Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-10T06:59:38.943Z Has data issue: false hasContentIssue false
  • English
  • Français

Regular or stochastic dynamics in families of higher-degree unimodal maps

Published online by Cambridge University Press:  03 April 2013

TREVOR CLARK
TREVOR CLARK*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, USA email tcclark@umich.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct a lamination of the space of unimodal maps with critical points of fixed degree $d\geq 2$ by the hybrid classes. The structure of the lamination yields a partition of the parameter space for one-parameter real analytic families of unimodal maps and allows us to transfer a priori bounds in the phase space to the parameter space. This implies that almost every map in such a family is either regular or stochastic.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 5 , October 2014 , pp. 1538 - 1566
Copyright
Copyright ©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avila, A. and Lyubich, M.. The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes. Publ. Math. Inst. Hautes. Études Sci. 114 (2011), 171–223.CrossRefGoogle Scholar
Avila, A., Lyubich, M. and de Melo, W.. Regular or stochastic dynamics in real analytic families of unimodal maps. Invent. Math. 154 (2003), 451–550.Google Scholar
Avila, A., Lyubich, M. and Shen, W.. Parapuzzle of the multibrot set and typical dynamics of unimodal maps. J. Eur. Math. Soc. 13 (1) (2008), 27–56.Google Scholar
Avila, A. and Moreira, C. G.. Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative. Astérisque 286 (2003), 81–118.Google Scholar
Avila, A. and Moreira, C.  G.. Statistical properties of unimodal maps: the quadratic family. Ann. of Math. (2) 161 (2005), 831–881.Google Scholar
Avila, A. and Moreira, C.  G.. Phase-parameter relation and sharp statistical properties for general families of unimodal maps. Contemp. Math. 389 (2005), 1–42.CrossRefGoogle Scholar
Avila, A. and Moreira, C.  G.. Statistical properties of unimodal maps: physical measures, periodic orbits and pathological laminations. Publ. Math. Inst. Hautes. Études Sci. 101 (2005), 1–67.Google Scholar
Bers, L. and Royden, H.  L.. Holomorphic families of injections. Acta. Math. 157 (1986), 259–286.Google Scholar
Collet, P. and Eckmann, J.-P.. Iterated Maps on the Interval as Dynamical Systems. Birkhäuser, Basel, 1980.Google Scholar
Graczyk, J., Sands, D. and Świątek, G.. Schwarzian derivative in unimodal dynamics. C. R. Acad. Sci. Paris 332 (4) (2001), 329–332.Google Scholar
Kozlovski, O.. Getting rid of the negative Schwarzian derivative condition. Ann. of Math. (2) 152 (2000), 743–762.Google Scholar
Kozlovski, O.. Axiom A maps are dense in the space of unimodal maps in the ${C}^{k} $ topology. Ann. of Math. (2) 157 (1) (2003), 1–43.Google Scholar
Kozlovski, O., Shen, W. and van Strien, S.. Rigidity for real polynomials. Ann. of Math. (2) 165 (3) (2007), 749–841.Google Scholar
Kozlovski, O. and van Strien, S.. Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials. Proc. Lond. Math. Soc. (2) 99 (2009), 275–296.Google Scholar
Levin, G. and van Strien, S.. Local connectivity of the Julia set of real polynomials. Ann. of Math. (2) 147 (1998), 471–541.Google Scholar
Levin, G. and van Strien, S.. Total disconnectedness of Julia sets and absence of invariant line fields for real polynomials. Astérisque 261 (2000), 161–172.Google Scholar
Levin, G. and van Strien, S.. Bounds for maps of an interval with one critical point of inflection type II. Invent. Math. 141 (2000), 399–465.Google Scholar
Lyubich, M.. Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. of Math. (2) 140 (2) (1994), 347–404.Google Scholar
Lyubich, M.. Dynamics of quadratic polynomials, I–II. Acta Math. 178 (2) (1997), 185–297.Google Scholar
Lyubich, M.. Dynamics of quadratic polynomials, III. Parapuzzle and SBR measures. Astérisque 261 (2000), 173–200.Google Scholar
Lyubich, M.. Feigenbaum–Coullet–Tresser universality and Milnor’s hairyness conjecture. Ann. of Math. (2) 149 (1999), 319–420.CrossRefGoogle Scholar
Lyubich, M.. Almost every real quadratic map is regular or stochastic. Ann. of Math. (2) 156 (2002), 1–78.Google Scholar
Mañé, R., Sad, R. and Sullivan, D.. On the dynamics of rational maps. Ann. Sci. Eć. Norm. Supér. 16 (1983), 193–217.Google Scholar
de Melo, W. and van Strien, S.. One-dimensional Dynamics. Springer, Berlin, 1993.Google Scholar
Palis, J.. A global view of dynamics and a conjecture of the denseness of finitude of attractors. Astérisque 261 (2000), 335–347.Google Scholar
Sands, D.. Misiurewicz maps are rare. Comm. Math. Phys. 197 (1) (1998), 109–129.Google Scholar
Shen, W.. On the metric properties of multimodal interval maps and ${C}^{2} $ density of Axiom A. Invent. Math. 156 (2004), 301–403.Google Scholar
Slodkowski, Z.. Holomorphic motions and polynomial hulls. Proc. Amer. Math. Soc. 111 (1991), 147–355.Google Scholar
Smania, D.. Puzzle geometry and rigidity: the Fibonacci cycle is hyperbolic. J. Amer. Math. Soc. 20 (3) (2007), 629–673.Google Scholar
van Strien, S. and Vargas, E.. Real bounds, ergodicity and negative Schwarzian for multimodal maps. J. Amer. Math. Soc. 4 (4) (2004), 749–782.Google Scholar
Yin, Y. and Zhai, Y.. No invariant line fields on Cantor Julia sets. Forum. Math. 22 (2010), 75–94.Google Scholar