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Ariel Rapaport

Publications and source records attributed to Ariel Rapaport.

At least 19 recordsLinked to original sources
arXiv2026

Dimension of self-conformal measures associated to an exponentially separated holomorphic IFS

Let $\Phi$ be a holomorphic IFS on a bounded domain in $\mathbb{C}$. Suppose that the following conditions hold: (1) the maps in $\Phi$ do not have a common fixed point; (2) there does not exist a regular real-analytic curve which is invariant under all of the maps in $\Phi$; (3) $\Phi$ is not holomorphically conjugate to a homothetic IFS; (4) $\Phi$ is exponentially separated. Under these assumptions, we show that the dimensions of the self-conformal measures associated to $\Phi$, as well as the Hausdorff dimension of the associated self-conformal set, attain their natural upper bounds. The proof combines recently developed methods from the dimension theory of stationary fractal measures with complex-analytic arguments.

math.DS↗
arXiv2025

Dimension of Furstenberg measures on $\mathbb{CP}^{1}$

Let $\theta$ be a finitely supported probability measure on $\mathrm{SL}(2,\mathbb{C})$, and suppose that the semigroup generated by $\mathcal{G}:=\mathrm{supp}(\theta)$ is strongly irreducible and proximal. Let $\mu$ denote the Furstenberg measure on $\mathbb{CP}^{1}$ associated to $\theta$. Assume further that no generalized circle is fixed by all M\"obius transformations corresponding to elements of $\mathcal{G}$, and that $\mathcal{G}$ satisfies a mild Diophantine condition. Under these assumptions, we prove that $\dim\mu=\min\left\{ 2,h_{\mathrm{RW}}/\left(2\chi\right)\right\} $, where $h_{\mathrm{RW}}$ and $\chi$ denote the random walk entropy and Lyapunov exponent associated to $\theta$, respectively. Since our result expresses $\dim\mu$ in terms of the random walk entropy rather than the Furstenberg entropy, and relies only on a mild Diophantine condition as a separation assumption, we are forced to directly confront difficulties arising from the ambient space $\mathbb{CP}^{1}$ having real dimension $2$ rather than $1$. Moreover, our analysis takes place in a projective, contracting-on-average setting. This combination of features introduces significant challenges and requires genuinely new ideas.

math.DS↗
arXiv2024

Dimension of self-conformal measures associated to an exponentially separated analytic IFS on $\mathbb{R}$

We extend Hochman's work on exponentially separated self-similar measures on $\mathbb{R}$ to the real analytic setting. More precisely, let $\Phi=\left\{ \varphi_{i}\right\} _{i\in\Lambda}$ be an iterated function system on $I:=[0,1]$ consisting of real analytic contractions, let $p=(p_{i})_{i\in\Lambda}$ be a positive probability vector, and let $\mu$ be the associated self-conformal measure. Suppose that the maps in $\Phi$ do not have a common fixed point, $0<\left|\varphi_{i}'(x)\right|<1$ for $i\in\Lambda$ and $x\in I$, and $\Phi$ is exponentially separated. Under these assumptions, we prove that $\dim\mu=\min\left\{ 1,H(p)/\chi\right\} $, where $H(p)$ is the entropy of $p$ and $\chi$ is the Lyapunov exponent. The main novelty of our work lies in an argument that reduces convolutions of $\mu$ with measures on the (infinite-dimensional) space of real analytic maps to convolutions with measures on vector spaces of polynomials of bounded degree. The reason for this reduction is that, for the latter convolutions, we can establish an entropy increase result, which plays a crucial role in the proof. We believe that our proof strategy has the potential to extend other significant recent results in the dimension theory of stationary fractal measures to the real analytic setting.

