Isomorphism and embedding of Borel systems on full sets

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 have measure zero for every invariant probability measure. For every t>0 we show that in this category there exists a unique free Borel system (Y,S) which is strictly t-universal in the sense that all invariant measures on Y have entropy <t, and if (X,T) is another free system obeying the same entropy condition then X embeds into Y off a null set. One gets a strictly t-universal system from mixing shifts of finite type of entropy at least t by removing the periodic points and "restricting" to the part of the system of entropy <t. As a consequence, after removing their periodic points the systems in the following classes are completely classified by entropy up to Borel isomorphism off null sets: mixing shifts of finite type, mixing positive-recurrent countable state Markov chains, mixing sofic shifts, beta shifts, synchronized subshifts, and axiom-A diffeomorphisms. In particular any two equal-entropy systems from these classes are entropy conjugate in the sense of Buzzi, answering a question of Boyle, Buzzi and Gomez.Comment: 17 pages, v2: correction to bibliograph

On Factor Universality in Symbolic Spaces

The study of factoring relations between subshifts or cellular automata is central in symbolic dynamics. Besides, a notion of intrinsic universality for cellular automata based on an operation of rescaling is receiving more and more attention in the literature. In this paper, we propose to study the factoring relation up to rescalings, and ask for the existence of universal objects for that simulation relation. In classical simulations of a system S by a system T, the simulation takes place on a specific subset of configurations of T depending on S (this is the case for intrinsic universality). Our setting, however, asks for every configurations of T to have a meaningful interpretation in S. Despite this strong requirement, we show that there exists a cellular automaton able to simulate any other in a large class containing arbitrarily complex ones. We also consider the case of subshifts and, using arguments from recursion theory, we give negative results about the existence of universal objects in some classes

On the zero-temperature limit of Gibbs states

We exhibit Lipschitz (and hence H\"older) potentials on the full shift $\{0,1\}^{\mathbb{N}}$ such that the associated Gibbs measures fail to converge as the temperature goes to zero. Thus there are "exponentially decaying" interactions on the configuration space $\{0,1\}^{\mathbb Z}$ for which the zero-temperature limit of the associated Gibbs measures does not exist. In higher dimension, namely on the configuration space $\{0,1\}^{\mathbb{Z}^{d}}$, $d\geq3$, we show that this non-convergence behavior can occur for finite-range interactions, that is, for locally constant potentials.Comment: The statement of Theorem 1.2 is more accurate and some new comment follow i

Subshifts, MSO Logic, and Collapsing Hierarchies

We use monadic second-order logic to define two-dimensional subshifts, or sets of colorings of the infinite plane. We present a natural family of quantifier alternation hierarchies, and show that they all collapse to the third level. In particular, this solves an open problem of [Jeandel & Theyssier 2013]. The results are in stark contrast with picture languages, where such hierarchies are usually infinite.Comment: 12 pages, 5 figures. To appear in conference proceedings of TCS 2014, published by Springe

Online Popularity and Topical Interests through the Lens of Instagram

Online socio-technical systems can be studied as proxy of the real world to investigate human behavior and social interactions at scale. Here we focus on Instagram, a media-sharing online platform whose popularity has been rising up to gathering hundred millions users. Instagram exhibits a mixture of features including social structure, social tagging and media sharing. The network of social interactions among users models various dynamics including follower/followee relations and users' communication by means of posts/comments. Users can upload and tag media such as photos and pictures, and they can "like" and comment each piece of information on the platform. In this work we investigate three major aspects on our Instagram dataset: (i) the structural characteristics of its network of heterogeneous interactions, to unveil the emergence of self organization and topically-induced community structure; (ii) the dynamics of content production and consumption, to understand how global trends and popular users emerge; (iii) the behavior of users labeling media with tags, to determine how they devote their attention and to explore the variety of their topical interests. Our analysis provides clues to understand human behavior dynamics on socio-technical systems, specifically users and content popularity, the mechanisms of users' interactions in online environments and how collective trends emerge from individuals' topical interests.Comment: 11 pages, 11 figures, Proceedings of ACM Hypertext 201

The Assouad dimensions of projections of planar sets

We consider the Assouad dimensions of orthogonal projections of planar sets onto lines. Our investigation covers both general and self-similar sets. For general sets, the main result is the following: if a set in the plane has Assouad dimension $s \in [0,2]$, then the projections have Assouad dimension at least $\min\{1,s\}$ almost surely. Compared to the famous analogue for Hausdorff dimension -- namely \emph{Marstrand's Projection Theorem} -- a striking difference is that the words `at least' cannot be dispensed with: in fact, for many planar self-similar sets of dimension $s < 1$, we prove that the Assouad dimension of projections can attain both values $s$ and $1$ for a set of directions of positive measure. For self-similar sets, our investigation splits naturally into two cases: when the group of rotations is discrete, and when it is dense. In the `discrete rotations' case we prove the following dichotomy for any given projection: either the Hausdorff measure is positive in the Hausdorff dimension, in which case the Hausdorff and Assouad dimensions coincide; or the Hausdorff measure is zero in the Hausdorff dimension, in which case the Assouad dimension is equal to 1. In the `dense rotations' case we prove that every projection has Assouad dimension equal to one, assuming that the planar set is not a singleton. As another application of our results, we show that there is no \emph{Falconer's Theorem} for Assouad dimension. More precisely, the Assouad dimension of a self-similar (or self-affine) set is not in general almost surely constant when one randomises the translation vectors.Comment: 29 pages, 2 figures, to appear in Proc. Lond. Math. So

Return times, recurrence densities and entropy for actions of some discrete amenable groups

Results of Wyner and Ziv and of Ornstein and Weiss show that if one observes the first k outputs of a finite-valued ergodic process, then the waiting time until this block appears again is almost surely asymptotic to $2^{hk}$, where $h$ is the entropy of the process. We examine this phenomenon when the allowed return times are restricted to some subset of times, and generalize the results to processes parameterized by other discrete amenable groups. We also obtain a uniform density version of the waiting time results: For a process on $s$ symbols, within a given realization, the density of the initial $k$-block within larger $n$-blocks approaches $2^{-hk}$, uniformly in $n>s^k$, as $k$ tends to infinity. Again, similar results hold for processes with other indexing groups.Comment: To appear in Journal d'Analyse Mathematiqu

Small union with large set of centers

Let $T\subset{\mathbb R}^n$ be a fixed set. By a scaled copy of $T$ around $x\in{\mathbb R}^n$ we mean a set of the form $x+rT$ for some $r>0$. In this survey paper we study results about the following type of problems: How small can a set be if it contains a scaled copy of $T$ around every point of a set of given size? We will consider the cases when $T$ is circle or sphere centered at the origin, Cantor set in ${\mathbb R}$, the boundary of a square centered at the origin, or more generally the $k$-skeleton ($0\le k<n$) of an $n$-dimensional cube centered at the origin or the $k$-skeleton of a more general polytope of ${\mathbb R}^n$. We also study the case when we allow not only scaled copies but also scaled and rotated copies and also the case when we allow only rotated copies