Non-uniform hyperbolicity in complex dynamics

Abstract

We study rational functions satisfying summability conditions - a family of weak conditions on the expansion along the critical orbits. Assuming their appropriate versions, we derive many nice properties: There exists a unique, ergodic, and non-atomic conformal measure with exponent equal to the Hausdorff dimension of the Julia set. There is an absolutely continuous invariant measure with respect to this conformal measure. The Minkowski dimension of the Julia set is strictly less than 2. Either the Julia set is the whole sphere, or the dynamics is unstable. For such polynomials and Blaschke products the Julia set is conformally removable. The connected components of the boundary of invariant Fatou components are locally connected. Finally, we derive a conformal analogue of Jakobson-Benedicks-Carleson theorem and prove the external continuity of the Hausdorff dimension of Julia sets for almost all points in the Mandelbrot set with respect to the harmonic measure. Some of the results extend to the multimodal maps of an interval.

Inventiones mathematicae, to appear. 85 pages, preprint from 1999 with minor corrcetions

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• Multifractal analysis for multimodal maps
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• Statistical properties of one-dimensional maps under weak hyperbolicity assumptions
• On non-uniformly hyperbolicity assumptions in one-dimensional dynamics
• Density of hyperbolicity for classes of real transcendental entire functions and circle maps
• Nice inducing schemes and the thermodynamics of rational maps
• Lyapunov spectrum for multimodal maps
• Metric definition of quasiconformality and exceptional sets
• Measures with positive Lyapunov exponent and conformal measures in rational dynamics
• Julia and John revisited
• A new family of exceptional rational functions
• Metric properties of mean wiggly continua
• Mixing and ergodicity of compositions of inner functions
• Markov extensions and lifting measures for complex polynomials
• Rigidity of non-renormalizable Newton maps
• Two counterexamples in rational and interval dynamics
• Rigidity of J-rotational rational maps and critical quasicircle maps
• Non-uniform hyperbolicity in polynomial skew products
• Lower bounds on the Hausdorff dimension of some Julia sets
• Nonuniform $(h,k,\mu ,\nu )$-dichotomy and Stability of Nonautonomous Discrete Dynamics
• Combinatorial rigidity of multicritical maps
• Dynamics of infinitely generated nicely expanding rational semigroups and the inducing method