Invariant measures for random expanding on average Saussol maps

In this paper, we investigate the existence of random absolutely continuous invariant measures (ACIP) for random expanding on average Saussol maps in higher dimensions. This is done by the establishment of a random Lasota–Yorke inequality for the transfer operators on the space of bounded oscillation. We prove that the number of ergodic skew product ACIPs is finite and will provide an upper bound for the number of these ergodic ACIPs. This work can be seen as a generalization of the work in [F. Batayneh and C. González-Tokman, On the number of invariant measures for random expanding maps in higher dimensions, Discrete Contin. Dyn. Syst. 41 (2021) 5887–5914] on admissible random Jabłoński maps to a more general class of higher-dimensional random maps.

Anosov diffeomorphisms, anisotropic BV spaces and regularity of foliations

Given any smooth Anosov map, we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that, in the case of expanding maps, it reduces exactly to the usual space of functions of bounded variation which has proved to be particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations, which provides new information that could prove useful in treating hyperbolic systems with singularities.

Inducing schemes for multi-dimensional piecewise expanding maps

<p style='text-indent:20px;'>We construct inducing schemes for general multi-dimensional piecewise expanding maps where the base transformation is Gibbs-Markov and the return times have exponential tails. Such structures are a crucial tool in proving statistical properties of dynamical systems with some hyperbolicity. As an application we check the conditions for the first return map of a class of multi-dimensional non-Markov, non-conformal intermittent maps.</p>

On the fractional susceptibility function of piecewise expanding maps

<p style='text-indent:20px;'>We associate to a perturbation <inline-formula><tex-math id="M1">\begin{document}$ (f_t) $\end{document}</tex-math></inline-formula> of a (stably mixing) piecewise expanding unimodal map <inline-formula><tex-math id="M2">\begin{document}$ f_0 $\end{document}</tex-math></inline-formula> a two-variable fractional susceptibility function <inline-formula><tex-math id="M3">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula>, depending also on a bounded observable <inline-formula><tex-math id="M4">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>. For fixed <inline-formula><tex-math id="M5">\begin{document}$ \eta \in (0,1) $\end{document}</tex-math></inline-formula>, we show that the function <inline-formula><tex-math id="M6">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula> is holomorphic in a disc <inline-formula><tex-math id="M7">\begin{document}$ D_\eta\subset \mathbb{C} $\end{document}</tex-math></inline-formula> centered at zero of radius <inline-formula><tex-math id="M8">\begin{document}$ &gt;1 $\end{document}</tex-math></inline-formula>, and that <inline-formula><tex-math id="M9">\begin{document}$ \Psi_\phi(\eta, 1) $\end{document}</tex-math></inline-formula> is the Marchaud fractional derivative of order <inline-formula><tex-math id="M10">\begin{document}$ \eta $\end{document}</tex-math></inline-formula> of the function <inline-formula><tex-math id="M11">\begin{document}$ t\mapsto \mathcal{R}_\phi(t): = \int \phi(x)\, d\mu_t $\end{document}</tex-math></inline-formula>, at <inline-formula><tex-math id="M12">\begin{document}$ t = 0 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M13">\begin{document}$ \mu_t $\end{document}</tex-math></inline-formula> is the unique absolutely continuous invariant probability measure of <inline-formula><tex-math id="M14">\begin{document}$ f_t $\end{document}</tex-math></inline-formula>. In addition, we show that <inline-formula><tex-math id="M15">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula> admits a holomorphic extension to the domain <inline-formula><tex-math id="M16">\begin{document}$ \{\, (\eta, z) \in \mathbb{C}^2\mid 0&lt;\Re \eta &lt;1, \, z \in D_\eta \,\} $\end{document}</tex-math></inline-formula>. Finally, if the perturbation <inline-formula><tex-math id="M17">\begin{document}$ (f_t) $\end{document}</tex-math></inline-formula> is horizontal, we prove that <inline-formula><tex-math id="M18">\begin{document}$ \lim_{\eta \in (0,1), \eta \to 1}\Psi_\phi(\eta, 1) = \partial_t \mathcal{R}_\phi(t)|_{t = 0} $\end{document}</tex-math></inline-formula>.</p>