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In this work we continue the exploration of affine and hyperbolic laminations associated with rational maps, which were introduced in LM. Our main goal is to construct natural geometric measures on these laminations: transverse conformal measures on the affine laminations and harmonic measures on the hyperbolic laminations. The exponent $\delta$ of the transverse conformal measure does not exceed 2, and is related to the eigenvalue of the harmonic measure by the formula $\lambda=\delta(\delta-2)$. In the course of the construction we introduce a number of geometric objects on the laminations: the basic cohomology class of an affine lamination (an obstruction to flatness), leafwise and transverse conformal streams, the backward and forward Poincaré series and the associated critical exponents. We discuss their relations to the Busemann and the Anosov--Sinai cocycles, the curvature form, currents and transverse invariant measures, $\lambda$-harmonic functions, Patterson--Sullivan and Margulis measures, etc. We also prove that the dynamical laminations in question are never flat except for several explicit special cases (rational functions with parabolic Thurston orbifold).
Parabolic renormalization of critical circle maps arises as a degenerate case of the usual renormalization when the periods of the renormalized maps become infinite. In the paper we give new proofs of the main renormalization conjectures for the parabolic case, which are notably simplier than those required in the usual case. The title of the paper refers to the attracting fixed point of the parabolic renormalization, whose existence we prov
We describe the order on the ratios that define the generic universal smooth period doubling Cantor set. We prove that this set of ratios forms itself a Cantor set, a Conjecture formulated by Coullet and Tresser in 1977. We also show that the two period doubling renormalization operators, acting on the codimension one space of period doubling maps, form an iterated function system whose limit set contains a Cantor set.
This paper presents evidence for a conjecture concerning the structure of the set of braid types of periodic orbits of Smale's horseshoe map, partially ordered by Boyland's forcing order. The braid types are partitioned into totally ordered subsets, which are defined by parsing the symbolic code of a periodic orbit into two segments, the prefix and the decoration: the set of braid types of orbits with each given decoration is totally ordered, the order being given by the unimodal order on symbol sequences. The conjecture is supported by computer experiment, by proofs of special cases, and by intuitive argument in terms of pruning theory.
In this paper we extend M. Lyubich's recent results on the global hyperbolicity of renormalization of quadratic-like germs to the space $\mathbb{U}^r$ of $C^r$ unimodal maps with quadratic critical point. We show that in $\mathbb{U}^r$ the bounded-type limit sets of the renormalization operator have an invariant hyperbolic structure provided $r \ge 2+\alpha$ with $\alpha$ close to one. As an intermediate step between Lyubich's results and ours, we prove that the renormalization operator is hyperbolic in a Banach space of real analytic maps. We construct the local stable manifolds and prove that they form a continuous lamination whose leaves are $C^1$ codimension one Banach submanifolds of $\mathbb{U}^r$, and whose holonomy is $C^{1+\beta}$ for some $\beta>0$. We also prove that the global stable sets are $C^1$ immersed (codimension one) submanifolds as well, provided $r \ge 3+\alpha$ with $\alpha$ close to one. As a corollary, we deduce that in generic one parameter families of $C^r$ unimodal maps, the set of parameters corresponding to infinitely renormalizable maps of bounded combinatorial type is a Cantor set with Hausdorff dimension less than one.
For diffeomorphisms on surfaces with basic sets, we show the following type of rigidity result: if a topological conjugacy between them is differentiable at a point in the basic set then the conjugacy has a smooth extension to the surface. These results generalize the similar ones of D. Sullivan, E. de Faria, and ours for one-dimensional expanding dynamics.
We study dynamics of rational maps that satisfy a decay of geometry condition. Well known conditions of non-uniform hyperbolicity, like summability condition with exponent one, imply this condition. We prove that Julia sets have zero Lebesgue measure, when not equal to the whole sphere, and in the polynomial case every connected component of the Julia set is locally connected. We show how rigidity properties of quasi-conformal maps that are conformal in a big dynamically defined part of the sphere, apply to dynamics. For example we give a partial answer to a problem posed by Milnor about Thurston's algorithm and we give a proof that the Mandelbrot set, and its higher degree analogues, are locally connected at parameters that satisfy the decay of geometry condition. Moreover we prove a theorem about similarities between the Mandelbrot set and Julia sets. In an appendix we prove a rigidity property that extends a key situation encountered by Yoccoz in his proof of local connectivity of the Mandelbrot set at at most finitely renormalizable parameters.
The earthquake flow and the Teichmüller horocycle flow are flows on bundles over the Riemann moduli space of a surface, and are similar in many respects to unipotent flows on homogeneous spaces of Lie groups. In analogy with results of Margulis, Dani and others in the homogeneous space setting, we prove strong nondivergence results for these flows. This extends previous work of Veech. As corollaries we obtain that every closed invariant set for the earthquake (resp. Teichmüller horocycle) flow contains a minimal set, and that almost every quadratic differential on a Teichmüller horocycle orbit has a uniquely ergodic vertical foliation.
We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a positive lower bound on the lengths of closed geodesics. When the surface is a once-punctured torus, the coefficients coincide with the continued fraction coefficients associated to the ending laminations.