Hausdorff dimension, strong hyperbolicity and complex dynamics

Discrete and Continuous Dynamical Systems

Volumn 4, Issue 3, 1998, Pages 405-430

  Friedland, Shmuel ^a   Ochs, Gunter ^b  

^a UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

^b UNIVERSITY OF BREMEN   (Germany)

Author keywords

Axiom A; Complex dynamics; Hausdorff dimension; H non map; Hyperbolicity

Indexed keywords

EID: 0032374768     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.1998.4.405     Document Type: Article

References (44)

1. L. Barreira, A non-additive thermodynamics formalism and applications to dimension theory of hyperbolic dynamical systems, Ergod. Th. & Dynam. Sys., to appear.
2. L. Barreira, Y. Pesin and J. Schmeling, Dimension of hyperbolic measures - a proof of the Eckmann-Ruelle conjecture, preprint 1996.
3. A. F. Beardon, Iteration of rational functions, Springer 1991.
4. 2: Currents, equilibrium measure and hyperbolicity, Invent. Math. 103 (1991), 69-99.
5. 2. II: Stable manifolds and recurrence, J. Am. Math. Soc. 4 (1991), 657-679.
6. 2. III: Ergodicity, exponents of entropy of the equilibrium measure, Math. Ann. 294 (1992), 395-420.
7. S. Besicovitch and P. A. Moran, The measure of product and cylindrical sets, J. London Math. Soc. 20 (1945), 110-120.
8. R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes Math. 470, Springer-Verlag, 1975.
9. R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. IHES 50 (1979), 11-25.
10. M. Brin and A. Katok, On local entropy, in Geometric Dynamics, Springer Lect. Notes 1007 (1983).
11. Z.-P. Chen, L.-F. He and S.-L. Yang, Orbit shift Ω-stability, Scientia Sinica (Ser. A) 31 (1988), 512-520.
12. K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, Wiley, 1990.
13. A. Fathi, M. R. Herman and J. C. Yoccoz, A proof of the Pesin stable manifold theorem, in Geometric Dynamics, Springer Lecture Notes in Math. 1007 (1983), 177-215.
14. H. M. Farkas and I. Kra, Riemann Surfaces, Springer-Verlag 1980.
15. J. E. Fornaess and N. Sibony, Complex dynamics in higher dimension, in Complex Potential Theory, 131-186, P.M. Gauthier (ed.), Kluwer 1994.
16. S. Friedland, Computing Hausdorff dimension of subshifts using matrices, Linear Algebra Appl., to appear.
17. S. Friedland, Discrete Lyapunov exponents and Hausdorff dimension, preprint 1997.
18. S. Friedland and J. Milnor, Dynamical properties of plane polynomial diffeomorphisms, Ergod. Th. & Dynam. Sys. 9 (1989), 67-99.
19. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley 1978.
20. n, Ergod. Th. & Dynam. Sys., to appear.
21. M. Hénon, Numerical study of area preserving mappings, Q. Appl. Math. 27 (1969), 291-312.
22. M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys. 50 (1976), 69-77.
23. n, Indiana Math. J. 43 (1994), 321-365.
24. F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms, I-II, Annals Math. 122 (1985), 509-574.
25. P.-D. Liu, Stability of orbit spaces of endomorphisms, Rep. #22, Peking Univ., 1996.
26. R. Mañe, A proof of Pesin's formula, Ergod. Th. & Dynam. Sys. 1 (1981), 95-102.
27. R. Mañe and C. Pugh, Stability of endomorphisms, Lecture Notes in Math. 468, pp. 175-184, Springer, 1975.
28. A. Manning, A relation between Lyapunov exponents, Hausdorff dimension and entropy, Ergod. Th. & Dynam. Sys. 1 (1981), 451-459.
29. H. McCluskey and A. Manning, Hausdorff dimension for horseshoes, Ergod. Th. & Dynam. Sys. 3 (1983), 251-260.
30. G. Ochs, Hyperbolische Limesmengen bei Iteration algebraischer Funktionen, Ph.D. thesis, University of Bremen, 1995.
31. V. I. Oseledec, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197-221.
32. Y. B. Pesin, Families of invariant manifolds correponding to non zero characteristic exponents, USSR Izvestia 10 (1976), 1261-1305.
33. Y. B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys 32 (1977), 55-114.
34. F. Przytycki, Anosov endomorphisms, Studia Math., 58 (1976), 249-285.
35. M. Qian and Z. Zhang, Ergodic theory for axiom A endomorphisms, Ergod. Th. & Dynam. Sys. 15 (1995), 161-174.
36. V. A. Rokhlin, On the fundamental ideas in measure theory, A.M.S. Trans. 1, 10 (1962), 1-54.
37. D. Ruelle, Ergodic theory of differential dynamical systems, Publ. Math. IHES 50 (1979), 27-58.
38. D. Ruelle, Repellers for real analytic maps, Ergod. Th. & Dynam. Sys. 2 (1982), 99-107.
39. D. Ruelle, Elements of Differentiate Dynamics and Bifurcation Theory, Academic Press, San Diego, 1989.
40. M. Shub, Global Stability of Dynamical Systems, Springer-Verlag 1987.
41. A. Verjovsky and H. Wu, Hausdorff dimension of Julia sets of complex Hénon maps, Ergod. Th. & Dynam. Sys. 16 (1996), 849-861.
42. P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982.
43. L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergod. Th. & Dynam. Sys. 2 (1982), 109-129.
44. Z.-S. Zhang, Expanding invariant sets of endomorphisms, Scientia Sinica (Ser. A) 27 (1984), 1035-1045.