We study the Julia sets for some periodic meromorphic maps, namely the maps of the form $f\left(z\right)=h\left(exp\frac{2\pi i}{T}z\right)$ where h is a rational function or, equivalently, the maps $˜f\left(z\right)=exp\left(\frac{2\pi i}{h}\left(z\right)\right)$. When the closure of the forward orbits of all critical and asymptotic values is disjoint from the Julia set, then it is hyperbolic and it is possible to construct the Gibbs states on J(˜f) for -α log |˜˜f|. For ˜α = HD(J(˜f)) this state is equivalent to the ˜α-Hausdorff measure or to the ˜α-packing measure provided ˜α is greater or smaller than 1....

Let $f$ be a dominant rational map of ${\mathbb{P}}^k$ such that there exists $s\<k$ with ${\lambda }_s\left(f\right)\>{\lambda }_l\left(f\right)$ for all $l$. Under mild hypotheses, we show that, for $A$ outside a pluripolar set of $\mathrm{Aut}\left({\mathbb{P}}^k\right)$, the map $f\circ A$ admits a hyperbolic measure of maximal entropy $log{\lambda }_s\left(f\right)$ with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of $f$ to ${\mathbb{P}}^{k+1}$. This provides many examples where non uniform hyperbolic dynamics is established.One of the key tools is to approximate the graph of a meromorphic...