zbMATH — the first resource for mathematics

Entropy structure. (English) Zbl 1151.37020
The author proposes an entropy structure as a worthwhile invariant between topological dynamical systems which capture the relationship between entropy and resolution. Roughly speaking, given a topological dynamical system \((X,T)\) with \(X\) a compact metric space and \(T\) a homeomorphism, an entropy structure is any nondecreasing sequence of nonnegative functions on the simplex of \(T\)-invariant measures which converges to the entropy function \(h\) and which is equivalent in a natural way to a specific sequence reflecting the imperfection of uniformity in convergence.
The author shows that the entropy structure (as a class) is a topological invariant and that any two entropy structures of the same system are uniformly equivalent. The author also proves that a slight modification of Bowen’s entropy, and the entropy theories by Romagnoli, Brin-Katok, Orntein-Weiss and Newhouse can be obtained as entropy structures. Some failing candidates are also presented. The results are applied to the computation of symbolic extension entropy (without reference to zero-dimensional extensions). Some considerations on noninvertible continuous maps are also given.

37B40 Topological entropy
28D20 Entropy and other invariants
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
Full Text: DOI
[1] [B] J. Buzzi,Intrinsic ergodicity of smooth interval maps, Israel J. Math.100 (1997), 125–161. · Zbl 0889.28009
[2] [B-D] M. Boyle and T. Downarowicz,The entropy theory of symbolic extensions, Invent. Math.,156 (2004), 119–161. · Zbl 1216.37004
[3] [B-F-F] M. Boyle, D. Fiebig and U. Fiebig,Residual entropy, conditional entropy and subshift covers, Forum Math.14 (2002), 713–757. · Zbl 1030.37012
[4] [B-K] M. Brin and A. Katok,On local entropy, inGeometric Dynamics (Rio de Janeiro 1981), Lecture Notes in Math., Vol. 1007, Springer-Verlag, Berlin, 1983, pp. 30–38.
[5] [D-G-S] M. Denker, C. Grillenberger and K. Sigmund,Ergodic Theory on Compact Spaces, Lecture Notes in Math., Vol. 527, Springer-Verlag, Berlin, 1976. · Zbl 0328.28008
[6] [D] T. Downarowicz,Entropy of a symbolic extension of a totally disconnected dynamical system, Ergodic Theory Dynam. Systems21 (2001), 1051–1070. · Zbl 1055.37022
[7] [D-F] T. Downarowicz and B. Frej,Topological and measure-theoretic entropy of a Markov operator, Ergodic Theory Dynam. Systems (2005), to appear. · Zbl 1088.47006
[8] [D-N] T. Downarowicz and S. Newhouse,Symbolic extensions and smooth dynamical systems, Invent. Math. (2005), to appear. · Zbl 1067.37018
[9] [D-S1] T. Downarowicz and J. Serafin,Fiber entropy and conditional variational principles in compact non-metrizable spaces, Fund. Math.172 (2002), 217–247. · Zbl 1115.37308
[10] [D-S2] T. Downarowicz and J. Serafin,Possible entropy functions, Israel J. Math.172 (2002), 217–247. · Zbl 1115.37308
[11] [D-W] T. Downarowicz and B. Weiss,Entropy theorems along times when x visits a set, Illinois J. Math.48 (2004), 59–69. · Zbl 1035.37004
[12] [G-L-W] E. Ghys, R. Langevin and P. G. Walczak,Entropie mesurée et partitions de l’unité, C. R. Acad. Sci. Paris, Sér. I303 (1986), 251–254. · Zbl 0595.60004
[13] [Gm] T. N. T. Goodman,Relating topological and measure entropy, Bull. London Math. Soc.3 (1971), 176–180. · Zbl 0219.54037
[14] [Gw] L. W. Goodwyn,Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc.23 (1969), 679–688. · Zbl 0186.09804
[15] [K] A. Katok,Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. I.H.E.S.51 (1980), 137–173. · Zbl 0445.58015
[16] [Kr] U. Krengel,Ergodic Theorems, de Gruyter, Berlin, New York, 1985.
[17] [Le] F. Ledrappier,A variational principle for the topological conditional entropy, Lecture Notes in Math., Vol. 729, Springer-Verlag, Berlin, 1979, pp. 78–88.
[18] [L-W] F. Ledrappier and P. Walters,A relativised variational principle for continuous transformations, J. London Math. Soc.16 (1977), 568–576. · Zbl 0388.28020
[19] [Li] E. Lindenstrauss,Mean dimension, small entropy factors and an imbedding theorem, Publ. Math. I.H.E.S.89 (1999), 227–262. · Zbl 0978.54027
[20] [Li-W] E. Lindenstrauss and B. Weiss,Mean topological dimension, Israel J. Math.115 (2000), 1–24. · Zbl 0978.54026
[21] [M1] M. Misiurewicz,A short proof of the variational principle for a Z + n action on a compact space, Asterisque40 (1976), 147–158.
[22] [M2] M. Misiurewicz,Topological conditional entropy, Studia Math.55 (1976), 175–200. · Zbl 0355.54035
[23] [N] S. Newhouse,Continuity properties of entropy, Ann. of Math. (2)129 (1989), 215–235;correction:131 (1990), 409–410. · Zbl 0676.58039
[24] [O-W] D. S. Ornstein and B. Weiss,Entropy and data compression schemes, IEEE Trans. Inform. Theory39 (1993), 78–83. · Zbl 0764.94003
[25] [P] W. Parry,Entropy and Generators in Ergodic Theory, W. A. Benjamin, New York, 1969.
[26] [R] P. P. Romagnoli,A local variational principle, for the topological entropy, Ergodic Theory Dynam. Systems23 (2003), 1601–1610. · Zbl 1056.37017
[27] [W] P. Walters,An Introduction to Ergodic Theory, Springer-Verlag, Berlin, 1982. · Zbl 0475.28009
[28] [Y] Y. Yomdin,Volume growth and entropy, Israel J. Math.57 (1987), 285–301. · Zbl 0641.54036
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.