  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.
