274 citations · 904 across the 27 of their papers we have counts for
34 papers
Dense times-b orbits from times-a invariant sets and measures, in zero entropy
Michael Hochman
Let be multiplicatively independent integers. We prove that if is a zero-entropy non-atomic -invariant measure on , then -a.e.~ has dense $\times b…
A short proof of Host's equidistribution theorem
Michael Hochman
This note contains a new proof of Host's equidistribution theorem for multiplicatively independent endomorphisms of . The method is a simplified version of o…
Hausdorff Dimension of Planar Self-Affine Sets and Measures with Overlaps
Michael Hochman, Ariel Rapaport
We prove that if is a self-affine measure in the plane whose defining IFS acts totally irreducibly on and satisfies an exponential separation condition, then it…
Hausdorff dimension of planar self-affine sets and measures
Balázs Bárány, Michael Hochman, Ariel Rapaport
Let be a strongly separated self-affine set in (or one satisfying the strong open set condition). Under mild non-compactness and irreducibility ass…
Smooth symmetries of -invariant sets
Michael Hochman
We study the smooth self-maps of -invariant sets . Under various assumptions we show that this forces at many points…
On the dimension of Furstenberg measure for random matrix products
Michael Hochman, Boris Solomyak
Let be a measure on generating a non-compact and totally irreducible subgroup, let denote its Lyapunov exponent, and let be the associated statio…
Some problems on the boundary of fractal geometry and additive combinatorics
Michael Hochman
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 , and derive, via a line…
Self embeddings of Bedford-McMullen carpets
Amir Algom, Michael Hochman
Let be a Bedford-McMullen carpet defined by multiplicatively independent exponents, and suppose that either is not a product set, or it is a product…
Every Borel automorphism without finite invariant measures admits a two-set generator
Michael Hochman
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 se…
Turing degree spectra of minimal subshifts
Michael Hochman, Pascal Vanier
Subshifts are shift invariant closed subsets of , minimal subshifts are subshifts in which all points contain the same patterns. It has been proved by Jeandel an…
Self similar sets, entropy and additive combinatorics
Michael Hochman
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…
Equidistribution from Fractals
Michael Hochman, Pablo Shmerkin
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 modul…
On self-similar sets with overlaps and inverse theorems for entropy
Michael Hochman
We study the Hausdorff dimension of self-similar sets and measures on the line. We show that if the dimension is smaller than the minimum of 1 and the similarity dimension, then at…
On the ratio ergodic theorem for group actions
Michael Hochman
We study the ratio ergodic theorem (RET) of Hopf for group actions. Under a certain technical condition, if a sequence of sets {F_n} in a group satisfy the RET, then there is a fin…
Slow entropy and differentiable models for infinite-measure preserving Z^k actions
Michael Hochman
We define "slow" entropy invariants for Z^2 actions on infinite measure spaces, which measures growth of itineraries at subexponential scales. We use this to construct infinite-mea…
On the finite-dimensional marginals of shift-invariant measures
J. -R. Chazottes, J. -M. Gambaudo, M. Hochman +1
Let be a finite alphabet, equipped with the shift action, and the simplex of shift-invariant measures on . We study the relation between…
Inverses, Powers and Cartesian products of topologically deterministic maps
Michael Hochman, Artur Siemaszko
We show that if (X,T) is a topological dynamical system with is deterministic in the sense of Kamiski, Siemaszko and Szymaski then T^{-1} and the product system need not be determi…
Examples of nonpolygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models
Michael Damron, Michael Hochman
We construct an edge-weight distribution for i.i.d. first-passage percolation on whose limit shape is not a polygon and whose extreme points are arbitrarily dense in…
Geometric rigidity of times-m invariant measures
Michael Hochman
Let b be an integer and mu a probability measure on [0,1] which is invariant and ergodic multiplication by b mod 1, and 0<dim(mu)<1. Let f be a diffeomorphism between open subsets…
Dynamics on fractals and fractal distributions
Michael Hochman
We study fractal measures on Euclidean space through the dynamics of "zooming in" on typical points. The resulting family of measures (the "scenery"), can be interpreted as an orbi…
Isomorphism and embedding of Borel systems on full sets
Michael Hochman
A Borel system consists of a measurable automorphism of a standard Borel space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which…
Local entropy averages and projections of fractal measures
Michael Hochman, Pablo Shmerkin
We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear ima…
On the zero-temperature limit of Gibbs states
J. -R. Chazottes, M. Hochman
We exhibit Lipschitz (and hence Hölder) potentials on the full shift such that the associated Gibbs measures fail to converge as the temperature goes to zero…
Non-expansive directions for -actions
Michael Hochman
We show that any direction in the plane occurs as the unique non-expansive direction of a \mathbb{Z}^{2} action, answering a question of Boyle and Lind. In the case of rational dir…