Abstract. We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of Fokina, Kalimullin, and R. Miller to show that for every computable ordinal α, 0 (α) is the degree of categoricity of some computable structure A. We show additionally that for α a computable successor ordinal, every degree 2-c.e. in and above 0 (α) is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures with degrees of categoricity is Π 1 1 complete.

We give several characterizations of Schnorr trivial sets, including a new lowness notion for Schnorr triviality based on truth-table reducibility. These characterizations allow us to see not only that some natural classes of sets, including maximal sets, are composed entirely of Schnorr trivials, but also that the Schnorr trivial sets form an ideal in the truth-table degrees but not the weak truth-table degrees. This answers a question of Downey. Griffiths and LaForte.

Abstract. We show that if a point in a computable probability space X satisfies the ergodic recurrence property for a computable measure-preserving T : X → X with respect to effectively closed sets, then it also satisfies Birkhoff's ergodic theorem for T with respect to effectively closed sets. As a corollary, every Martin-Löf random sequence in the Cantor space satisfies Birkhoff's ergodic theorem for the shift operator with respect to Π 0 1 classes. This answers a question of Hoyrup and Rojas.Several theorems in ergodic theory state that almost all points in a probability space behave in a regular fashion with respect to an ergodic transformation of the space. For example, if T : X → X is ergodic, 1 then almost all points in X recur in a set of positive measure:. Let (X, μ) be a probability space, and let T : X → X be ergodic. For all E ⊆ X of positive measure, for almost all x ∈ X, T n (x) ∈ E for infinitely many n.

Abstract. In this paper, we define new notions of randomness based on the difference hierarchy. We consider various ways in which a real can avoid all effectively given tests consisting of n-r.e. sets for some given n. In each case, the n-r.e. randomness hierarchy collapses for n ≥ 2. In one case, we call the resulting notion difference randomness and show that it results in a class of random reals that is a strict subclass of the Martin-Löf random reals and a proper superclass of both the Demuth random and weakly 2-random reals. In particular, we are able to characterize the difference random reals as the Turing incomplete Martin-Löf random reals. We also provide a martingale characterization for difference randomness.