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Michael Hochman

Publications and source records attributed to Michael Hochman.

At least 19 recordsLinked to original sources
arXiv2025

A threshold for Poisson behavior of non-stationary product measures

Let $\gamma_{n}= O (\log^{-c}n)$ and let $\nu$ be the infinite product measure whose $n$-th marginal is Bernoulli$(1/2+\gamma_{n})$. We show that $c=1/2$ is the threshold, above which $\nu$-almost every point is simply Poisson generic in the sense of Peres-Weiss, and below which this can fail. This provides a range in which $\nu$ is singular with respect to the uniform product measure, but $\nu$-almost every point is simply Poisson generic.

math.DS↗
arXiv2024

On the dimension of $\alpha \beta$-sets

We show that the Feng-Xiong lower bound of $1/2$ for the box dimension of $\alpha\beta$-sets is tight. We also study how much of an $\alpha\beta$-orbit ``carries the dimension'': deleting an arbitararily small positive density set of times can cause the box dimension to drop to zero, but the Assouad dimension cannot drop below $1/4$.

math.DS↗
arXiv2024

New results on embeddings of self-similar sets via renormalization

For self-similar sets $X,Y\subseteq \mathbb{R}$, we obtain new results towards the affine embeddings conjecture of Feng-Huang-Rao (2014), and the equivalent weak intersections conjecture. We show that the conjecture holds when the defining maps of $X,Y$ have algebraic contraction ratios, and also for arbitrary $Y$ when the maps defining $X$ have algebraic-contraction ratios and there are sufficiently many of them relative to the number of maps defining $Y$. We also show that it holds for a.e. choice of the corresponding contraction ratios, and obtain bounds on the packing dimension of the exceptional parameters. Key ingredients in our argument include a new renormalization procedure in the space of embeddings, and Orponen's projection Theorem for Assouad dimension (2021).

math.DS↗
arXiv2023

Some remarks about self-products and entropy

Let $(X,\mathcal{B},\mu,T)$ be a probability-preserving system with $X$ compact and $T$ a homeomorphism. We show that if every point in $X\times X$ is two-sided recurrent, then $h_{\mu}(T)=0$, resolving a problem of Benjamin Weiss, and that if $h_{\mu}(T)=\infty$ then every full-measure set in $X$ contains mean-asymptotic pairs (i.e. the associated process is not tight), resolving a problem of Ornstein and Weiss.

math.DS↗
arXiv2023

A short proof of Host's equidistribution theorem

This note contains a new proof of Host's equidistribution theorem for multiplicatively independent endomorphisms of $\mathbb{R}/\mathbb{Z}$. The method is a simplified version of our upcoming work on equidistribution under toral automorphisms and is related to the argument in [HochmanShmerkin2015], but avoids the use of the scenery flow and of Marstrand's projection theorem, using instead a direct Fourier argument to establish smoothness of the limit measure.

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

Smooth symmetries of $\times a$-invariant sets

We study the smooth self-maps $f$ of $\times a$-invariant sets $X\subseteq[0,1]$. Under various assumptions we show that this forces $\log f'(x)/\log a\in\mathbb{Q}$ at many points in $X$. Our method combines scenery flow methods and equidistribution results in the positive entropy case, where we improve previous work of the author and Shmerkin, with a new topological variant of the scenery flow which applies in the zero-entropy case.

math.DS↗
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

On the dimension of Furstenberg measure for $SL_2(R)$ random matrix products

Let $μ$ be a measure on $SL_{2}(\mathbb{R})$ generating a non-compact and totally irreducible subgroup, let $χ>0$ denote its Lyapunov exponent, and let $ν$ be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that if $μ$ is supported on finitely many matrices with algebraic entries, then \[ \dimν=\min\{1,\frac{h_{\textrm{RW}}(μ)}{2χ}\} \] where $h_{\textrm{RW}}(μ)$ is the random walk entropy of $μ$, and $\dim$ denotes pointwise dimension. In particular, for every $δ>0$, there is a neighborhood $U$ of the identity in $SL_{2}(\mathbb{R})$ such that if a measure $μ\in\mathcal{P}(U)$ is supported on algebraic matrices with all atoms of size at least $δ$, and generates a group which is non-compact and totally irreducible, then its stationary measure $ν$ satisfies $\dimν=1$.

math.DS↗
arXiv2017

On self-similar sets with overlaps and inverse theorems for entropy in $\mathbb{R}^d$

