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Peter Burton

Publications and source records attributed to Peter Burton.

At least 19 recordsLinked to original sources
arXiv2024

Formulations of Furstenberg's $\times 2 \times 3$ conjecture in complex analysis and operator algebras

Furstenberg's $\times 2 \times 3$ conjecture has remained a central open problem in ergodic theory for over $50$ years, and it serves as the basic test case for a broad class of rigidity phenomena which are believed to hold in number-theoretic dynamics. More recently, two related statements have appeared in the literature: a question about periodic approximation raised by Levit and Vigdorovich in the context of approximate group theory and a periodic equidistribution conjecture formulated by Lindenstrauss. The purpose of this article is to provide equivalent formulations for these three statements in a complex-analytic setting and an operator-algebraic setting, giving nine conjectures grouped into three triples. The complex-analytic conjectures involve so-called Carath\'{e}odory functions on the unit disk that satisfy a certain functional identity, and we find that Furstenberg's conjecture is equivalent to the assertion that every such function is a convex combination of rational functions. The operator-algebraic conjectures involve tracial states on the full group $C^\ast$-algebra of a certain semidirect product, which is related to Baumslag-Solitar groups.

math.DS↗
arXiv2024

Computable Gelfand Duality

We establish a computable version of Gelfand Duality. Under this computable duality, computably compact presentations of metrizable spaces uniformly effectively correspond to computable presentations of unital commutative $C^*$ algebras.

math.LO↗
arXiv2023

Hyperlinear approximations to amenable groups come from sofic approximations

We provide a quantitative formulation of the equivalence between hyperlinearity and soficity for amenable groups, effectively showing how every hyperlinear approximation to such a group is simulated by a suitable sofic approximation. The proof is probabilistic, using the concentration of measure in high-dimensional spheres to control the deviation of an operator's matrix coefficients from its trace. As a corollary, we obtain a result connecting stability of sofic approximations with stability of hyperlinear approximations.

math.GR↗
arXiv2023

Synergodic actions of product groups

This article studies a structural aspect of measure-preserving actions of products of countable discrete groups, involving a so-called 'synergodic decomposition' in terms of the ergodic components of the actions of the two factor groups. We show that this construction provides a canonical way to detect whether the action is built as a product of actions on independent measure spaces, and we use it to prove a result about convergence of ergodic averages on product groups which are highly imbalanced between the factors. Defining an action to be synergodic if it is isomorphic to its synergodic decomposition, we show that if a countable group $G$ is amenable then every action of $G \times G$ can be approximated by synergodic actions and that this statement fails if $G$ is a nonabelian free group. The last result relies on the refutation of Connes' embedding conjecture.

math.DS↗
arXiv2023

The extension problem in free harmonic analysis

This paper studies certain aspects of harmonic analysis on nonabelian free groups. We focus on the concept of a positive definite function on the free group and our primary goal is to understand how such functions can be extended from balls of finite radius to the entire group. Previous work showed that such extensions always exist and we study the problem of simultaneous extension of multiple positive definite functions. More specifically, we define a concept of 'relative energy' which measures the proximity between a pair of positive definite functions, and show that a pair of positive definite functions on a finite ball can be extended to the entire group without increasing their relative energy. The proof is analytic, involving differentiation of noncommutative Szego parameters.

math.FA↗
arXiv2023

Statistical constructions in quantum information theory

We introduce a notion of strategies based on averaging for nonlocal games in quantum information theory. These so-called statistical strategies come in a commuting type and a more specific spatial type, which are respectively special cases of the quantum commuting and quantum spatial strategies commonly considered in the field. We prove a theorem that the sets of statistical commuting strategies and statistical spatial strategies are respectively equal to the sets of quantum commuting strategies and quantum spatial strategies for any nonlocal game. Thus we are able to use the recent negative solution of Tsirelson's problem to obtain a statistical analog showing that there exists a nonlocal game where the set of statistical commuting strategies properly contains the closure of the set of statistical spatial strategies. The proof of this theorem involves development of statistical replicas for numerous constructions in quantum information theory, in particular for the Fourier-type duality between observation structures and dynamical structures. The main point of the argument is to apply the established theory of approximating unitary representations of countable discrete groups by ergodic measure preserving actions of such groups. We note that the relevant groups are nonamenable. We also give an explicit description of a statistical strategy to win the CHSH game from Aspect's experiment with a probability exceeding the maximum possible value for a classical strategy.

quant-ph↗
arXiv2021

Hyperlinear approximations to amenable groups come from sofic approximations

We provide a quantitative formulation of the equivalence between hyperlinearity and soficity for amenable groups, showing that every hyperlinear approximation to such a group is essentially produced from a sofic approximation. This translates to a quantitative relationship between Hilbert-Schmidt and permutation stability for approximate homomorphisms which appropriately separate the elements of the group.

