New examples of Bernoulli algebraic actions

We give an example of a principal algebraic action of the non-commutative free group ${\mathbb {F}}$ of rank two by automorphisms of a connected compact abelian group for which there is an explicit measurable isomorphism with the full Bernoulli 3-shift action of ${\mathbb {F}}$ . The isomorphism is defined using homoclinic points, a method that has been used to construct symbolic covers of algebraic actions. To our knowledge, this is the first example of a Bernoulli algebraic action of ${\mathbb {F}}$ without an obvious independent generator. Our methods can be generalized to a large class of acting groups.

On Baire measurable colorings of group actions

The field of descriptive combinatorics investigates to what extent classical combinatorial results and techniques can be made topologically or measure-theoretically well behaved. This paper examines a class of coloring problems induced by actions of countable groups on Polish spaces, with the requirement that the desired coloring be Baire measurable. We show that the set of all such coloring problems that admit a Baire measurable solution for a particular free action $\unicode[STIX]{x1D6FC}$ is complete analytic (apart from the trivial situation when the orbit equivalence relation induced by $\unicode[STIX]{x1D6FC}$ is smooth on a comeager set); this result confirms the ‘hardness’ of finding a topologically well-behaved coloring. When $\unicode[STIX]{x1D6FC}$ is the shift action, we characterize the class of problems for which $\unicode[STIX]{x1D6FC}$ has a Baire measurable coloring in purely combinatorial terms; it turns out that closely related concepts have already been studied in graph theory with no relation to descriptive set theory. We remark that our framework permits a wholly dynamical interpretation (with colorings corresponding to equivariant maps to a given subshift), so this article can also be viewed as a contribution to generic dynamics.

Cyclically presented groups with length four positive relators

We classify cyclically presented groups of the form {G=G_{n}(x_{0}x_{j}x_{k}x_{l})} for finiteness and, modulo two unresolved cases, we classify asphericity for the underlying presentations. We relate finiteness and asphericity to the dynamics of the shift action by the cyclic group of order n on the nonidentity elements of G and show that the fixed point subgroup of the shift is always finite.

Uniformity, universality, and computability theory

We prove a number of results motivated by global questions of uniformity in computabi- lity theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal. We also show the existence of refinements of Martin’s ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations.

Thermodynamic formalism for a modified shift map

We introduce a transfer operator and use it to prove some theorems of a classical flavor from thermodynamic formalism (including existence and uniqueness of appropriately defined Gibbs states and equilibrium states for potential functions satisfying Dini's condition and stochastic laws for Hölder continuous potential and observable functions) in a novel setting: the "alphabet" E is a compact metric space equipped with an a priori probability measure ν and an endomorphism T. The "modified shift map" S is defined on the product space Eℕ by the rule (x1x2x3…) ↦ (T(x2)x3…). The greatest novelty is found in the variational principle, where a term must be added to the entropy to reflect the transformation of the first coordinate by T after shifting. Our motivation is that this system, in its full generality, cannot be treated by the existing methods of either rigorous statistical mechanics of lattice gases (where only the true shift action is used) or dynamical systems theory (where the a priori measure is always implicitly taken to be the counting measure).

On the finite-dimensional marginals of shift-invariant measures

Let Σ be a finite alphabet, Ω=Σℤdequipped with the shift action, and ℐ the simplex of shift-invariant measures on Ω. We study the relation between the restriction ℐnof ℐ to the finite cubes {−n,…,n}d⊂ℤd, and the polytope of ‘locally invariant’ measures ℐlocn. We are especially interested in the geometry of the convex set ℐn, which turns out to be strikingly different whend=1 and whend≥2 . A major role is played by shifts of finite type which are naturally identified with faces of ℐn, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of ℐn, although in dimensiond≥2 there are also extreme points which arise in other ways. We show that ℐn=ℐlocnwhend=1 , but in higher dimensions they differ fornlarge enough. We also show that while in dimension one ℐnare polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of ℐnfor all large enoughn.

On the automorphism groups of multidimensional shifts of finite type

We investigate algebraic properties of the automorphism group of multidimensional shifts of finite type (SFTs). We show that positive entropy implies that the automorphism group contains every finite group and, together with transitivity, implies that the center of the automorphism group is trivial (i.e. consists only of the shift action). We also show that positive entropy and dense minimal points (in particular, dense periodic points) imply that the automorphism group of X contains a copy of the automorphism group of the one-dimensional full shift, and hence contains non-trivial elements of infinite order. On the other hand we construct a mixing, positive-entropy SFT whose automorphism group is, modulo the shift action, a union of finite groups.

A coloring property for countable groups

Motivated by research on hyperfinite equivalence relations we define a coloring property for countable groups. We prove that every countable group has the coloring property. This implies a compactness theorem for closed complete sections of the free part of the shift action ofGon 2G. Our theorems generalize known results about.