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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.

##### MSC:
 37B40 Topological entropy 28D20 Entropy and other invariants 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
##### Keywords:
topological dynamical system; entropy; entropy structure
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##### References:
 [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
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