Invariant Measures Exist Without a Growth Condition

Communications in Mathematical Physics

Volumn 241, Issue 2-3, 2003, Pages 287-306

  Bruin, Henk ^a   Shen, Weixiao ^b   Van Strien, Sebastian ^b  

^a UNIVERSITY OF GRONINGEN   (Netherlands)

^b UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0242370750     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00220-003-0928-z     Document Type: Article

References (24)

1. Bowen, R.: Invariant measures for Markov maps of the interval. Commun. Math. Phys. 69, 1-17 (1979)
2. Bruin, H.: Topological conditions for the existence of invariant measures for unimodal maps. Ergod. Th. & Dynam. Sys. 14, 433-451 (1994)
3. Bruin, H.: Topological conditions for the existence of Cantor attractors. Trans. Am. Math. Soc. 350, 2229-2263 (1998)
4. Bruin, H., Keller, G., Nowicki, T., van Strien, S.: Wild Cantor attractors exist. Ann. Math. 143. 97-130 (1996)
5. Collet, P., Eckmann, J.-P.: Positive Liapunov exponents and absolute continuity for maps of the interval. Ergod. Th. & Dyn. Sys. 3, 13-46 (1983)
6. Graczyk, J., Świa̧tek, G.: Induced expansion for quadratic polynomials. Ann. Sci ÉC. Norm. Súp. 29, 399-482 (1996)
7. Graczyk, J., Sands, D., Świa̧tek, G.: Decay of geometry for unimodal maps: Negative Schwarzian case. Preprint, 2000
8. Jakobson, M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81, 39-88 (1981)
9. Jakobson, M., Świa̧tek, G.: Metric properties of non-renormalizable S-unimodaI maps. I. Induced expansion and invariant measures. Ergod. Th. & Dynam. Sys. 14, 721-755 (1994)
10. Johnson, S.: Singular measures without restrictive intervals. Commun. Math. Phys. 110, 185-190 (1987)
11. Keller, G., Nowicki, T.: Fibonacci maps re(al)visited. Ergod. Th. & Dyn. Sys. 15, 99-120 (1995)
12. Lyubich, M.: Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. of Math. 140, 347-404 (1994) and Erratum Manuscript, 2000
13. Lyubich, M., Milnor, J.: The Fibonacci unimodal map. J. Am. Math. Soc. 6, 425-457 (1993)
14. Mañé, R.: Hyperbolicity, sinks and measure in one-dimensional dynamics. Commun. Math. Phys. 100(4), 495-524 (1985)
15. Martens. M.: Interval dynamics. Ph.D. Thesis, Delft, 1990
16. Martens, M.: Distortion results and invariant Cantor sets of unimodal maps. Ergod. Th. & Dynam. Sys. 14, 331-349 (1994)
17. Martens, M., Nowicki, T.: Invariant measures for typical quadratic maps, Géométrie complexe et systèmes dynamiques (Orsay, 1995). Astérisque 261, 239-252 (2000)
18. de Melo, W., van Strien, S.: One-dimensional dynamics. Berlin-Heidelberg-New York: Springer, 1993
19. Misiurewicz, M.: Absolutely continuous measures for certain maps of an interval. Publ. Math. I.H.E.S. 53, 17-51 (1981)
20. Nowicki, T.: A positive Liapunov exponent for the critical value of an S-unimodal mapping implies uniform hyperbolicity. Ergod. Th. & Dynam. Sys. 8, 425-435 (1988)
21. Nowicki, T., van Strien, S.: Invariant measures exist under a summability condition. Invent. Math. 105, 123-136 (1991)
22. n). Boll. Un. Mat. Ital. 16, 364-378 (1979)
23. Shen, W.: Decay geometry for unimodal maps: An elementary proof. Preprint Warwick, 2002
24. Straube, E.: On the existence of invariant absolutely continuous measures. Commun. Math. Phys. 81, 27-30 (1981)