1,955 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
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
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