Related papers: On normal numbers and self-similar measures

Let $T$ be a bijective map on $\mathbb{R}^n$ such that both $T$ and $T^{-1}$ are Borel measurable. For any $\btheta \in \mathbb{R}^n$ and any real $n \times n$ positive definite matrix $\Sigma,$ let $N (\btheta, \Sigma)$ denote the…

For $C^0$ generic continuous maps or homeomorphisms on compact Riemannian manifold, we prove that (1) the space of physical-like measures coincides with the set of invariant measures supported on chain recurrent classes, (2) every point in…

Fix $c\in (1,23/22)$. Let $\alpha$ and $\beta$ be two distinct non-zero real numbers with $|\alpha|\neq |\beta|$. It is shown that for any measure preserving system $(X,\mathcal{X},\mu,T)$ and any $f,g\in L^{\infty}(\mu)$, the limit…

For any C1 diffeomorphism with dominated splitting we consider a nonempty set of invariant measures which describes the asymptotic statistics of Lebesgue-almost all orbits. They are the limits of convergent subsequences of averages of the…

Consider an iterated function system consisting of similarities on the complex plane of the form $g_{i}(z) = \lambda_i z + t_i,\ \lambda_i, t_i \in \mathbb{C},\ |\lambda_i|<1, i=1,\ldots, k$. We prove that for almost every choice of…

Suppose $\alpha$ is a rotationally symmetric norm on $L^{\infty}\left(\mathbb{T}\right) $ and $\beta$ is a "nice" norm on $L^{\infty}\left(\Omega,\mu \right) $ where $\mu$ is a $\sigma$-finite measure on $\Omega$. We prove a version of…

We study generalizations of Reifenberg's Theorem for measures in $\mathbb R^n$ under assumptions on the Jones' $\beta$-numbers, which appropriately measure how close the support is to being contained in a subspace. Our main results, which…

Let $\{f_i\}_{i=1}^N$ be a set of equi-contractive similitudes on $\mathbb{R}^1$ satisfying the finite-type condition. We study the asymptotic quantization error for self-similar measures $\mu$ associated with $\{f_i\}_{i=1}^N$ and a…

For the Fourier transform $\mathcal{F}\mu$ of a general (non-trivial) self-similar measure $\mu$ on the real line $\mathbb{R}$, we prove a large deviation estimate \[ \lim_{c\to +0} \varlimsup_{t\to \infty}\frac{1}{t}\log…

A $C^\infty$ surface diffeomorphism admits a SRB measure if and only if the set \left \{x, \limsup_n \frac{1}{ n} \log \|d_xf^n \|> 0\right\} has positive Lebesgue measure. Moreover the basins of the ergodic SRB measures are covering this…

If $S$ is an infinite sequence over a finite alphabet $\Sigma$ and $\beta$ is a probability measure on $\Sigma$, then the {\it dimension} of $ S$ with respect to $\beta$, written $\dim^\beta(S)$, is a constructive version of Billingsley…

It is known that if $x\in[0,1]$ is polynomial time random (i.e. no polynomial time computable martingale succeeds on the binary fractional expansion of $x$) then $x$ is normal in any integer base greater than one. We show that if $x$ is…

We study how physical measures vary with the underlying dynamics in the open class of $C^r$, $r>1$, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs $u$-state is positive. If transitive,…

Let $\overline{\mathfrak{S}}_\infty$ denote the set of all bijections of natural numbers. Consider the action of $\overline{\mathfrak{S}}_\infty$ on a measure space $\left( X,\mathfrak{M},\mu \right)$, where $\mu$ is…

We study periodic expansions in positional number systems with a base $\beta\in\C,\ |\beta|>1$, and with coefficients in a finite set of digits $\A\subset\C.$ We are interested in determining those algebraic bases for which there exists…

The first aim of this article is to give information about the algebraic properties of alternate bases $\boldsymbol{\beta}=(\beta_0,\dots,\beta_{p-1})$ determining sofic systems. We show that a necessary condition is that the product…

If $S$ is an infinite sequence over a finite alphabet $\Sigma$ and $\beta$ is a probability measure on $\Sigma$, then the {\it dimension} of $ S$ with respect to $\beta$, written $\dim^\beta(S)$, is a constructive version of Billingsley…

Given a closed orientable surface (\Sigma) of genus at least two, we establish an affine isomorphism between the convex compact set of isotopy-invariant topological measures on (\Sigma) and the convex compact set of additive functions on…

We consider the map $T_{\alpha,\beta}(x):= \beta x + \alpha \mod 1$, which admits a unique probability measure of maximal entropy $\mu_{\alpha,\beta}$. For $x \in [0,1]$, we show that the orbit of $x$ is $\mu_{\alpha,\beta}$-normal for…

The irrationality exponent $\mu(t)$ of an irrational number t, defined using the irrationality measure $1/q^\mu$, distinguishes among non-Liouville numbers and is infinite for Liouville numbers. Using the irrationality measure $1/\beta^q$,…