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We consider the set of points with high pointwise emergence for $C^{1+\alpha}$ diffeomorphisms preserving a hyperbolic measure. We find a lower bound on the Hausdorff dimension of this set in terms of unstable Hausdorff dimension of the…

We study the one-dimensional expanding Lorenz maps and show the existence of dense subset D of Lorens maps such that each f in D has an uncountable set of ergodic invariant probabilities with infinite Lyapunov exponent and positive entropy.…

We establish the existence and fundamental properties of the equilibrium measure in uniformly quasiregular dynamics. We show that a uniformly quasiregular endomorphism $f$ of degree at least 2 on a closed Riemannian manifold admits an…

Let $I\subset\mathbb{R}$ be an interval and $T_a:[0,1]\to[0,1]$, $a\in I$, a one-parameter family of piecewise expanding maps such that for each $a\in I$ the map $T_a$ admits a unique absolutely continuous invariant probability measure…

Let $(M,d,\mu)$ be a uniformly discrete metric measure space satisfying space homogeneous volume doubling condition. We consider discrete time Markov chains on $M$ symmetric with respect to $\mu$ and whose one-step transition density is…

Let $X$ be a real-valued random variable with distribution function $F$. Set $X_1,\dots, X_m$ to be independent copies of $X$ and let $F_m$ be the corresponding empirical distribution function. We show that there are absolute constants…

In this paper we consider torus homeomorphisms $f$ homotopic to Dehn twists. We prove that if the vertical rotation set of $f$ is reduced to zero, then there exists a compact connected essential "horizontal" set K, invariant under $f$. In…

A sufficient geometrical condition for the existence of absolutely continuous invariant probability measures for S-unimodal maps will be discussed. The Lebesgue typical existence of such measures in the quadratic family will be a…

A von Neumann flow is a special flow over an irrational rotation of the circle and under a piecewise $C^1$ roof function with a non-zero sum of jumps. We prove that the absolute value of the slope is a (measure theoretic) invariant in the…

Let $T$ be the map defined on $\N=\{1,2,3, ...\}$ by $T(n) = \frac{n}{2} $ if $n$ is even and by $T(n) = \frac{3n+1}{2}$ if $n$ is odd. Consider the dynamical system $(\N, 2^{\N}, T,\mu)$ where $\mu$ is the counting measure. This dynamical…

Let $\Omega\subset\er^3$ be a domain and let $f\colon\Omega\to\er^3$ be a bi-$BV$ homeomorphism. Very recently in \cite{HKL} it was shown that the distributional adjugate of $Df$ (and thus also of $Df^{-1}$) is a matrix-valued measure. In…

Let $M$ be a smooth compact manifold (maybe with boundary, maybe disconnected) of any dimension $d \ge 1$. We consider the set of $C^1$ maps $f:M\to M$ which have no absolutely continuous (with respect to Lebesgue) invariant probability…

We consider an abundant class of non-uniformly hyperbolic $C^2$-H\'enon like diffeomorphisms called strongly regular and which corresponds to Benedicks-Carleson parameters. We prove the existence of $m>0$ such that for any such…

We use height arguments to prove two results about the dynamical Mordell-Lang problem. (i) For an endomorphism of a projective variety, the return set of a dense orbit into a curve is finite if any cohomological Lyapunov multiplier of any…

Suppose that f and g are Markov surjections, each defined on a wedge of circles, each fixing the branch point and having the branch point as the only critical value. We show that if the points in the inverse limit spaces associated with f…

For regime-switching diffusions processes with singular drifts, we introduce integrability conditions involving a nice reference probability measure and the $Q$-matrix of the jump part to study the existence of the invariant probability…

We consider the rotation number $\rho(t)$ of a diffeomorphism $f_t=R_t\circ f$, where $R_t$ is the rotation by $t$ and $f$ is an orientation preserving $C^\infty$ diffeomorphism of the circle $S^1$. We shall show that if $\rho(t)$ is…

Suppose that the critical locus $\Sigma$ of a complex analytic function $f$ on affine space is, itself, a space with an isolated singular point at the origin $\0$, and that the Milnor number of $f$ restricted to normal slices of…

We consider families of geometries of D--dimensional space, described by a finite number of parameters. Starting from the De Witt metric we extract a unique integration measure which turns out to be a geometric invariant, i.e. independent…

Given a finite-dimensional real vector space $V$, a probability measure $\mu$ on $\operatorname{PGL}(V)$ and a $\mu$-invariant subspace $W$, under a block-Lyapunov contraction assumption, we prove existence and uniqueness of lifts to…