math.DS↗
arXiv2024

Dimension of Bernoulli Convolutions in $\mathbb{R}^{d}$

For $(\lambda_{1},...,\lambda_{d})=\lambda\in(0,1)^{d}$ with $\lambda_{1}>...>\lambda_{d}$, denote by $\mu_{\lambda}$ the Bernoulli convolution associated to $\lambda$. That is, $\mu_{\lambda}$ is the distribution of the random vector $\sum_{n\ge0}\pm\left(\lambda_{1}^{n},...,\lambda_{d}^{n}\right)$, where the $\pm$ signs are chosen independently and with equal weight. Assuming for each $1\le j\le d$ that $\lambda_{j}$ is not a root of a polynomial with coefficients $\pm1,0$, we prove that the dimension of $\mu_{\lambda}$ equals $\min\left\{ \dim_{L}\mu_{\lambda},d\right\} $, where $\dim_{L}\mu_{\lambda}$ is the Lyapunov dimension. More generally, we obtain this result in the context of homogeneous diagonal self-affine systems on $\mathbb{R}^{d}$ with rational translations. The proof extends to higher dimensions the works of Breuillard and Varj\'u and Varj\'u regarding Bernoulli convolutions on the real line. The main novelty and contribution of the present work lies in an extension of an entropy increase result, due to Varj\'u, in which the amount of increase in entropy is given explicitly. The extension of this result to the higher-dimensional non-conformal case requires significant new ideas.

math.DS↗
arXiv2023

Dimension of diagonal self-affine sets and measures via non-conformal partitions

Let $Φ:=\left\{ (x_{1},...,x_{d})\rightarrow\left(r_{i,1}x_{1}+a_{i,1},...,r_{i,d}x_{d}+a_{i,d}\right)\right\} _{i\inΛ}$ be an affine diagonal IFS on $\mathbb{R}^{d}$. Suppose that for each $1\le j_{1}<j_{2}\le d$ there exists $i\inΛ$ so that $|r_{i,j_{1}}|\ne|r_{i,j_{2}}|$, and that for each $1\le j\le d$ the IFS $\left\{ t\rightarrow r_{i,j}t+a_{i,j}\right\} _{i\inΛ}$ on the real line is exponentially separated. Under these assumptions we show that the Hausdorff dimension of the attractor of $Φ$ is equal to $\min\left\{ \dim_{A}Φ,d\right\} $, where $\dim_{A}Φ$ is the affinity dimension. This follows from a result regarding self-affine measures, which says that, under the additional assumption that the linear parts of the maps in $Φ$ are all contained in a $1$-dimensional subgroup, the dimension of an associated self-affine measure $μ$ is equal to the minimum of its Lyapunov dimension and $d$. Most of the proof is dedicated to an entropy increase result for convolutions of $μ$ with general measures $θ$ of non-negligible entropy, where entropy is measured with respect to non-conformal partitions corresponding to the Lyapunov exponents of $μ$. It turns out that with respect to these partitions, the entropy across scales of repeated self-convolutions of $θ$ behaves quite differently compared to the conformal case. The analysis of this non-conformal multi-scale entropy is the main ingredient of the proof, and is also the main novelty of this paper.

math.DS↗
arXiv2023

Self-similar measures associated to a homogeneous system of three maps

We study the dimension of self-similar measures associated to a homogeneous iterated function system of three contracting similarities on $\bf R$ and other more general IFS's. We extend some of the theory recently developed for Bernoulli convolutions to this setting. In the setting of three maps a new phenomenon occurs, which has been highlighted by recent examples of Baker, and Bárány, Käenmäki. To overcome the difficulties stemming form these, we develop novel techniques, including an extension of Hochman's entropy increase method to a function field setup.

