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Peter Burton

Publications and source records attributed to Peter Burton.

22 records · Page 2Linked to original sources
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↗
arXiv2015

A quotient-like construction involving elementary submodels

This article is an investigation of a method of deriving a topology from a space and an elementary submodel containing it. We first define and give the basic properties of this construction, known as $X/M$. In the next section, we construct some examples and analyse the topological relationship between $X$ and $X/M$. In the final section, we apply $X/M$ to get novel results about Lindelöf spaces, giving partial answers to a question of F.D. Tall and another question of Tall and M. Scheepers.

math.LO↗