We consider a nonlinear area preserving Anosov map $M$ on the torus phase space, which is the simplest example of a fully chaotic dynamics. We are interested in the quantum dynamics for long time, generated by the unitary quantum propagator $\widehat{M}$. The usual semi-classical Trace formula expresses $Tr\left({\widehat{M}}^t\right)$ for finite time $t$, in the limit $\hslash \to 0$, in terms of periodic orbits of $M$ of period $t$. Recent work reach time $t\ll t_E/6$ where $t_E=log(1/\hslash )/\lambda$ is the Ehrenfest time, and $\lambda$ is the Lyapounov coefficient. Using a semi-classical normal form...

It is well known that any continuous piecewise monotone interval map f with positive topological entropy $h_{top}\left(f\right)$ is semiconjugate to some piecewise affine map with constant slope $e^{h_{top}\left(f\right)}$. We prove this result for a class of Markov countably piecewise monotone continuous interval maps.