Borel and countably determined reducibility in nonstandard domain

We consider reducibility of equivalence relations (ERs, for brevity), in a nonstandard domain, in terms of the Borel reducibility and the countably determined (CD, for brevity) reducibility. This reveals phenomena partially analogous to those discovered in descriptive set theory. The Borel reducibility structure of Borel sets and (partially) CD reducibility structure of CD sets in *N is described. We prove that all CD ERs with countable equivalence classes are CD-smooth, but not all are B-smooth, for instance, the ER of having finite difference on *N. Similarly to the Silver dichotomy theorem in Polish spaces, any CD ER on *N either has at most continuum-many classes or there is an infinite internal set of pairwise inequivalent elements. Our study of monadic ERs on *N, i.e., those of the form x E y iff |x-y| belongs to a given additive Borel cut in *N, shows that these ERs split in two linearly families, associated with countably cofinal and countably coinitial cuts, each of which is linearly ordered by Borel reducibility. The relationship between monadic ERs and the ER of finite symmetric difference on hyperfinite subsets of *N is studied.Comment: 34 page

The rapid points of a complex oscillation

By considering a counting-type argument on Brownian sample paths, we prove a result similar to that of Orey and Taylor on the exact Hausdorff dimension of the rapid points of Brownian motion. Because of the nature of the proof we can then apply the concepts to so-called complex oscillations (or 'algorithmically random Brownian motion'), showing that their rapid points have the same dimension.Comment: 11 page

The complexity of the topological conjugacy problem for Toeplitz subshifts

In this paper, we analyze the Borel complexity of the topological conjugacy relation on Toeplitz subshifts. More specifically, we prove that topological conjugacy of Toeplitz subshifts with separated holes is hyperfinite. Indeed, we show that the topological conjugacy relation is hyperfinite on a larger class of Toeplitz subshifts which we call Toeplitz subshifts with growing blocks. This result provides a partial answer to a question asked by Sabok and Tsankov

Pointwise ergodic theorem for locally countable quasi-pmp graphs

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, analogous to pointwise ergodic theorems for group actions, replacing the group with a Schreier graph of the action. For any quasi-pmp graph, the theorem gives an increasing sequence of Borel subgraphs with finite connected components along which the averages of $L^1$ functions converge to their expectations. Equivalently, it states that any (not necessarily pmp) locally countable Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. The pmp version of this theorem was first proven by R. Tucker-Drob using probabilistic methods. Our proof is different: it is descriptive set theoretic and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant, a method of producing finite equivalence subrelations with large domain, and a simple method of exploiting nonamenability of a measured graph. The non-pmp setting additionally requires a new gadget for analyzing the interplay between the underlying cocycle and the graph.Comment: Added to the introduction a discussion of existing results about pointwise ergodic theorems for quasi-action

Cardinal characteristics and countable Borel equivalence relations

Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number $\mathfrak b$. We analyze some of the basic behavior of these properties, showing for instance that the property corresponding to the splitting number $\mathfrak s$ coincides with smoothness. We then settle many of the implication relationships between the properties; these relationships turn out to be closely related to (but not the same as) the Borel Tukey ordering on cardinal characteristics