58 research outputs found
Thurston obstructions and Ahlfors regular conformal dimension
Let be an expanding branched covering map of the sphere to
itself with finite postcritical set . Associated to is a canonical
quasisymmetry class \GGG(f) of Ahlfors regular metrics on the sphere in which
the dynamics is (non-classically) conformal. We show \inf_{X \in \GGG(f)}
\hdim(X) \geq Q(f)=\inf_\Gamma \{Q \geq 2: \lambda(f_{\Gamma,Q}) \geq 1\}.
The infimum is over all multicurves . The map
is defined by where the second sum is over all preimages
of freely homotopic to in , and is its Perron-Frobenius leading eigenvalue. This
generalizes Thurston's observation that if , then there is no
-invariant classical conformal structure.Comment: Minor revisions are mad
Rigidity and Expansion for Rational Maps
A general approach is proposed to prove that the combination of expansion with bounded distortion yields strong rigidity of conjugacie
Finite type coarse expanding conformal dynamics
We continue the study of non-invertible topological dynamical systems with
expanding behavior. We introduce the class of {\em finite type} systems which
are characterized by the condition that, up to rescaling and uniformly bounded
distortion, there are only finitely many iterates. We show that subhyperbolic
rational maps and finite subdivision rules (in the sense of Cannon, Floyd,
Kenyon, and Parry) with bounded valence and mesh going to zero are of finite
type. In addition, we show that the limit dynamical system associated to a
selfsimilar, contracting, recurrent, level-transitive group action (in the
sense of V. Nekrashevych) is of finite type. The proof makes essential use of
an analog of the finiteness of cone types property enjoyed by hyperbolic
groups.Comment: Updated versio
Hyperbolic groups with planar boundaries
This new version contains a proof of the quasi-isometric rigidity of the class of convex-cocompact Kleinian groups. The structure of the text has been reorganized.We prove that the class of convex-cocompact Kleinian groups is quasi-isometrically rigid. We also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually a convex-cocompact Kleinian group provided that its boundary has Ahlfors regular conformal dimension strictly less than or if it acts geometrically on a CAT(0) cube complex
Asymptotic entropy and green speed for random walks on countable groups
We study asymptotic properties of the Green metric associated with transient
random walks on countable groups. We prove that the rate of escape of the
random walk computed in the Green metric equals its asymptotic entropy. The
proof relies on integral representations of both quantities with the extended
Martin kernel. In the case of finitely generated groups, where this result is
known (Benjamini and Peres [Probab. Theory Related Fields 98 (1994) 91--112]),
we give an alternative proof relying on a version of the so-called fundamental
inequality (relating the rate of escape, the entropy and the logarithmic volume
growth) extended to random walks with unbounded support.Comment: Published in at http://dx.doi.org/10.1214/07-AOP356 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Empilements de cercles et modules combinatoires
Version revisée et corrigée.International audienceThe purpose of this paper is to try and explain the relationships between circle-packings and combinatorial moduli, and to compare the different approaches to J.W. Cannon's conjecture which follow.Le but cette note est de tenter d'expliquer les liens étroits qui unissent la théorie des empilements de cercles et des modules combinatoires, et de comparer les approches à la conjecture de J.W. Cannon qui en découlent
Invariant Jordan curves of Sierpiski carpet rational maps
In this paper, we prove that if
is a postcritically finite
rational map with Julia set homeomorphic to the Sierpi\'nski carpet, then there
is an integer , such that, for any , there exists an
-invariant Jordan curve containing the postcritical set of .Comment: 16 pages, 1 figu
An algebraic characterization of expanding Thurston maps
Let be a postcritically finite branched covering map without periodic branch points. We give necessary and sufficient algebraic conditions for to be homotopic, relative to its postcritical set, to an expanding map
Examples of coarse expanding conformal maps
In previous work, a class of noninvertible topological dynamical systems was introduced and studied; we called these {\em topologically coarse expanding conformal} systems. To such a system is naturally associated a preferred quasisymmetry (indeed, snowflake) class of metrics in which arbitrary iterates distort roundness and ratios of diameters by controlled amounts; we called this {\em metrically coarse expanding conformal}. In this note we extend the class of examples to several more familiar settings, give applications of our general methods, and discuss implications for the computation of conformal dimension
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