arXiv subjects

Bryna Kra

Publications and source records attributed to Bryna Kra.

At least 37 records · Page 2Linked to original sources

arXiv2019

Realizing ergodic properties in zero entropy subshifts

A subshift with linear block complexity has at most countably many ergodic measures, and we continue of the study of the relation between such complexity and the invariant measures. By constructing minimal subshifts whose block complexity is arbitrarily close to linear but has uncountably many ergodic measures, we show that this behavior fails as soon as the block complexity is superlinear. With a different construction, we show that there exists a minimal subshift with an ergodic measure whose slow entropy grows slower than any given rate tending to infinitely but faster than any other rate majorizing this one yet still growing subexponentially. These constructions lead to obstructions in using subshifts in applications to properties of the prime numbers and in finding a measurable version of the complexity gap that arises for shifts of sublinear complexity.

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arXiv2018

Equidistribution of dilated curves on nilmanifolds

Generalizing classic results for a family of measures in the torus, for a family $(μ_t)_{t\geq 0}$ of measures defined on a nilmanifold $X$, we study conditions under which the family equidistributes, meaning conditions under which the measures $μ_t$ converge as $t\to\infty$ in the weak$^\ast$ topology to the Haar measure on $X$. We give general conditions on a family of measures defined by a dilation process, showing necessary and sufficient conditions for equidistribution as the family dilates, along with conditions such that this holds for all dilates outside some set of density zero. Furthermore, we show that these two types of equidistribution are different.

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arXiv2017

The automorphism group of a shift of slow growth is amenable

Suppose $(X,σ)$ is a subshift, $P_X(n)$ is the word complexity function of $X$, and ${\rm Aut}(X)$ is the group of automorphisms of $X$. We show that if $P_X(n)=o(n^2/\log^2 n)$, then ${\rm Aut}(X)$ is amenable (as a countable, discrete group). We further show that if $P_X(n)=o(n^2)$, then ${\rm Aut}(X)$ can never contain a nonabelian free semigroup (and, in particular, can never contain a nonabelian free subgroup). This is in contrast to recent examples, due to Salo and Schraudner, of subshifts with quadratic complexity that do contain such a semigroup.

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arXiv2017

The spacetime of a shift endomorphism

The automorphism group of a one dimensional shift space over a finite alphabet exhibits different types of behavior: for a large class with positive entropy, it contains a rich collection of subgroups, while for many shifts of zero entropy, there are strong constraints on the automorphism group. We view this from a different perspective, considering a single automorphism (and sometimes endomorphism) and studying the naturally associated two dimensional shift system. In particular, we describe the relation between nonexpansive subspaces in this two dimensional system and dynamical properties of an automorphism of the shift.

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arXiv2017

Distortion and the automorphism group of a shift

The set of automorphisms of a one-dimensional \shift $(X, σ)$ forms a countable, but often very complicated, group. For zero entropy shifts, it has recently been shown that the automorphism group is more tame. We provide the first examples of countable groups that cannot embed into the automorphism group of any zero entropy \shiftno. In particular, we show that the Baumslag-Solitar groups ${\rm BS}(1,n)$ and all other groups that contain exponentially distorted elements cannot embed into ${\rm Aut}(X)$ when $h_{\rm top}(X)=0$. We further show that distortion in nilpotent groups gives a nontrivial obstruction to embedding such a group in any low complexity shift.

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arXiv2016

Free ergodic $\mathbb{Z}^2$-systems and complexity

Using results relating the complexity of a two dimensional subshift to its periodicity, we obtain an application to the well-known conjecture of Furstenberg on a Borel probability measure on $[0,1)$ which is invariant under both $x\mapsto px \pmod 1$ and $x\mapsto qx \pmod 1$, showing that any potential counterexample has a nontrivial lower bound on its complexity.

