Abstract
In this paper, we shall prove that Axiom A maps are dense in the space of C 2 interval maps (endowed with the C 2 topology). As a step of the proof, we shall prove real and complex a priori bounds for (first return maps to certain small neighborhoods of the critical points of) real analytic multimodal interval maps with non-degenerate critical points. We shall also discuss rigidity for interval maps without large bounds.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ahlfors, L.: Lectures on quasiconformal mappings. Van Nostrand Co. 1966
Avila, A., Lyubich, M., de Melo, W.: Regular or stochastic dynamics in real analytic families of unimodal maps. Invent. Math. 154, 451–550 (2003)
Blokh, A., Lyubich, M.: Nonexistence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems. II. The smooth case. Ergodic Theory Dyn. Syst. 9, 751–758 (1989)
Blokh, A., Lyubich, M.: Measure and dimension of solenoidal attractors of one dimensional dynamical systems. Commun. Math. Phys. 1, 573–583 (1990)
Blokh, A., Misiurewicz, M.: Typical limit sets of critical points for smooth interval maps. Ergodic Theory Dyn. Syst. 20, 15–45 (2000) and Erratum. Ergodic Theory Dyn. Syst. 23, 661–661 (2003)
Bruin, H., Keller, G., Nowicki, T., van Strien, S.: Wild Cantor attractors exist. Ann. Math. (2) 143, 97–130 (1996)
Carleson, L., Gamelin, T.: Complex dynamics. Springer 1993
Douady, A., Hubbard, J.: On the dynamics of polynomial-like mappings. Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 287–343 (1985)
Douady, A., Hubbard, J.: A proof of Thurston’s topological characterization of rational functions. Acta Math. 171, 263–297 (1993)
de Faria, E., de Melo, W.: Rigidity of critical circle mappings. I. J. Eur. Math. Soc. (JEMS) 1, 339–392 (1999)
Graczyk, J., Światek, G.: Generic hyperbolicity in the logistic family. Ann. Math. (2) 146, 1–52 (1997)
Graczyk, J., Światek, G.: Polynomial-like property for real quadratic polynomials. Topol. Proc. 21, 33–112 (1996)
Graczyk, J., Światek, G.: Smooth unimodal maps in 1990s. Ergodic Theory Dyn. Syst. 19, 263–287 (1999)
Guckenheimer, J.: Sensitive dependence on initial conditions for one dimensional maps. Commun. Math. Phys. 70, 133–160 (1979)
Hu, J.: Renormalization, rigidity, and universality in bifurcation theory. Thesis 1995
Hu, J., Jiang Y.: The Julia set of the Feigenbaum quadratic polynomial. In: Proceedings of the International Conference on Dynamical Systems in Honor of Professor Shantao Liao, (Beijing, August 1998), pp. 99–124. World Scientific 1999
Hubbard, J.: Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz. In: Topological methods in modern mathematics (Stony Brook, NY, 1991), pp. 467–511. Houston, TX: Publish or Perish 1993
Jakobson, M.: Smooth mappings of the circle into itself. Math. Sb. (N.S.) 85 (127), 163–188 (1971)
Keller, G., Nowicki, T.: Fibonacci maps re(aℓ)visited. Ergodic Theory Dyn. Syst. 15, 99–120 (1995)
Kozlovski, O.: Getting rid of the negative Schwarzian derivative condition. Ann. Math. (2) 152, 743–762 (2000)
Kozlovski, O.: Axiom A maps are dense in the space of unimodal maps in the C k topology. Ann. Math. (2) 157, 1–43 (2003)
Levin, G., van Strien, S.: Local connectivity of the Julia set of real polynomials. Ann. Math. (2) 147, 471–541 (1998)
Levin, G., van Strien, S.: Total disconnectedness of Julia sets and absence of invariant linefields for real polynomials. Astèrisque 261, 161–172 (2000)
