Dimension of weakly expanding points for quadratic maps

Bulletin de la Societe Mathematique de France

Volumn 131, Issue 3, 2003, Pages 399-420

  Senti, Samuel ^a  

^a PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

Absolutely continuous invariant probability measure; Bernoulli map; Hausdorff dimension; Induced expansion; Jakobson's theorem; Markov partition; Quadratic map

Indexed keywords

EID: 2342461693     PISSN: 00379484     EISSN: None     Source Type: Journal    
DOI: 10.24033/bsmf.2448     Document Type: Article

References (17)

1. AVILA (A.), LYUBICH (M.) & DE MELO (W.) - Regular or stochastic dynamics in real analytic families of unimodal maps, Preprint, 2001.
2. AVILA (A.) & MOREIRA (C.G.) - Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative, in Geometric Methods in Dynamics (I) (de Melo (W.), Viana (M.) & Yoccoz (J.-C.), eds.), Astérisque, vol. 286, Soc. Math. France, Paris, 2003, pp. 81-118.
3. _, Statistical properties of unimodal maps: the quadratic family, to appear in Ann. of Math., 2003.
4. 2 on (-1, 1), Ann. of Math. (2), t. 122 (1985), no. 1, pp. 1-25.
5. _, The dynamics of the Hénon map, Ann. of Math. (2), t. 133 (1991), no. 1, pp. 73-169.
6. BOWEN (R.) - Equilibrium states, Lect. Notes in Math., vol. 470, Springer Verlag, 1975.
7. FALCONER (K.) - Fractal Geometry; Mathematical Foundations and Applications, John Wiley, 1990.
8. GRACZYK (J.) & ŚWIA̧TEK (G.) - Generic hyperbolicity in the logistic family, Ann. of Math. (2), t. 146 (1997), no. 1, pp. 1-52.
9. JAKOBSON (M. V.) - Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys., t. 81 (1981), no. 1, pp. 39-88.
10. LUZZATTO (S.) - Bounded recurrence of critical points and Jakobson's theorem, in The Mandelbrot set, theme and variations, Cambridge Univ. Press, Cambridge, 2000, pp. 173-210.
11. LYUBICH (M.) - Dynamics of quadratic polynomials. I, II, Acta Math., t. 178 (1997), no. 2, pp. 185-247, 247-297.
12. DE MELO (W.) & VAN STRIEN (S.) - One-dimensional dynamics, Springer-Verlag, Berlin, 1993.
13. RÉNYI (A.) - Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar, t. 8 (1957), pp. 477-493.
14. RYCHLIK (M. R.) - Another proof of Jakobson's theorem and related results, Ergodic Theory Dynamical Systems, t. 8 (1988), no. 1, pp. 93-109.
15. SENTI (S.) - Dimension de Hausdorff de l'ensemble exceptionnel dans le théorème de Jakobson, Ph.D. Thesis, Université de Paris-Sud, 2000, available at http://www.math.psu.edu/senti.
16. TSUJII (M.) - A proof of Benedicks-Carleson-Jacobson theorem, Tokyo J. Math., t. 16 (1993), no. 2, pp. 295-310.
17. YOCCOZ (J.-C.) - Jakobson's Theorem, Manuscript of Course at Collège de France, 1997.