Anosov families, renormalization and non-stationary subshifts

Ergodic Theory and Dynamical Systems

Volumn 25, Issue 3, 2005, Pages 661-709

  Arnoux, Pierre ^a   Fisher, Albert M ^b  

^a AIX MARSEILLE UNIVERSITÉ   (France)

^b UNIVERSITY OF SÃO PAULO   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 19844370367     PISSN: 01433857     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0143385704000641     Document Type: Review

References (41)

1. [Adl98] R. L. Adler. Symbolic dynamics and Markov partitions. Bull Amer. Math. Soc. (N.S.) 35(1) (1998), 1-56.
2. [AF01] P. Arnoux and A. M. Fisher. The scenery flow for geometric structures on the torus: the linear setting. Chinese Ann. Math. B 22(4) (2001), 427-470.
3. [AF05] P. Arnoux and A. M. Fisher. Anosov families, the scenery flow, and the boundary at infinity of the modular group. Technical Report, 2005.
4. [AF02b] P. Arnoux and A. M. Fisher. The Ruelle-Perron-Frobenius theorem for non-stationary and random subshifts, 2002.
5. [Arn94] P. Arnoux. Le codage du flot géodésique sur la surface modulaire. Enseign. Math. 40 (1994), 29-48.
6. [AW70] R. L. Adler and B. Weiss. Similarity of automorphisms of the torus. Mem. Amer. Math. Soc. 98 (1970), 1-43.
7. [Bak95a] V. I. Bakhtin. Random processes generated by a hyperbolic sequence of mappings I. Russian Acad. Sci. Izv. Math. 44(2) (1995), 247-279.
8. [Bak95b] V. I. Bakhtin. Random processes generated by a hyperbolic sequence of mappings II. Russian Acad. Sci. Izv. Math. 44(2) (1995), 617-627.
9. [BG95] T. Bogenschütz and M. Gundlach. Ruelle's transfer operator for random subshifts of finite type. Ergod. Th. & Dynam. Sys. 15 (1995), 413-447.
10. [BH94] M. Boyle and D. Handelman. Entropy versus orbit equivalence for minimal homeomorphisms. Pacific J. Math. 164(1) (1994), 1-13.
11. [Bog93] T. Bogenschütz. Entropy, pressure and a variational principle for random dynamical systems. Random Comp. Dyn. 1 (1992-93), 99-116.
12. [Bow75] R. Bowen. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.
13. [Bow77] R. Bowen. On Axiom A diffeomorphisms. Conf. Board Math. Sci. 35 (1977).
14. [DHS99] F. Durand, B. Host and C. Skau. Substitution dynamical systems, Bratteli diagrams and dimension groups. Ergod. Th. & Dynam. Sys. 19 (1999), 953-993.
15. [Dur98] F. Durand. A characterization of substitutive sequences using return words. Discrete Math. 179 (1998), 89-101.
16. [ES80] E. G. Effros and C.-L. Shen. Approximately finite C*-algebras and continued fractions. Indiana Univ. Math. J. 29(2) (1980), 191-204.
17. [FB88] P. Ferrero and B.Schmitt. Produits aléatoires d'operateurs matrices de transfert. Prob. Theory 79 (1988), 227-248.
18. [Fis03] A. M. Fisher. Small-scale structure and randomness: the scenery flow in dynamics, geometry and probability. Thesis for livre docencia, University of São Paulo, March 2003. http://ime.usp.br/~afisher.
19. [Fis04] A. M. Fisher. Small-scale structure via flows. Progress in Probability (Conference Proceedings, Fractal Geometry and Stochastics III, Friedrichroda, Germany, March 2003). Birkhäuser, Basel, 2004, pp. 59-78.
20. [Fis92] A. M. Fisher. Integer Cantor sets and an order-two ergodic theorem. Ergod. Th. & Dynam. Sys. 13 (1992), 45-64.
21. [Fra82] J. Franks. Homology and Dynamical Systems (CBMS, 49). American Mathematical Society, 1982.
22. [Fur63] H. Furstenberg. A Poisson formula for semi-simple Lie groups. Ann. Math. 77 (1963), 335-386.
23. [Fur71] H. Furstenberg. Random Walks and Discrete Subgroups of Lie Groups. Vol. 1. Dekker, New York, 1971, pp. 3-63.
24. [GJ98] R. Gjerde and O. Johansen. Bratteli-Vershik models for Cantor minimal systems. Technical Report, 1998.
25. [Hun74] T. W. Hungerford. Algebra. Holt, Rinehardt and Winston, New York, 1974.
26. [Kif00] Y. Kifer. Random f-expansions. Proceedings Symp. Pure Math., 00, 2000. To appear.
27. [Led77] F. Ledrappier. Mesures d'equilibre d'entropie compétement positive. Astérisque 50 (1977), 251-272.
28. [LM95] D. Lind and B. Marcus. Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.
29. [Man02] A. Manning. A Markov partition that reflects the geometry of a hyperbolic toral automorphism. Trans. Amer. Math. Soc. 354 (2002), 2865-2895.
30. [Pin91] A. Pinto. Convergence of renormalisation and rigidity of dynamical systems. PhD Thesis, Warwick University, 1991.
31. [PP90] W. Parry and M. Pollicott. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187-188 (1990), 1-268.
32. 1+ structures on dynamical fractals: 1. The moduli space of solenoid functions for Markov maps on train tracks. Ergod. Th. & Dynam. Sys. 15 (1995), 697-734.
33. 1+ structures on dynamical fractals: 2. Embedded trees. Ergod. Th. & Dynam. Sys. 15 (1995), 969-992.
34. 1+α case. Nonlinearity 1 (1988), 181-202.
35. [Riv02] J. Rival. Prime matrix in dimension 3, in N. Pytheas-Fogg. Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics, 1794). Eds. V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. Springer, 2002.
36. [Rue72] D. Ruelle. On the use of 'small external fields' in the problem of symmetry breakdown in statistical mechanics. Ann. Phys. 69 (1972), 364-374.
37. [Shu87] M. Shub. Global Stability of Dynamical Systems. Springer, Berlin, 1987.
38. [Sta88] J. Stark. Smooth conjugacy and renormalization for diffeomorphisms of the circle. Nonlinearity 1 (1988), 541-575.
39. [Ver94] A. M. Vershik. Locally transversal symbolic dynamics. Algebra i Analiz 6 (1994), 94-106 (in Russian).
40. [VL92] A. M. Vershik and A. N. Livshits. Adic models of ergodic transformations, spectral theory, substitutions, and related topics. Adv. Sov. Math. 9 (1992), 185-204.
41. [You86] L.-S. Young. Stochastic stability of hyperbolic attractors. Ergod. Th. & Dynam. Sys. 6 (1986), 311-319.