Related papers: On the ergodic theory of the real Rel foliation

In this paper, we conduct a comprehensive study on ergodic properties of the geodesic flow on a $C^\infty$ uniform visibility manifold $M$ without conjugate points. If $M$ is a closed surface of genus at least two without conjugate points,…

In this paper we introduce and explore the notion of rigidity group, associated with a collection of finitely many sequences, and show that this concept has many, somewhat surprising characterizations of algebraic, spectral, and unitary…

We consider the density properties of divergence-free vector fields $ b \in L^1([0,1],\textit{BV}([0,1]^2)) $ which are ergodic/weakly mixing/strongly mixing: this means that their Regular Lagrangian Flow $X_t$ is an ergodic/weakly…

In this article, we study the ergodicity of the geodesic flows on surfaces with no focal points. Let $M$ be a smooth connected and closed surface equipped with a $C^\infty$ Riemannian metric $g$, whose genus $\mathfrak{g} \geq 2$. Suppose…

Given $\a \in [0,1]$ and $\varphi: \T \to \R$ measurable, the {\it cylindircal cascade} $S_{\a,\varphi}$ is the map from $\T \times \R$ to itself given by $S_{\a,\varphi} (x,y) = (x+\a,y+\varphi(x))$ that naturally appears in the study of…

We prove some ergodic theorems for flat surfaces of finite area. The first result concerns such surfaces whose Teichmuller orbits are recurrent to a compact subset of $SL(2;R)/SL(S)$, where $SL(S)$ is the Veech group of the surface. In this…

Ten years ago A. Zorich discovered, by computer experiments on interval exchange transformations, some striking new power laws for the ergodic integrals of generic non-exact Hamiltonian flows on higher genus surfaces. In Zorich's later work…

We study dynamics of the horocycle flow on strata of translation surfaces, introduce new invariants for ergodic measures, and analyze the interaction of the horocycle flow and real Rel surgeries. We use this analysis to complete and extend…

We study the horocycle flow on the stratum of translation surfaces $\mathcal{H}(2)$. We show that there is a sequence of horocycle ergodic measures, each supported on a periodic horocycle orbit, which weakly converges to an invariant, but…

Given an ergodic flow $T=(T_t)_{t\in\Bbb R}$, let $I(T)$ be the set of reals $s\ne 0$ for which the flows $(T_{st})_{t\in\Bbb R}$ and $T$ are isomorphic. It is proved that $I(T)$ is a Borel subset of $\Bbb R^*$. It carries a natural Polish…

We consider smooth area-preserving flows (also known as locally Hamiltonian flows) on surfaces of genus $g\geq 1$ and study ergodic integrals of smooth observables along the flow trajectories. We show that these integrals display a…

We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure…

In this paper, we study ergodic optimization of continuous functions for flows by concentrating on the entropy spectrum of their maximizing measures. Precisely, over a wide family of flows with non-uniformly hyperbolic structure, we obtain…

We investigate specific examples of locally-defined real vector-fields on strata of translation surfaces. Integrating SL(2,R)-loci of Veech surfaces along these vector-fields yield interesting new examples of horocyle-invariant ergodic…

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.

We consider the horospherical foliation on any invariant subvariety in the moduli space of translation surfaces. This foliation can be described dynamically as the strong unstable foliation for the geodesic flow on the invariant subvariety,…

Let $K/\mathbf Q$ be a degree $d$ extension. Inside the ring of integers $\mathcal O_K$ we define the set of $k$-free integers $\mathcal F_k$ and a natural $\mathcal O_K$-action on the space of binary $\mathcal O_K$-indexed sequences,…

Fix a translation surface $X$, and consider the measures on $X$ coming from averaging the uniform measures on all the saddle connections of length at most $R$. Then as $R\to\infty$, the weak limit of these measures exists and is equal to…

We prove that for closed surfaces $M$ with Riemannian metrics without conjugate points and genus $\geq 2$ the geodesic flow on the unit tangent bundle $T^1M$ has a unique measure of maximal entropy. Furthermore, this measure is fully…

We give a short proof of the unique ergodicity of the strong stable foliation of the geodesic flow on the frame bundle of a hyperbolic manifold admitting a finite measure of maximal entropy. Equivalently, let G = S0o(n, 1), $\Gamma$…