrum of asymptotically sub-additive potentials, Commun. Math. Phys., 297, 1, 1-43, 2010 ·Zbl 1210.37007 |
| [16] | Gelfert, K.; Kwietniak, D., On density of ergodic measures and generic points, Ergod. Theory Dyn. Syst., 38, 1745-1767, 2018 ·Zbl 1403.37021 |
| [17] | Glasner, E.; Weiss, B., Strictly ergodic, uniform positive entropy models, Bull. Soc. Math. Fr., 122, 3, 399-412, 1994 ·Zbl 0833.54022 |
| [18] | Guan, L.; Sun, P.; Wu, W., Measures of intermediate entropies and homogeneous dynamics, Nonlinearity, 30, 3349-3361, 2017 ·Zbl 1388.37004 |
| [19] | Hahn, F.; Katznelson, Y., On the entropy of uniquely ergodic transformations, Transl. Am. Math. Soc., 126, 335-360, 1967 ·Zbl 0191.21502 |
| [20] | Huang, W.; Xu, L.; Xu, S., Ergodic measures of intermediate entropy for affine transformations of nilmanifolds, Electron. Res. Arch., 29, 4, 2819-2827, 2021 ·Zbl 1477.37012 |
| [21] | Katok, A., Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. IHES, 51, 137-173, 1980 ·Zbl 0445.58015 |
| [22] | Konieczny, J.; Kupsa, M.; Kwietniak, D., Arcwise connectedness of the set of ergodic measures of hereditary shifts, Proc. Am. Math. Soc., 146, 8, 3425-3438, 2018 ·Zbl 1390.37006 |
| [23] | Kwietniak, D., Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts, Discrete Contin. Dyn. Syst., Ser. A, 33, 6, 2451-2467, 2013 ·Zbl 1271.37013 |
| [24] | Kwietniak, D.; Lacka, M.; Oprocha, P., A panorama of specification-like properties and their consequences, Contemp. Math., 669, 155-186, 2016 ·Zbl 1376.37024 |
| [25] | Kwietniak, D.; Oprocha, P.; Rams, M., On entropy of dynamical systems with almost specification, Isr. J. Math., 213, 1, 475-503, 2016 ·Zbl 1365.37011 |
| [26] | Li, J.; Oprocha, P., Properties of invariant measures in dynamical systems with the shadowing property, Ergod. Theory Dyn. Syst., 38, 2257-2294, 2018 ·Zbl 1397.37093 |
| [27] | Li, M.; Shi, Y.; Wang, S.; Wang, X., Measures of intermediate entropies for star vector fields, Isr. J. Math., 240, 791-819, 2020 ·Zbl 1483.37029 |
| [28] | Liao, G.; Viana, M.; Yang, J., The entropy conjecture for diffeomorphisms away from tangencies, J. Eur. Math. Soc., 15, 6, 2043-2060, 2013 ·Zbl 1325.37031 |
| [29] | Lindenstrauss, J.; Olsen, G.; Sternfeld, Y., The Poulsen simplex, Ann. Inst. Fourier (Grenoble), 28, 1, 1978, vi, 91-114 ·Zbl 0363.46006 |
| [30] | Marcus, B., A note on periodic points for ergodic toral automorphisms, Monatshefte Math., 89, 121-129, 1980 ·Zbl 0422.28013 |
| [31] | Misiurewicz, M., Topological conditional entropy, Stud. Math., 55, 175-200, 1976 ·Zbl 0355.54035 |
| [32] | Pavlov, R., On intrinsic ergodicity and weakenings of the specification property, Adv. Math., 295, 250-270, 2016 ·Zbl 1358.37032 |
| [33] | Pfister, C.-E.; Sullivan, W. G., Large deviations estimates for dynamical systems without the specification property. Application to the β-shifts, Nonlinearity, 18, 237-261, 2005 ·Zbl 1069.60029 |
| [34] | Phelps, R. R., Lectures on Choquet’s Theorem, Lecture Notes in Mathematics, vol. 1757, 2001, Springer-Verlag: Springer-Verlag Berlin ·Zbl 0997.46005 |
| [35] | Quas, A.; Soo, T., Ergodic universality of some topological dynamical systems, Trans. Am. Math. Soc., 368, 6, 4137-4170, 2016 ·Zbl 1354.37010 |
| [36] | Sigmund, K., Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math., 11, 99-109, 1970 ·Zbl 0193.35502 |
| [37] | Sun, P., Zero-entropy invariant measures for skew product diffeomorphisms, Ergod. Theory Dyn. Syst., 30, 923-930, 2010 ·Zbl 1202.37010 |
| [38] | Sun, P., Measures of intermediate entropies for skew product diffeomorphisms, Discrete Contin. Dyn. Syst., Ser. A, 27, 3, 1219-1231, 2010 ·Zbl 1192.37036 |
| [39] | Sun, P., Density of metric entropies for linear toral automorphisms, Dyn. Syst., 27, 2, 197-204, 2012 ·Zbl 1244.37007 |
| [40] | Sun, P., Minimality and gluing orbit property, Discrete Contin. Dyn. Syst., Ser. A, 39, 7, 4041-4056, 2019 ·Zbl 1429.37008 |
| [41] | Sun, P., Denseness of intermediate pressures for systems with the Climenhaga-Thompson structures, J. Math. Anal. Appl., 487, 2, Article 124027 pp., 2020 ·Zbl 1439.37042 |
| [42] | Sun, P., Zero-entropy dynamical systems with gluing orbit property, Adv. Math., 372, Article 107294 pp., 2020 ·Zbl 1450.37016 |
| [43] | Sun, P., Equilibrium states of intermediate entropies, Dyn. Syst., 36, 1, 69-78, 2021 ·Zbl 1475.37037 |
| [44] | Sun, P., Unique ergodicity for zero-entropy dynamical systems with the approximate product property, Acta Math. Sin., 37, 2, 362-376, 2021 ·Zbl 1466.37011 |
| [45] | Sun, P., Non-dense orbits of systems with the approximate product property, Nonlinearity, 35, 5, 2682-2694, 2022 ·Zbl 1504.37027 |
| [46] | Tian, X.; Sun, W., Diffeomorphisms with various \(C^1\)-stable properties, Acta Math. Sci., 32B, 2, 552-558, 2012 ·Zbl 1265.37005 |
| [47] | Tian, X.; Wang, S.; Wang, X., Intermediate Lyapunov exponents for system with periodic gluing orbit property, Discrete Contin. Dyn. Syst., 39, 2, 1019-1032, 2019 ·Zbl 1404.37023 |
| [48] | Ures, R., Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part, Proc. Am. Math. Soc., 140, 6, 1973-1985, 2012 ·Zbl 1258.37033 |
| [49] | Walters, P., An Introduction to Ergodic Theory, 1982, Springer-Verlag ·Zbl 0475.28009 |
| [50] | Yang, D.; Zhang, J., Non-hyperbolic ergodic measures and horseshoes in partially hyperbolic homoclinic classes, J. Inst. Math. Jussieu, 19, 5, 1765-1792, 2020 ·Zbl 1451.37044 |
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.