math.DS↗
arXiv2022

On self-affine measures associated to strongly irreducible and proximal systems

Let $μ$ be a self-affine measure on $\mathbb{R}^{d}$ associated to an affine IFS $Φ$ and a positive probability vector $p$. Suppose that the maps in $Φ$ do not have a common fixed point, and that standard irreducibility and proximality assumptions are satisfied by their linear parts. We show that $\dimμ$ is equal to the Lyapunov dimension $\dim_{L}(Φ,p)$ whenever $d=3$ and $Φ$ satisfies the strong separation condition (or the milder strong open set condition). This follows from a general criteria ensuring $\dimμ=\min\{d,\dim_{L}(Φ,p)\}$, from which earlier results in the planar case also follow. Additionally, we prove that $\dimμ=d$ whenever $Φ$ is Diophantine (which holds e.g. when $Φ$ is defined by algebraic parameters) and the entropy of the random walk generated by $Φ$ and $p$ is at least $(χ_{1}-χ_{d})\frac{(d-1)(d-2)}{2}-\sum_{k=1}^{d}χ_{k}$, where $0>χ_{1}\ge...\geχ_{d}$ are the Lyapunov exponents. We also obtain results regarding the dimension of orthogonal projections of $μ$.

math.DS↗
arXiv2021

On the Rajchman property for self-similar measures on $\mathbb{R}^{d}$

We establish a complete algebraic characterization of self-similar iterated function systems $Φ$ on $\mathbb{R}^{d}$, for which there exists a positive probability vector $p$ so that the Fourier transform of the self-similar measure corresponding to $Φ$ and $p$ does not tend to $0$ at infinity.

math.DS↗
arXiv2020

Exact dimensionality and Ledrappier-Young formula for the Furstenberg measure

Assuming strong irreducibility and proximality, we prove that the Furstenberg measure, corresponding to a finitely supported measure on the general linear group of a finite dimensional real vector space, is exact dimensional. We also establish a Ledrappier-Young type formula for its dimension. The general strategy of the proof is based on the argument given by Feng for the exact dimensionality of self-affine measures.

math.DS↗
arXiv2020

Dimension of ergodic measures projected onto self-similar sets with overlaps

For self-similar sets on $\mathbb{R}$ satisfying the exponential separation condition we show that the natural projections of shift invariant ergodic measures is equal to $\min\{1,\frac{h}{-χ}\}$, where $h$ and $χ$ are the entropy and Lyapunov exponent respectively. The proof relies on Shmerkin's recent result on the $L^{q}$ dimension of self-similar measures. We also use the same method to give results on convolutions and orthogonal projections of ergodic measures projected onto self-similar sets.

math.DS↗
arXiv2019

Hausdorff Dimension of Planar Self-Affine Sets and Measures with Overlaps

We prove that if $μ$ is a self-affine measure in the plane whose defining IFS acts totally irreducibly on $\mathbb{RP}^1$ and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension. We also treat a class of reducible systems. This extends our previous work on the subject with Bárány to the overlapping case.

math.DS↗
arXiv2018

On self-similar measures with absolutely continuous projections and dimension conservation in each direction

Relying on results due to Shmerkin and Solomyak, we show that outside a $0$-dimensional set of parameters, for every planar homogeneous self-similar measure $ν$, with strong separation, dense rotations and dimension greater than $1$, there exists $q>1$ such that $\{P_{z}ν\}_{z\in S}\subset L^{q}(\mathbb{R})$. Here $S$ is the unit circle and $P_{z}w=\left\langle z,w\right\rangle $ for $w\in\mathbb{R}^{2}$. We then study such measures. For instance, we show that $ν$ is dimension conserving in each direction and that the map $z\rightarrow P_{z}ν$ is continuous with respect to the weak topology of $L^{q}(\mathbb{R})$.