We study self-similar sets and measures on $\mathbb{R}^{d}$. Assuming that the defining iterated function system $Φ$ does not preserve a proper affine subspace, we show that one of the following holds: (1) the dimension is equal to the trivial bound (the minimum of $d$ and the similarity dimension $s$); (2) for all large $n$ there are $n$-fold compositions of maps from $Φ$ which are super-exponentially close in $n$; (3) there is a non-trivial linear subspace of $\mathbb{R}^{d}$ that is preserved by the linearization of $Φ$ and whose translates typically meet the set or measure in full dimension. In particular, when the linearization of $Φ$ acts irreducibly on $\mathbb{R}^{d}$, either the dimension is equal to $\min\{s,d\}$ or there are super-exponentially close $n$-fold compositions. We give a number of applications to algebraic systems, parametrized systems, and to some classical examples. The main ingredient in the proof is an inverse theorem for the entropy growth of convolutions of measures on $\mathbb{R}^{d}$, and the growth of entropy for the convolution of a measure on the orthogonal group with a measure on $\mathbb{R}^{d}$. More generally, this part of the paper applies to smooth actions of Lie groups on manifolds.

math.CA↗
arXiv2017

Every Borel automorphism without finite invariant measures admits a two-set generator

We show that if an automorphism of a standard Borel space does not admit finite invariant measures, then it has a two-set generator modulo the sigma-ideal generated by wandering sets. This implies that if the entropies of invariant probability measures of a Borel system are all less than log(k), then the system admits a k-set generator, and that a wide class of hyperbolic-like systems are classified completely at the Borel level by entropy and periodic points counts.

math.DS↗
arXiv2017

Some problems on the boundary of fractal geometry and additive combinatorics

This paper is an exposition, with some new applications, of our results on the growth of entropy of convolutions. We explain the main result on $\mathbb{R}$, and derive, via a linearization argument, an analogous result for the action of the affine group on $\mathbb{R}$. We also develop versions of the results for entropy dimension and Hausdorff dimension. The method is applied to two problems on the border of fractal geometry and additive combinatorics. First, we consider attractors $X$ of compact families $Φ$ of similarities of $\mathbb{R}$. We conjecture that if $Φ$ is uncountable and $X$ is not a singleton (equivalently, $Φ$ is not contained in a 1-parameter semigroup) then $\dim X=1$. We show that this would follow from the classical overlaps conjecture for self-similar sets, and unconditionally we show that if $X$ is not a point and $\dimΦ>0$ then $\dim X=1$. Second, we study a problem due to Shmerkin and Keleti, who have asked how small a set $\emptyset\neq Y\subseteq\mathbb{R}$ can be if at every point it contains a scaled copy of the middle-third Cantor set $K$. Such a set must have dimension at least $\dim K$ and we show that its dimension is at least $\dim K+δ$ for some constant $δ>0$.

math.DS↗
arXiv2016

Self embeddings of Bedford-McMullen carpets

Let $F \subseteq \mathbb{R}^2$ be a Bedford-McMullen carpet defined by multiplicatively independent exponents, and suppose that either $F$ is not a product set, or it is a product set with marginals of dimension strictly between $0$ and $1$. We prove that any similarity $g$ such that $g(F) \subseteq F$ is an isometry composed of reflections about lines parallel to the axes. Our approach utilizes the structure of tangent sets of $F$, obtained by "zooming in" on points of $F$, projection theorems for products of self-similar sets, and logarithmic commensurability type results for self similar sets in the line.

math.DS↗
arXiv2014

Equidistribution from Fractals

We give a fractal-geometric condition for a measure on [0,1] to be supported on points x that are normal in base n, i.e. such that the sequence x,nx,n^2 x,... equidistributes modulo 1. This condition is robust under C^1 coordinate changes, and it applies also when n is a Pisot number and equidistribution is understood with respect to the beta-map and Parry measure. As applications we obtain new results (and strengthen old ones) about the prevalence of normal numbers in fractal sets, and new results on measure rigidity, specifically completing Host's theorem to multiplicatively independent integers and proving a Rudolph-Johnson-type theorem for certain pairs of beta transformations.

math.DS↗
arXiv2014

Turing degree spectra of minimal subshifts

Subshifts are shift invariant closed subsets of $Σ^{\mathbb{Z}^d}$ , minimal subshifts are subshifts in which all points contain the same patterns. It has been proved by Jeandel and Vanier that the Turing degree spectra of non-periodic minimal subshifts always contain the cone of Turing degrees above any of its degree. It was however not known whether each minimal subshift's spectrum was formed of exactly one cone or not. We construct inductively a minimal subshift whose spectrum consists of an uncountable number of cones with disjoint base.

cs.FL↗
arXiv2014

Self similar sets, entropy and additive combinatorics

This article is an exposition of recent results on self-similar sets, asserting that if the dimension is smaller than the trivial upper bound then there are almost overlaps between cylinders. We give a heuristic derivation of the theorem using elementary arguments about covering numbers. We also give a short introduction to additive combinatorics, focusing on inverse theorems, which play a pivotal role in the proof. Our elementary approach avoids many of the technicalities in the original proof but also falls short of a complete proof. In the last section we discuss how the heuristic argument is turned into a rigorous one.

math.CA↗