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arXiv2015

The automorphism group of a minimal shift of stretched exponential growth

The group of automorphisms of a symbolic dynamical system is countable, but often very large. For example, for a mixing subshift of finite type, the automorphism group contains isomorphic copies of the free group on two generators and the direct sum of countably many copies of $\mathbb{Z}$. In contrast, the group of automorphisms of a symbolic system of zero entropy seems to be highly constrained. Our main result is that the automorphism group of any minimal subshift of stretched exponential growth with exponent $<1/2$, is amenable (as a countable discrete group). For shifts of polynomial growth, we further show that any finitely generated, torsion free subgroup of Aut(X) is virtually nilpotent.

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arXiv2015

Counting generic measures for a subshift of linear growth

In 1984 Boshernitzan proved an upper bound on the number of ergodic measures for a minimal subshift of linear block growth and asked if it could be lowered without further assumptions on the shift. We answer this question, showing that Boshernitzan's bound is sharp. We further prove that the same bound holds for the, a priori, larger set of nonatomic generic measures, and that this bound remains valid even if one drops the assumption of minimality. Applying these results to interval exchange transformations, we give an upper bound on the number of nonatomic generic measures of a minimal IET, answering a question recently posed by Chaika and Masur.

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arXiv2014

The automorphism group of a shift of linear growth: beyond transitivity

For a finite alphabet $\mathcal{A}$ and shift $X\subseteq\mathcal{A}^{\mathbb{Z}}$ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group ${\rm Aut}(X)$. For such systems, we show that every finitely generated subgroup of ${\rm Aut}(X)$ is virtually ${\mathbb Z}^d$, in contrast to the behavior when the complexity function grows more quickly. With additional dynamical assumptions we show more: if $X$ is transitive, then ${\rm Aut}(X)$ is virtually $\mathbb Z$; if $X$ has dense aperiodic points, then ${\rm Aut}(X)$ is virtually ${\mathbb Z}^d$. We also classify all finite groups that arise as the automorphism group of a shift.

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arXiv2014

Complexity and directional entropy in two dimensions

We study the directional entropy of the dynamical system associated to a $\Z^2$ configuration in a finite alphabet. We show that under local assumptions on the complexity, either every direction has zero topological entropy or some direction is periodic. In particular, we show that all nonexpansive directions in a $\Z^2$ system with the same local assumptions have zero directional entropy.

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arXiv2014

Variations on topological recurrence

Recurrence properties of systems and associated sets of integers that suffice for recurrence are classical objects in topological dynamics. We describe relations between recurrence in different sorts of systems, study ways to formulate finite versions of recurrence, and describe connections to combinatorial problems. In particular, we show that sets of Bohr recurrence (meaning sets of recurrence for rotations) suffice for recurrence in nilsystems. Additionally, we prove an extension of this property for multiple recurrence in affine systems.

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arXiv2014

Rudolph's Two-Step Coding Theorem and Alpern's Lemma for R^d Actions

Rudolph showed that the orbits of any measurable, measure preserving $\mathbb R^d$ action can be measurably tiled by $2^d$ rectangles and asked if this number of tiles is optimal for $d>1$. In this paper, using a tiling of $\mathbb R^d$ by notched cubes, we show that $d+1$ tiles suffice. Furthermore, using a detailed analysis of the set of invariant measures on tilings of $\mathbb R^2$ by two rectangles, we show that while for $\mathbb R^2$ actions with completely positive entropy this bound is optimal there exist mixing $\mathbb R^2$ actions whose orbits can be tiled by 2 tiles.

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arXiv2014

The automorphism group of a shift of subquadratic growth

For a subshift over a finite alphabet, a measure of the complexity of the system is obtained by counting the number of nonempty cylinder sets of length $n$. When this complexity grows exponentially, the automorphism group has been shown to be large for various classes of subshifts. In contrast, we show that subquadratic growth of the complexity implies that for a topologically transitive shift $X$, the automorphism group $\Aut(X)$ is small: if $H$ is the subgroup of $\Aut(X)$ generated by the shift, then $\Aut(X)/H$ is periodic.

