Related papers: Recurrence of integral zeros for ergodic flows

We prove that the geodesic flow for the Weil-Petersson metric on the moduli space of Riemann surfaces is ergodic (in fact Bernoulli) and has finite, positive metric entropy.

Let the coefficients $a_{ij}$ and $b_i$, $i,j \leq d$, of the linear Fokker-Planck-Kolmogorov equation (FPK-eq.) $$\partial_t\mu_t = \partial_i\partial_j(a_{ij}\mu_t)-\partial_i(b_i\mu_t)$$ be Borel measurable, bounded and continuous in…

We prove that every homogeneous flow on a finite-volume homogeneous manifold has countably many independent invariant distributions unless it is conjugate to a linear flow on a torus. We also prove that the same conclusion holds for every…

We study convergence of return- and hitting-time distributions of small sets $E_{k}$ with $\mu(E_{k})\rightarrow0$ in recurrent ergodic dynamical systems preserving an infinite measure $\mu$. Some properties which are easy in finite measure…

We study Poincar\'e recurrence for flows and observations of flows. For Anosov flow, we prove that the recurrence rates are linked to the local dimension of the invariant measure. More generally, we give for the recurrence rates for the…

We show that a certain type of quasi finite, conservative, ergodic, measure preserving transformation always has a maximal zero entropy factor, generated by predictable sets. We also construct a conservative, ergodic, measure preserving…

Let $(X,\nu,T)$ be a measure-preserving system, and let $P_1,\ldots, P_k$ be polynomials with integer coefficients. We prove that, for any $f_1,\ldots, f_k\in L^{\infty}(X)$, the M\"obius-weighted polynomial multiple ergodic averages…

We prove that many, but not all injective factors arise as crossed products by nonsingular Bernoulli actions of the group $\mathbb{Z}$. We obtain this result by proving a completely general result on the ergodicity, type and Krieger's…

In this paper we give a criterion for a special flow to be not isomorphic to its inverse which is a refine of a result in \cite{Fr-Ku-Le}. We apply this criterion to special flows $T^f$ built over ergodic interval exchange transformations…

Consider a Markov process $\{\Phi(t) : t\geq 0\}$ evolving on a Polish space ${\sf X}$. A version of the $f$-Norm Ergodic Theorem is obtained: Suppose that the process is $\psi$-irreducible and aperiodic. For a given function $f\colon{\sf…

In this article we prove that the Hausdorff dimension of geodesic directions that are recurrent and diverge on average coincides with the entropy at infinity of the geodesic flow for any complete, pinched negatively curved Riemannian…

Given ergodic p-invariant measures {\mu_i} on the 1-torus T=R/Z, we give a sharp condition on their entropies, guaranteeing that the entropy of the convolution \muon converges to \log p. We also prove a variant of this result for joinings…

We consider straight line flows on a translation surface that are minimal but not uniquely ergodic. We give bounds for the number of generic invariant probability measures.

Given sequence of measure preserving transformations $\{U_k:\,k=1,2,\ldots, n\}$ on a measurable space $(X,\mu)$. We prove a.e. convergence of the ergodic means \begin{equation} \frac{1}{s_1\cdots…

The paper is devoted to the existence of integral functionals $\int_0^\infty f(X(t))\,{\mathrm{d}t}$ for several classes of processes in $\mathbb{R}$ with $d\ge 3$. Some examples such as Brownian motion, fractional Brownian motion, compound…

We consider dynamics of scalar semilinear parabolic equations on bounded intervals with periodic boundary conditions, and on the entire real line, with a general nonlinearity $g(t,x,u,u_x)$ either not depending on $t$, or periodic in $t$.…

Let T be an ergodic automorphism of the d-dimensional torus T^d, and f be a continuous function from T^d to R^l. On the probability space T^d equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical…

Given a $\mathbb Z^r$-action $\alpha$ on a nilmanifold $X$ by automorphisms and an ergodic $\alpha$-invariant probability measure $\mu$, we show that $\mu$ is the uniform measure on $X$, unless modulo finite index modification, one of the…

Let $(X,\mathcal{B},\mu,T)$ be a measure preserving system. We say that a function $f\in L^2(X,\mu)$ is $\mu$-mean equicontinuous if for any $\epsilon>0$ there is $k\in \mathbb{N}$ and measurable sets ${A_1,A_2,\cdots,A_k}$ with…

We establish that $C^\infty$ three-dimensional flows with positive topological entropy admit only finitely many ergodic measures of maximal entropy, even when singularities (zero-velocity points) are present. Furthermore, every ergodic…