Topological entropy for noncompact sets
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- by Rufus Bowen PDF
- Trans. Amer. Math. Soc. 184 (1973), 125-136 Request permission
Abstract:
For $f:X \to X$ continuous and $Y \subset X$ a topological entropy $h(f,Y)$ is defined. For X compact one obtains results generalizing known theorems about entropy for compact Y and about Hausdorff dimension for certain $Y \subset X = {S^1}$ . A notion of entropy-conjugacy is proposed for homeomorphisms.References
- R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. MR 175106, DOI 10.1090/S0002-9947-1965-0175106-9
- Roy L. Adler and Benjamin Weiss, Similarity of automorphisms of the torus, Memoirs of the American Mathematical Society, No. 98, American Mathematical Society, Providence, R.I., 1970. MR 0257315
- Patrick Billingsley, Hausdorff dimension in probability theory, Illinois J. Math. 4 (1960), 187–209. MR 131903
- Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0192027
- Rufus Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414. MR 274707, DOI 10.1090/S0002-9947-1971-0274707-X
- Rufus Bowen, Markov partitions for Axiom $\textrm {A}$ diffeomorphisms, Amer. J. Math. 92 (1970), 725–747. MR 277003, DOI 10.2307/2373370
- C. M. Colebrook, The Hausdorff dimension of certain sets of nonnormal numbers, Michigan Math. J. 17 (1970), 103–116. MR 260697, DOI 10.1307/mmj/1029000420
- E. I. Dinaburg, A correlation between topological entropy and metric entropy, Dokl. Akad. Nauk SSSR 190 (1970), 19–22 (Russian). MR 0255765
- H. G. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser. 20 (1949), 31–36. MR 31026, DOI 10.1093/qmath/os-20.1.31
- Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. MR 213508, DOI 10.1007/BF01692494
- Donald Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math. 5 (1970), 339–348 (1970). MR 274716, DOI 10.1016/0001-8708(70)90008-3
- T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc. 3 (1971), 176–180. MR 289746, DOI 10.1112/blms/3.2.176
- L. Wayne Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc. 23 (1969), 679–688. MR 247030, DOI 10.1090/S0002-9939-1969-0247030-3
- William Parry, Entropy and generators in ergodic theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0262464 D. Ruelle, Statistical mechanics on a compact set with ${Z^\nu }$ action satisfying expansiveness and specification (preprint).
- Ja. G. Sinaĭ, Markov partitions and U-diffeomorphisms, Funkcional. Anal. i Priložen 2 (1968), no. 1, 64–89 (Russian). MR 0233038
- B. Weiss, Intrinsically ergodic systems, Bull. Amer. Math. Soc. 76 (1970), 1266–1269. MR 267076, DOI 10.1090/S0002-9904-1970-12632-5
- K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR 0226684
- Robert Ash, Information theory, Interscience Tracts in Pure and Applied Mathematics, No. 19, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1965. MR 0229475
- Karl Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc. 190 (1974), 285–299. MR 352411, DOI 10.1090/S0002-9947-1974-0352411-X
- Benjamin Weiss, The isomorphism problem in ergodic theory, Bull. Amer. Math. Soc. 78 (1972), 668–684. MR 304616, DOI 10.1090/S0002-9904-1972-12979-3
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 184 (1973), 125-136
- MSC: Primary 28A65; Secondary 54H20
- DOI: https://doi.org/10.1090/S0002-9947-1973-0338317-X
- MathSciNet review: 0338317