Rotation and entropy
Authors:
William Geller and Michał 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 $f$ 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.
- 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
- 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, DOI https://doi.org/10.1017/S0143385797084927
- F. Botelho, Rotational entropy for annulus endomorphisms, Pacific J. Math. 151 (1991), no. 1, 1–19. MR 1127583
- Rufus Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc. 184 (1973), 125–136. MR 338317, DOI https://doi.org/10.1090/S0002-9947-1973-0338317-X
- 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
- J. Kwapisz, Rotation sets and entropy, PhD Thesis, SUNY at Stony Brook, 1995.
- J. Kwapisz and R. Swanson, Asymptotic entropy, periodic orbits, and pseudo-Anosov maps, Ergod. Th. & Dynam. Sys. 18 (1998), 425–439.
- Jaume Llibre and Michał Misiurewicz, Horseshoes, entropy and periods for graph maps, Topology 32 (1993), no. 3, 649–664. MR 1231969, DOI https://doi.org/10.1016/0040-9383%2893%2990014-M
- 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, DOI https://doi.org/10.1017/S0143385700006052
- 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
- M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), no. 1, 45–63. MR 579440, DOI https://doi.org/10.4064/sm-67-1-45-63
- 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
- Richard Swanson, Periodic orbits and the continuity of rotation numbers, Proc. Amer. Math. Soc. 117 (1993), no. 1, 269–273. MR 1112502, DOI https://doi.org/10.1090/S0002-9939-1993-1112502-X
- Krystyna Ziemian, Rotation sets for subshifts of finite type, Fund. Math. 146 (1995), no. 2, 189–201. MR 1314983
Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 54H20, 58F99, 58F11
Retrieve articles in all journals with MSC (1991): 54H20, 58F99, 58F11
Additional Information
William Geller
Affiliation:
Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
Email:
wgeller@math.iupui.edu
Michał Misiurewicz
Affiliation:
Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
MR Author ID:
125475
Email:
mmisiure@math.iupui.edu
Keywords:
Rotation sets,
entropy
Received by editor(s):
February 22, 1997
Published electronically:
March 29, 1999
Additional Notes:
The second author was partially supported by NSF grant DMS-9305899.
Article copyright:
© Copyright 1999
American Mathematical Society