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arXiv2013

Nonexpansive Z^2 subdynamics and Nivat's conjecture

For a finite alphabet $\A$ and $η\colon \Z\to\A$, the Morse-Hedlund Theorem states that $η$ is periodic if and only if there exists $n\in\N$ such that the block complexity function $P_η(n)$ satisfies $P_η(n)\leq n$, and this statement is naturally studied by analyzing the dynamics of a $\Z$-action associated to $η$. In dimension two, we analyze the subdynamics of a $\ZZ$-action associated to $η\colon\ZZ\to\A$ and show that if there exist $n,k\in\N$ such that the $n\times k$ rectangular complexity $P_η(n,k)$ satisfies $P_η(n,k)\leq nk$, then the periodicity of $η$ is equivalent to a statement about the expansive subspaces of this action. As a corollary, we show that if there exist $n,k\in\N$ such that $P_η(n,k)\leq \frac{nk}{2}$, then $η$ is periodic. This proves a weak form of a conjecture of Nivat in the combinatorics of words.

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arXiv2013

Complexity of short rectangles and periodicity

The Morse-Hedlund Theorem states that a bi-infinite sequence $η$ in a finite alphabet is periodic if and only if there exists $n\in\N$ such that the block complexity function $P_η(n)$ satisfies $P_η(n)\leq n$. In dimension two, Nivat conjectured that if there exist $n,k\in\N$ such that the $n\times k$ rectangular complexity $P_η(n,k)$ satisfies $P_η(n,k)\leq nk$, then $η$ is periodic. Sander and Tijdeman showed that this holds for $k\leq2$. We generalize their result, showing that Nivat's Conjecture holds for $k\leq3$. The method involves translating the combinatorial problem to a question about the nonexpansive subspaces of a certain $\ZZ$ dynamical system, and then analyzing the resulting system.

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arXiv2012

Complexity of Nilsystems and systems lacking nilfactors

Nilsystems are a natural generalization of rotations and arise in various contexts, including in the study of multiple ergodic averages in ergodic theory, in the structural analysis of topological dynamical systems, and in asymptotics for patterns in certain subsets of the integers. We show, however, that many natural classes in both measure preserving systems and topological dynamical systems contain no higher order nilsystems as factors, meaning that the only nilsystems they contain as factors are rotations. In the ergodic setting, we show that there are spectral obstructions that give rise to this behavior. In the topological setting, nilsystems have a particular type of complexity of polynomial growth, where the polynomial (with explicit degree) is an asymptotic both from below and above. We also deduce several ergodic and topological applications of these results.

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arXiv2011

The polynomial multidimensional Szemerédi Theorem along shifted primes

If $\vf_1, ... \vf_m\colon\Z\to\Z^\ell$ are polynomials with zero constant terms and $E\subset\Z^\ell$ has positive upper Banach density, then we show that the set $E\cap (E-\vf_1(p-1))\cap\...\cap (E-\vf_m(p-1))$ is nonempty for some prime $p$. We also prove mean convergence for the associated averages along the prime numbers, conditional to analogous convergence results along the full integers. This generalizes earlier results of the authors, of Wooley and Ziegler, and of Bergelson, Leibman and Ziegler.

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arXiv2010

A point of view on Gowers uniformity norms

Gowers norms have been studied extensively both in the direct sense, starting with a function and understanding the associated norm, and in the inverse sense, starting with the norm and deducing properties of the function. Instead of focusing on the norms themselves, we study associated dual norms and dual functions. Combining this study with a variant of the Szemeredi Regularity Lemma, we give a decomposition theorem for dual functions, linking the dual norms to classical norms and indicating that the dual norm is easier to understand than the norm itself. Using the dual functions, we introduce higher order algebras that are analogs of the classical Fourier algebra, which in turn can be used to further characterize the dual functions.

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