Related papers: Quantitative equidistribution of horocycle push-fo…

Let $(h_t)_{t\in \mathbb{R}}$ be the horocycle flow acting on $(M,\mu)=(\Gamma \backslash \text{SL}(2,\mathbb{R}),\mu)$, where $\Gamma$ is a co-compact lattice in $\text{SL}(2,\mathbb{R})$ and $\mu$ is the homogeneous probability measure…

Let $\Gamma$ be a lattice in $G=\mathrm{SL}(2,\mathbb{C})$. We give an effective equidistribution result with precise error terms for expanding translates of pieces of horospherical orbits in $\Gamma\backslash G$. Our method of proof relies…

We show that if $\Gamma$ is a co-compact arithmetic lattice in $SL(2,\mathbb{R})$ or $\Gamma=SL(2,\mathbb{Z})$ then the horocycle orbit of every non-periodic point $x\in SL(2,\mathbb{R})/\Gamma$ equidistributes (with respect to Haar…

In this paper, it is shown that for every lattice $\Gamma \subset PSL_2(\mathbb{R})$ there exists a $c>0$ such that for any $0 \leq \gamma<c$ the sequence $p h(n^{1+\gamma})$ equidistributes for any $p \in \Gamma \backslash…

We study the invariant distributions for the horocycle map on $\Gamma\backslash SL(2, \mathbb{R})$ and prove Sobolev estimates for the cohomological equation of the horocycle map. As an application, we obtain an estimate for the rate of…

We provide a self-contained, accessible introduction to Ratner's Equidistribution Theorem in the special case of horocyclic flow on a complete hyperbolic surface of finite area. This equidistribution result was first obtained in the early…

We give a rate of equidistribution of lifts of horocycles from the space $\mathrm{SL}(2,\mathbb Z)\backslash \mathrm{SL}(2, \mathbb R)$ to the space $\mathrm{ASL}(2,\mathbb Z)\backslash \mathrm{ASL}(2,\mathbb R)$, making effective a theorem…

We prove bounds for twisted ergodic averages for horocycle flows of hyperbolic surfaces, both in the compact and in the non-compact finite area case. From these bounds we derive effective equidistribution results for horocycle maps. As an…

Let $(X,\mathfrak{B},\mu)$ be a Borel probability space. Let $T_n: X\rightarrow X$ be a sequence of continuous transformations on $X$. Let $\nu$ be a probability measure on $X$ such that $\frac{1}{N}\sum_{n=1}^N (T_n)_\ast \nu \rightarrow…

Let $G=SL_2(\mathbb R)^d$ and $\Gamma=\Gamma_0^d$ with $\Gamma_0$ a lattice in $SL_2(\mathbb R)$. Let $S$ be any "curved" submanifold of small codimension of a maximal horospherical subgroup of $G$ relative to an $\mathbb R$-diagonalizable…

Let $\Gamma\subset PSL(2,\mathbb{R})$ be such that the space $X=\Gamma\backslash PSL(2,\mathbb{R})$ is not compact. Let $(h_t)$ be the horocycle flow acting on $X$. We show that for every $x\in X$ that is not periodic for $(h_t)$ and for…

\noindent In [1] L. Polterovich and Z. Rudnick considered the behavior of a one-parameter subgroup of a Lie group under the influence of a sequence of kicks. Among others they raise the following problem: {\it is the horocycle flow stably…

We consider a random walk $S_k$ with i.i.d. steps on a compact group equipped with a bi-invariant metric. We prove quantitative ergodic theorems for the sum $\sum_{k=1}^N f(S_k)$ with H\"older continuous test functions $f$, including the…

We prove effective equidistribution of horospherical flows in $\operatorname{SO}(n,1)^\circ / \Gamma$ when $\Gamma$ is geometrically finite and the frame flow is exponentially mixing for the Bowen-Margulis-Sullivan measure. We also discuss…

In this article, we give explicit conditions for compact group extensions of hyperbolic flows (including geodesic flows on negatively curved manifolds) to exhibit quantifiable rates of mixing (or decay of correlations) with respect to the…

The main results of this paper are limit theorems for horocycle flows on compact surfaces of constant negative curvature. One of the main objects of the paper is a special family of horocycle-invariant finitely-additive Hoelder measures on…

Let $G$ be a semisimple Lie group and $\Gamma$ a lattice in $G$. We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of horospheres in the homogeneous space $\Gamma\backslash G$.

We prove effective equidistribution of expanding horocycles in $\mathrm{SL}_2(\mathbb{Z})\backslash\mathrm{SL}_2(\mathbb{R})$ with respect to various classes of Borel probability measures on $\mathbb{R}$ having certain Fourier asymptotics.…

Let $M=\Gamma\backslash\mathrm{PSL}(2,\mathbb{R})$ be a compact manifold, and let $f\in C^\infty(M)$ be a function of zero average. We use spectral methods to get uniform (i.e. independent of spectral gap) bounds for twisted averages of $f$…

Let $G$ be a semisimple Lie group without compact factor and $\Gamma < G$ a torsion-free, cocompact, irreducible lattice. According to Selberg, periodic orbits of regular Weyl chamber flows live on maximal flat periodic tori of the space of…