Related papers: Absolutely Continuous Invariant Measure for Genera…
Consider piecewise linear Lorenz maps on $[0, 1]$ of the following form \[ f_{a,b,c}(x)= {ll} ax+1-ac & x \in [0, c) b(x-c) & x \in (c, 1].\] We prove that $f_{a,b,c}$ admits an absolutely continuous invariant probability measure (acim)…
We consider the non autonomous dynamical system $\{\tau_{n}\},$ where $\tau_{n}$ is a continuous map $X\rightarrow X,$ and $X$ is a compact metric space. We assume that $\{\tau_{n}\}$ converges uniformly to $\tau .$ The inheritance of…
We prove that a partially hyperbolic attracting set for a C2 vector field, having slow recurrence to equilibria, supports an ergodic physical/SRB measure if, and only if, the trapping region admits non-uniform sectional expansion on a…
We give a description of ergodic components of SRB measures in terms of ergodic homoclinic classes associated to hyperbolic periodic points. For transitive surface diffeomorphisms, we prove that there exists at most one SRB measure.
We compute explicitly the density of the invariant measure for the Reverse algorithm which is absolutely continuous with respect to Lebesgue measure, using a method proposed by Arnoux and Nogueira. We also apply the same method on the…
We define the speed measure $\nu$ for mappings $\gamma:I\to X$ from an interval to a metric space that are locally of bounded variation. We characterize continuity and absolute continuity of $\gamma$ in terms of $\nu$ and identify the…
For random compositions of independent and identically distributed measurable maps on a Polish space, we study the existence and finitude of absolutely continuous ergodic stationary probability measures (which are, in particular, physical…
We study some special classes of piecewise continuous maps on a finite smooth partition of a compact manifold and look for invariant measures for such maps. We show that in the simplest one-dimensional case (so-called interval translation…
We study Markov processes generated by iterated function systems (IFS). The constituent maps of the IFS are monotonic transformations of the interval. We first obtain an upper bound on the number of SRB (Sinai-Ruelle-Bowen) measures for the…
A finitely-additive measure $\lambda $ on an infinite-dimensional real Hilbert space $E$ which is invariant with respect to shifts and orthogonal mappings has been defined. This measure can be considered as the analog of the Lebesgue…
We establish a transversality theorem for multiple-point crossings under generic linear perturbations with explicit Hausdorff measure estimates for the exceptional parameter set, and hence explicit upper bounds on its Hausdorff dimension.…
Given a surface $M$ and a Borel probability measure $\nu$ on the group of $C^2$-diffeomorphisms of $M$, we study $\nu$-stationary probability measures on $M$. Assuming the positivity of a certain entropy, the following dichotomy is proved:…
An important class of `physically relevant' measures for dynamical systems with hyperbolic behavior is given by Sinai-Ruelle-Bowen (SRB) measures. We survey various techniques for constructing SRB measures and studying their properties,…
We study measure-theoretical aspects of torus piecewise isometries. Not much is known about this type of dynamical systems, except for the special case of one-dimensional interval exchange mappings. The last case is fundamentally different…
In this paper, we study the smoothness of the density function of absolutely continuous measures supported on random self-similar sets on the line. We show that the natural projection of a measure with symbolic local dimension greater than…
We prove that the (B) conjecture and the Gardner-Zvavitch conjecture are true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures. Actually, our result apply beyond the case…
Given a general Anosov $\mathbb{R}^\kappa$ action on a closed manifold, we study properties of certain invariant measures that have recently been introduced in \cite{BGHW20} using the theory of Ruelle-Taylor resonances. We show that these…
In this paper, we study the limit measures of the empirical measures of Lebesgue almost every point in the basin of a partially hyperbolic attractor. They are strongly related to a notion named Gibbs u-state, which can be defined in a large…
Let $M$ be a Riemannian, boundaryless, and compact manifold, with $\dim M\geq 2$ and let $f$ be a $C^{1+}$ diffeomorphism. We show that there is a hyperbolic SRB measure if and only if there exists an unstable leaf with a subset of positive…
We study transversality for Lipschitz-Fredholm maps in the context of bounded Fr\'{e}chet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fr\'{e}chet spaces has the transverse stability property. We…