math.DS↗
arXiv2018

Geometric Law for Multiple Returns until a Hazard

For a $ψ$-mixing stationary process $ξ_0,ξ_1,ξ_2,...$ we consider the number $\mathcal N_N$ of multiple recurrencies $\{ξ_{q_i(n)}\inΓ_N,\, i=1,...,\ell\}$ to a set $Γ_N$ for $n$ until the moment $τ_N$ (which we call a hazard) when another multiple recurrence $\{ξ_{q_i(n)}\inΔ_N,\, i=1,...,\ell\}$ takes place for the first time where $Γ_N\capΔ_N= \emptyset$ and $q_i(n)<q_{i+1}(n),\, i=1,...,\ell$ are nonnegative increasing functions taking on integer values on integers. It turns out that if $P\{ξ_0\inΓ_N\}$ and $P\{ξ_0\inΔ_N\}$ decay in $N$ with the same speed then $\mathcal N_N$ converges weakly to a geometrically distributed random variable. We obtain also a similar result in the dynamical systems setup considering a $ψ$-mixing shift $T$ on a sequence space $Ω$ and study the number of multiple recurrencies $\{ T^{q_i(n)}ω\in A_n^b,\, i=1,...,\ell\}$ until the first occurence of another multiple recurrence $\{ T^{q_i(n)}ω\in A_m^a,\, i=1,...,\ell\}$ where $A_m^a,\, A_n^b$ are cylinder sets of length $m$ and $n$ constructed by sequences $a,b\inΩ$, respectively, and chosen so that their probabilities have the same order. This work is motivated by a number of papers on asymptotics of numbers of single and multiple returns to shrinking sets, as well as by the papers on open systems studying their behavior until an exit through a "hole".

math.PR↗
arXiv2017

Hausdorff dimension of planar self-affine sets and measures

Let $X=\bigcupφ_{i}X$ be a strongly separated self-affine set in $\mathbb{R}^2$ (or one satisfying the strong open set condition). Under mild non-compactness and irreducibility assumptions on the matrix parts of the $φ_{i}$, we prove that $\dim X$ is equal to the affinity dimension, and similarly for self-affine measures and the Lyapunov dimension. The proof is via analysis of the dimension of the orthogonal projections of the measures, and relies on additive combinatorics methods.

math.MG↗
arXiv2017

A self-similar measure with dense rotations, singular projections and discrete slices

We construct a planar homogeneous self-similar measure, with strong separation, dense rotations and dimension greater than $1$, such that there exist lines for which dimension conservation does not hold and the projection of the measure is singular. In fact, the set of such directions is residual and the typical slices of the measure, perpendicular to these directions, are discrete.

math.DS↗
arXiv2017

A dimension gap for continued fractions with independent digits - the non stationary case

We show there exists a constant $0<c_{0}<1$ such that the dimension of every measure on $[0,1]$, which makes the digits in the continued fraction expansion independent, is at most $1-c_{0}$. This extends a result of Kifer, Peres and Weiss from 2001, which established this under the additional assumption of stationarity. For $k\ge1$ we prove an analogues statement for measures under which the digits form a $*$-mixing $k$-step Markov chain. This is also generalized to the case of $f$-expansions. In addition, we construct for each $k$ a measure, which makes the continued fraction digits a stationary and $*$-mixing $k$-step Markov chain, with dimension at least $1-2^{3-k}$.

math.DS↗
arXiv2015

On self-affine measures with equal Hausdorff and Lyapunov dimensions

Let $μ$ be a self-affine measure on $\mathbb{R}^{d}$ associated to a self-affine IFS $\{φ_λ(x) = A_λx + v_λ\}_{λ\inΛ}$ and a probability vector $p=(p_λ)_λ>0$. Assume the strong separation condition holds. Let $γ_{1}\ge...\geγ_{d}$ and $D$ be the Lyapunov exponents and dimension corresponding to $\{A_λ\}_{λ\inΛ}$ and $p^{\mathbb{N}}$, and let $\mathbf{G}$ be the group generated by $\{A_λ\}_{λ\inΛ}$. We show that if $γ_{m+1}>γ_{m}=...=γ_{d}$, if $\mathbf{G}$ acts irreducibly on the vector space of alternating $m$-forms, and if the Furstenberg measure $μ_{F}$ satisfies $\dim_{H}μ_{F}+D>(m+1)(d-m)$, then $μ$ is exact dimensional with $\dimμ=D$.

math.DS↗