## Institute for Mathematical Sciences

## Preprint ims96-2

** E. Prado**
* Teichmuller distance for some polynomial-like maps*

Abstract: In this work we will show that the Teichm\"{u}ller distance for all elements of a certain class of generalized polynomial-like
maps (the class of off-critically hyperbolic generalized
polynomial-like maps) is actually a distance, as in the case of
real polynomials with connected Julia set, as studied by
Sullivan. This class contains several important classes of
generalized polynomial-like maps, namely: Yoccoz, Lyubich,
Sullivan and Fibonacci. In our proof we can not use external
arguments (like external classes). Instead we use hyperbolic
sets inside the Julia sets of our maps. Those hyperbolic sets
will allow us to use our main analytic tool, namely Sullivan's
rigidity Theorem for non-linear analytic hyperbolic systems.
Lyubich has constructed a measure of maximal entropy measure $m$
on the Julia set of any rational function $f$. Zdunik
classified exactly when the Hausdorff dimension of $m$ equals
the Hausdorff dimension of the Julia set. We show that the
strict inequality holds if $f$ is off-crititcally hyperbolic,
except for Chebyshev polynomials. This result is a particular
case of Zdunik's result if we consider $f$ as a polynomial, but
is an extension of Zdunik's result if $f$ is a generalized
polynomial-like map. The proof follows from the non-existence
of invariant affine structure.

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