We consider a class of endomorphisms which contains a set of piecewise partially hyperbolic skew-products with a non-uniformly expanding base map. The aimed transformation preserves a foliation which is almost everywhere uniformly contracted with possible discontinuity sets, which are parallel to the contracting direction. We prove that the associated transfer operator, acting on suitable anisotropic normed spaces, has a spectral gap (on which we have quantitative estimation) and the disintegration of the unique invariant physical measure, along the stable leaves, is ζ -Hölder. We use this fact to obtain exponential decay of correlations on the set of ζ -Hölder functions.

We consider a class of endomorphisms that contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps. Our goal is to study a class of endomorphisms that preserve a foliation that is almost everywhere uniformly contracted, with possible discontinuity sets parallel to the contracting direction. We apply the spectral gap property and the $ \zeta $-Hölder regularity of the disintegration of its equilibrium states to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size $ \delta $, we show that the $ F $-invariant measure varies continuously with respect to a suitable anisotropic norm. Furthermore, we establish that certain interesting classes of perturbations exhibit a modulus of continuity estimated by $ D_2\delta^\zeta \log \delta $, where $ D_2 $ is a constant.

We consider a random dynamical system arising as a model of the behavior of a macrovariable related to a more complicated model of associative memory. This system can be seen as a small (stochastic and deterministic) perturbation of a determinstic system having two weak attractors which are destroyed after the perturbation. We show, with a computer aided proof, that the system has a kind of chaotic itineracy. Typical orbits are globally chaotic, while they spend a relatively long time visiting the attractor’s ruins.

We consider a class of endomorphisms which contains a set of piecewise partially hyperbolic skew-products with a non-uniformly expanding base map. The aimed transformation preserves a foliation which is almost everywhere uniformly contracted with possible discontinuity sets, which are parallel to the contracting direction. We prove that the associated transfer operator, acting on suitable anisotropic normed spaces, has a spectral gap (on which we have quantitative estimation) and the disintegration of the unique invariant physical measure, along the stable leaves, is $\zeta$-H\"older. We use this fact to obtain exponential decay of correlations on the set of $\zeta$-H\"older functions.

We consider a class of endomorphisms that contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps. Our goal is to study a class of endomorphisms that preserve a foliation that is almost everywhere uniformly contracted, with possible discontinuity sets parallel to the contracting direction. We apply the spectral gap property and the $\zeta$-H\"older regularity of the disintegration of its equilibrium states to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size $\delta$, we show that the $F$-invariant measure varies continuously with respect to a suitable anisotropic norm. Moreover, we prove that for certain interesting classes of perturbations, its modulus of continuity is $O(\delta^\zeta \log \delta)$. This article has been accepted for publication in the Discrete and Continuous Dynamical Systems journal.