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DCDS

We deal with the Fatou functions $f_\lambda(z)=z+e^{-z}+\lambda$, Re$\lambda\ge
1$. We consider the set $J_r(f_\lambda)$ consisting of those points
of the Julia set of $f_\lambda$ whose real parts do not escape to infinity under
positive iterates of $f_\lambda$. Our ultimate result is that the function
$\lambda\mapsto$HD$(J_r(f_\lambda))$ is real-analytic. In order to prove it we
develop the thermodynamic formalism of potentials of the form
$-t$log$|F_\lambda'|$, where $F_\lambda$ is the projection of $f_\lambda$ to the
infinite cylinder. It includes appropriately defined
topological pressure, Perron-Frobenius operators, geometric and
invariant generalized conformal measures (Gibbs states). We show that
our Perron-Frobenius operators are quasicompact, that they embed into a
family of operators depending holomorphically on an appropriate parameter
and we obtain several
other properties of these operators. We prove an appropriate
version of Bowen's formula that the Hausdorff dimension of the set
$J_r(f_\lambda)$ is equal to the unique zero of the pressure
function. Since the formula
for the topological pressure is independent of the set $J_r(f_\lambda)$,
Bowen's formula also indicates that $J_r(f_\lambda)$ is the right set to deal
with. What concerns geometry of the set $J_r(f_\lambda)$ we also prove
that the HD$(J_r(f_\lambda))$-dimensional Hausdorff measure of the set $J_r(F_\lambda)$
is positive and finite whereas its HD$(J_r(f_\lambda))$-dimensional
packing measure is locally infinite. This last property allows us to
conclude that HD$(J_r(f_\lambda))<2$.
We also study in detail the properties of
quasiconformal conjugations between the maps $f_\lambda$. As a byproduct of
our main course
of reasoning we prove stochastic properties of the dynamical system
generated by $F_\lambda$ and the invariant Gibbs states $\mu_t$ such as the
Central Limit Theorem and the exponential decay of correlations.

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