Geometry of limit sets for expansive Markov systems

Gelfert, Katrin; Rams, Michał

doi:10.1090/S0002-9947-08-04759-4

Geometry of limit sets for expansive Markov systems
HTML articles powered by AMS MathViewer

by Katrin Gelfert and Michał Rams PDF

Trans. Amer. Math. Soc. 361 (2009), 2001-2020 Request permission

Abstract:

We describe the geometric and dynamical properties of expansive Markov systems.

References

A. Biś and M. Urbański, Geometry of Markov systems and codimension-one foliations, preprint.
Rufus Bowen, A horseshoe with positive measure, Invent. Math. 29 (1975), no. 3, 203–204. MR 380890, DOI 10.1007/BF01389849
John Cantwell and Lawrence Conlon, Foliations and subshifts, Tohoku Math. J. (2) 40 (1988), no. 2, 165–187. MR 943817, DOI 10.2748/tmj/1178228024
Manfred Denker and Mariusz Urbański, On the existence of conformal measures, Trans. Amer. Math. Soc. 328 (1991), no. 2, 563–587. MR 1014246, DOI 10.1090/S0002-9947-1991-1014246-4
M. Denker and M. Urbański, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc. (2) 43 (1991), no. 1, 107–118. MR 1099090, DOI 10.1112/jlms/s2-43.1.107
Neil Dobbs, Hyperbolic dimension for interval maps, Nonlinearity 19 (2006), no. 12, 2877–2894. MR 2273763, DOI 10.1088/0951-7715/19/12/008
Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
Franz Hofbauer and Peter Raith, The Hausdorff dimension of an ergodic invariant measure for a piecewise monotonic map of the interval, Canad. Math. Bull. 35 (1992), no. 1, 84–98. MR 1157469, DOI 10.4153/CMB-1992-013-x
Steven Hurder, Dynamics and the Godbillon-Vey class: a history and survey, Foliations: geometry and dynamics (Warsaw, 2000) World Sci. Publ., River Edge, NJ, 2002, pp. 29–60. MR 1882764, DOI 10.1142/9789812778246_{0}003
S. Matsumoto, Measure of exceptional minimal sets of codimension one foliations, A fête of topology, Academic Press, Boston, MA, 1988, pp. 81–94. MR 928398
Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
Yakov B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. Contemporary views and applications. MR 1489237, DOI 10.7208/chicago/9780226662237.001.0001
V. A. Pliss, On a conjecture of Smale, Differencial′nye Uravnenija 8 (1972), 268–282 (Russian). MR 0299909
Arthur J. Schwartz, A generalization of a Poincaré-Bendixson theorem to closed two-dimensional manifolds, Amer. J. Math. 85 (1963), 453-458; errata, ibid 85 (1963), 753. MR 0155061
Maximilian Thaler, Transformations on $[0,\,1]$ with infinite invariant measures, Israel J. Math. 46 (1983), no. 1-2, 67–96. MR 727023, DOI 10.1007/BF02760623
Mariusz Urbański, Parabolic Cantor sets, Fund. Math. 151 (1996), no. 3, 241–277. MR 1424576
Peter Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978), 121–153. MR 466493, DOI 10.1090/S0002-9947-1978-0466493-1
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
PawełWalczak, Dynamics of foliations, groups and pseudogroups, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 64, Birkhäuser Verlag, Basel, 2004. MR 2056374, DOI 10.1007/978-3-0348-7887-6
Michiko Yuri, Zeta functions for certain non-hyperbolic systems and topological Markov approximations, Ergodic Theory Dynam. Systems 18 (1998), no. 6, 1589–1612. MR 1658607, DOI 10.1017/S0143385798117972
Michiko Yuri, Thermodynamic formalism for certain nonhyperbolic maps, Ergodic Theory Dynam. Systems 19 (1999), no. 5, 1365–1378. MR 1721626, DOI 10.1017/S0143385799141695