math.GR↗
arXiv2021

Existence of a zero-free strip for the Riemann zeta function

In a recent article a breakthrough was made in quantum computer science which established the existence of a phenomenon commonly known as $\mathsf{MIP}^\ast = \mathsf{RE}$. We show that the existence of this phenomenon implies that the supremum of the real parts of the zeroes of the Riemann zeta function is less than one. Our main tool is an assertion about the representation theory of the rank two free group that we refer to as the charged mean ergodic theorem. Our method also involves the construction of novel modular functions we refer to as complete electromagnetic functions.

math.NT↗
arXiv2019

Flexible stability and nonsoficity

A sofic group $G$ is said to be flexibly stable if every sofic approximation to $G$ can converted to a sequence of disjoint unions of Schreier graphs by modifying an asymptotically vanishing proportion of edges. We establish that if $\mathrm{PSL}_d(\mathbb{Z})$ is flexibly stable for some $d \geq 5$ then there exists a group which is not sofic.

math.GR↗
arXiv2016

Weak equivalence of stationary actions and the entropy realization problem

We introduce the notion of weak containment for stationary actions of a countable group and define a natural topology on the space of weak equivalence classes. We prove that Furstenberg entropy is an invariant of weak equivalence, and moreover that it descends to a continuous function on the space of weak equivalence classes.

math.DS↗
arXiv2016

Uniform mixing and completely positive sofic entropy

Let $G$ be a countable discrete sofic group. We define a concept of uniform mixing for measure-preserving $G$-actions and show that it implies completely positive sofic entropy. When $G$ contains an element of infinite order, we use this to produce an uncountable family of pairwise nonisomorphic $G$-actions with completely positive sofic entropy. None of our examples is a factor of a Bernoulli shift.

math.DS↗
arXiv2016

Completely positive entropy actions of sofic groups with $\mathbb{Z}$ in their center

Let $Γ$ be a sofic group with a copy of $\mathbb{Z}$ in its center. We construct an uncountable family of pairwise nonisomorphic measure-preserving $Γ$ actions with completely positive entropy, none of which is a factor of a Bernoulli shift. Our construction shows that the relation of isomorphism among completely positive entropy $Γ$ actions is not smooth, in contrast with the relation of isomorphism among Bernoulli shifts.

math.DS↗
arXiv2016

Naive entropy of dynamical systems

We study an invariant of dynamical systems called naive entropy, which is defined for both measurable and topological actions of any countable group. We focus on nonamenable groups, in which case the invariant is two-valued, with every system having naive entropy either zero or infinity. Bowen has conjectured that when the acting group is sofic, zero naive entropy implies sofic entropy at most zero for both types of systems. We prove the topological version of this conjecture by showing that for every action of a sofic group by homeomorphisms of a compact metric space, zero naive entropy implies sofic entropy at most zero. This result and the simple definition of naive entropy allow us to show that the generic action of a free group on the Cantor set has sofic entropy at most zero. We observe that a distal $Γ$-system has zero naive entropy in both senses, if $Γ$ has an element of infinite order. We also show that the naive entropy of a topological system is greater than or equal to the naive measure entropy of the same system with respect to any invariant measure.

math.DS↗
arXiv2016

Topology and convexity in the space of actions modulo weak equivalence

We analyse the structure of the quotient $\mathrm{A}_\sim(Γ,X,μ)$ of the space of measure-preserving actions of a countable discrete group by the relation of weak equivalence. This space carries a natural operation of convex combination. We show that the convex structure of $\mathrm{A}_\sim(Γ,X,μ)$ is compatible with the topology, and as a consequence deduce that $\mathrm{A}_\sim(Γ,X,μ)$ is path connected. Using ideas of Tucker-Drob we are able to give a complete description of the topological and convex structure of $\mathrm{A}_\sim(Γ,X,μ)$ for amenable $Γ$ by identifying it with the simplex of invariant random subgroups. In particular we conclude that $\mathrm{A}_\sim(Γ,X,μ)$ can be represented as a compact convex subset of a Banach space if and only if $Γ$ is amenable. We consider the space $\mathrm{A}_{\sim_s}(Γ,X,μ)$ of stable weak equivalence classes and show that is always a compact convex subset of a Banach space. For a free group $\mathbb{F}_N$, we show that if one restricts to the compact convex set $\mathrm{FR}_{\sim_s}(\mathbb{F}_N,X,μ) \subseteq \mathrm{A}_{\sim_s}(\mathbb{F}_N,X,μ)$ of the stable weak equivalence classes of free actions, the extreme points are dense in $\mathrm{FR}_{\sim_s}(\mathbb{F}_N,X,μ)$.

math.DS↗