Markov cell structures for expanding maps in dimension two

Farrell, F. T.; Jones, L. E.

doi:10.1090/S0002-9947-1979-0542883-4

Markov cell structures for expanding maps in dimension two
HTML articles powered by AMS MathViewer

by F. T. Farrell and L. E. Jones

Trans. Amer. Math. Soc. 255 (1979), 315-327

DOI: https://doi.org/10.1090/S0002-9947-1979-0542883-4

PDF | Request permission

Abstract:

Let $f: {M^2} \to {M^2}$ be an expanding self-immersion of a closed 2-manifold, then for some positive integer n, ${f^n}$ leaves invariant a cell structure on ${M^2}$. A similar result is true when M is a branched 2-manifold.

References

R. L. Adler and B. Weiss, Entropy, a complete metric invariant for automorphisms of the torus, Proc. Nat. Acad. Sci. U.S.A. 57 (1967), 1573–1576. MR 212156, DOI 10.1073/pnas.57.6.1573
Rufus Bowen, Markov partitions for Axiom $\textrm {A}$ diffeomorphisms, Amer. J. Math. 92 (1970), 725–747. MR 277003, DOI 10.2307/2373370

Markov partitions are not smooth

F. T. Farrell and L. E. Jones, Markov cell structures, Bull. Amer. Math. Soc. 83 (1977), no. 4, 739–740. MR 436217, DOI 10.1090/S0002-9904-1977-14374-7
F. T. Farrell and L. E. Jones, New attractors in hyperbolic dynamics, J. Differential Geometry 15 (1980), no. 1, 107–133 (1981). MR 602444, DOI 10.4310/jdg/1214435388
John M. Franks, Invariant sets of hyperbolic toral automorphisms, Amer. J. Math. 99 (1977), no. 5, 1089–1095. MR 482846, DOI 10.2307/2374001
S. G. Hancock, Orbits of paths under hyperbolic toral automorphisms, Dynamical systems, Vol. I—Warsaw, Astérisque, No. 49, Soc. Math. France, Paris, 1977, pp. 93–96. MR 0488166
John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
K. Krzyżewski, On connection between expanding mappings and Markov chains, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 291–293 (English, with Russian summary). MR 289747
Feliks Przytycki, Construction of invariant sets for Anosov diffeomorphisms and hyperbolic attractors, Studia Math. 68 (1980), no. 2, 199–213. MR 599146, DOI 10.4064/sm-68-2-199-213
Michael Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175–199. MR 240824, DOI 10.2307/2373276
Ja. G. Sinaĭ, Markov partitions and U-diffeomorphisms, Funkcional. Anal. i Priložen. 2 (1968), no. 1, 64–89 (Russian). MR 0233038
R. F. Williams, Classification of one dimensional attractors, Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 341–361. MR 0266227
R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 169–203. MR 348794, DOI 10.1007/BF02684369