Levin, G., van Strien, S.: Bounds for maps of an interval with one critical point of inflection type. II. Invent. Math. 141, 399–465 (2000)
Lyubich, M.: Nonexistence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems. I. The case of negative Schwarzian derivative. Ergodic Theory Dyn. Syst. 9, 737–749 (1989)
Lyubich, M.: Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. Math. (2) 140, 347–404 (1994)
Lyubich, M.: Dynamics of quadratic polynomials, I, II. Acta Math. 178, 185–247, 247–297 (1997)
Lyubich, M.: Renormalization ideas in conformal dynamics. In: Current developments in mathematics, 1995 (Cambridge, MA), pp. 155–190. Cambridge, MA: Internat. Press 1994
Lyubich, M.: Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture. Ann. Math. (2) 149, 319–420 (1999)
Lyubich, M., Milnor, J.: The Fibonacci unimodal map. J. Am. Math. Soc. 6, 425–457 (1993)
Lyubich, M., Yampolsky, M.: Dynamics of quadratic polynomials: complex bounds for real maps. Ann. Inst. Fourier 47, 1219–1255 (1997)
Mañé, R.: Hyperbolicity, sinks and measure in one-dimensional dynamics. Commun. Math. Phys. 100, 495–524 (1985)
Mañé, R., Sad, P., Sullivan, D.: On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 193–217 (1983)
Martens, M.: Distortion results and invariant Cantor sets of unimodal maps. Ergodic Theory Dyn. Syst. 14, 331–349 (1994)
de Melo, W., van Strien, S.: One-dimensional dynamics. Berlin: Springer 1993
de Melo, W., van Strien, S.: A structure theorem in one-dimensional dynamics. Ann. Math. (2) 129, 519–546 (1989)
Martens, M., de Melo, W., van Strien, S.: Julia-Fatou-Sullivan theory for real one-dimensional dynamics. Acta Math. 168, 273–318 (1992)
McMullen, C.: Complex dynamics and renormalization. Ann. Math. Stud. 135. Princeton, NJ: Princeton University Press 1994
McMullen, C.: Renormalization and 3-manifolds which fiber over the circle. Ann. Math. Stud. 142. Princeton, NJ: Princeton University Press 1996
Milnor, J.: Dynamics in one complex variable. Introductory lectures. Braunschweig: Friedr. Vieweg and Sohn 1999
Milnor, J., Thurston, W.: On iterated maps of the interval. In: Dynamical systems (College Park, MD, 1986–87), pp. 465–563. Lecture Notes in Math. 1342. Berlin, New York: Springer 1988
Shen, W.: Bounds for one-dimensional maps without inflection critical points. J. Math. Sci., Tokyo 10, 41–88 (2003)
Shen, W.: On the measurable dynamics of real rational functions. Ergodic Theory Dyn. Syst. 23, 957–983 (2003)
Shishikura, M.: Yoccoz puzzles, τ-functions and their applications. Unpublished
Smania, D.: Complex bounds for multimodal maps: bounded combinatorics. Nolinearity 14, 1311–1330 (2001)
Sullivan, D.: Bounds, quadratic differentials, and renormalization conjectures. In: Am. Math. Soc. Centen. Publ., Vol. II (Providence, RI, 1988), pp. 417–466. Providence, RI: Am. Math. Soc. 1992
van Strien, S., Vargas, E.: Real bounds, ergodicity and negative Schwarzian for multimodal maps. Preprint 2002
Światek, G., Vargas, E.: Decay of geometry in the cubic family. Ergodic Theory Dyn. Syst. 18, 1311–1329 (1998)
Vargas, E.: Measure of minimal sets of polymodal maps. Ergodic Theory Dyn. Syst. 16, 159–178 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000)
Primary 37E05; Secondary 37F25
Rights and permissions
About this article
Cite this article
Shen, W. On the metric properties of multimodal interval maps and C 2 density of Axiom A. Invent. math. 156, 301–403 (2004). https://doi.org/10.1007/s00222-003-0343-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-003-0343-2