Rotation and entropy

Geller, William; Misiurewicz, Michał

doi:10.1090/S0002-9947-99-02344-2

Rotation and entropy

Authors: William Geller and Michal Misiurewicz
Journal: Trans. Amer. Math. Soc. 351 (1999), 2927-2948
MSC (1991): Primary 54H20, 58F99, 58F11
DOI: https://doi.org/10.1090/S0002-9947-99-02344-2
Published electronically: March 29, 1999
MathSciNet review: 1615967
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a given map $f: X \to X$ and an observable $\varphi : X \to \mathbb{R} ^{d},$ rotation vectors are the limits of ergodic averages of $\varphi .$ We study which part of the topological entropy of is associated to a given rotation vector and which part is associated with many rotation vectors. According to this distinction, we introduce directional and lost entropies. We discuss their properties in the general case and analyze them more closely for subshifts of finite type and circle maps.

[ALM] Lluís Alsedà, Jaume Llibre, and Michał Misiurewicz, Combinatorial dynamics and entropy in dimension one, Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1255515
[BM] Alexander Blokh and Michał Misiurewicz, New order for periodic orbits of interval maps, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 565–574. MR 1452180, https://doi.org/10.1017/S0143385797084927
[Bot] F. Botelho, Rotational entropy for annulus endomorphisms, Pacific J. Math. 151 (1991), no. 1, 1–19. MR 1127583
[Bow] Rufus Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc. 184 (1973), 125–136. MR 338317, https://doi.org/10.1090/S0002-9947-1973-0338317-X
[DGS] Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675
[Kw] J. Kwapisz, Rotation sets and entropy, PhD Thesis, SUNY at Stony Brook, 1995.
[KS] J. Kwapisz and R. Swanson, Asymptotic entropy, periodic orbits, and pseudo-Anosov maps, Ergod. Th. & Dynam. Sys. 18 (1998), 425-439. CMP 98:12
[LM] Jaume Llibre and Michał Misiurewicz, Horseshoes, entropy and periods for graph maps, Topology 32 (1993), no. 3, 649–664. MR 1231969, https://doi.org/10.1016/0040-9383(93)90014-M
[MT] Brian Marcus and Selim Tuncel, The weight-per-symbol polytope and scaffolds of invariants associated with Markov chains, Ergodic Theory Dynam. Systems 11 (1991), no. 1, 129–180. MR 1101088, https://doi.org/10.1017/S0143385700006052
[M] Michał Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 2, 167–169 (English, with Russian summary). MR 542778
[MS] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), no. 1, 45–63. MR 579440, https://doi.org/10.4064/sm-67-1-45-63
[PP] Ya. B. Pesin and B. S. Pitskel′, Topological pressure and the variational principle for noncompact sets, Funktsional. Anal. i Prilozhen. 18 (1984), no. 4, 50–63, 96 (Russian, with English summary). MR 775933
[Sw] Richard Swanson, Periodic orbits and the continuity of rotation numbers, Proc. Amer. Math. Soc. 117 (1993), no. 1, 269–273. MR 1112502, https://doi.org/10.1090/S0002-9939-1993-1112502-X
[Z] Krystyna Ziemian, Rotation sets for subshifts of finite type, Fund. Math. 146 (1995), no. 2, 189–201. MR 1314983