Related papers: Singular measures of circle homeomorphisms with tw…

In this paper we construct a paramaterized family of annular homeomorphisms with Birkhoff-like rotational attractors that vary continuously with the parameter, are all homeomorphic to the pseudo-circle, display interesting boundary dynamics…

We consider C^2 families of C^4 unimodal maps f_t whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure of f_t depends differentiably on t, as a distribution of order 1. The proof uses…

Let $\pi:X\to Y$ be a factor map, where $(X,\sigma_X)$ and $(Y,\sigma_Y)$ are subshifts over finite alphabets. Assume that $X$ satisfies weak specification. Let $\ba=(a_1,a_2)\in \R^2$ with $a_1>0$ and $a_2\geq 0$. Let $f$ be a continuous…

Starting from the De Witt supermetric and limiting ourselves to a family of geometries characterized by a finite number of geometric invariants we extract the unique integration measure. Such a measure turns out to be a geometric invariant,…

This note explains how the two measures used to define the $\mu$-deformed Segal-Bargmann space are natural and essentially unique structures. As is well known, the density with respect to Lebesgue measure of each of these measures involves…

Let $T:[0,1]^d \rightarrow[0,1]^d$ be a piecewise expanding map with an absolutely continuous (with respect to the $d$-dimensional Lebesgue measure $m_d$) $T$-invariant probability measure $\mu$. Let $\left\{\mathbf{r}_n\right\}$ be a…

In this paper, we study the uniqueness of the direct decomposition of a toric manifold. We first observe that the direct decomposition of a toric manifold as \emph{algebraic varieties} is unique up to order of the factors. An algebraically…

In this paper, we show that the Gibbs measure of the stochastic hyperbolic sine-Gordon equation on the circle is the unique invariant measure for the Markov process. Moreover, the Markov transition probabilities converge exponentially fast…

For a $\mathcal{C}^1$ diffeomorphism $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ isotopic to the identity, we prove that for any value $l\in\mathbb{R}$ of the linking number at finite time of the orbits of two points there exists at least a…

We relate the Mather invariant of diffeomorphisms of the (closed) interval to their asymptotic distortion. For maps with only parabolic fixed points, we show that the former is trivial if and only if the latter vanishes. As a consequence,…

Fix an irrational number $\alpha$ and a smooth, positive, real function $\mathfrak{p}$ on the circle. If current position is $x\in \mathbb R/\mathbb Z$ then in the next step jump to $x+\alpha$ with probability $\mathfrak{p}(x)$ or to…

We investigate the existence of invariant measures for self-stabilizing diffusions. These stochastic processes represent roughly the behavior of some Brownian particle moving in a double-well landscape and attracted by its own law. This…

We study geometric and statistical properties of complex rational maps satisfying the Topological Collet-Eckmann Condition. We show that every such a rational map possesses a unique conformal probability measure of minimal exponent, and…

Let $A$ be an $n\times n$ random matrix with independent, identically distributed mean 0, variance 1 subgaussian entries. We prove that $$ \mathbb{P}(A\text{ has distinct singular values})\geq 1-e^{-cn} $$ for some $c>0$, confirming a…

We consider a smooth area-preserving Anosov diffeomorphism $f\colon \mathbb T^2\rightarrow \mathbb T^2$ homotopic to an Anosov automorphism $L$ of $\mathbb T^2$. It is known that the positive Lyapunov exponent of $f$ with respect to the…

This paper generalizes the results of [13] and then provides an interesting example. We construct a family of $W$-like maps $\{W_a\}$ with a turning fixed point having slope $s_1$ on one side and $-s_2$ on the other. Each $W_a$ has an…

For $C^2$ vector fields, we study regular ergodic measures whose supports admit singular dominated splittings with one of the bundles having dimension $1$. For such a measure $\mu$, we prove that if any periodic orbit within the support of…

We study the regularity of exceptional actions of groups by $C^{1,\alpha}$ diffeomorphisms on the circle, i.e. ones which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with concave modulus of…

We call a dynamical system on a measurable metric space {\em measure-expansive} if the probability of two orbits remain close each other for all time is negligible (i.e. zero). We extend results of expansive systems on compact metric spaces…

We consider a complete noncompact smooth metric measure space $(M^n,g,e^{-f} dv)$ and the associated drifting Laplacian. We find sufficient conditions on the geometry of the space so that every nonnegative $f$-subharmonic function with…