In this article, we show two related results on circle diffeomorphisms. The first result is on quasi-reducibility: for a Baire-dense set of α, for any diffeomorphism f of rotation number α, it is possible to accumulate Rα with a sequence hn f h−1 n , hn being a diffeomorphism. The second result is: for a Baire-dense set of α, given two commuting diffeomorphisms f and g, such that f has α for ro...

The purpose of this chapter is to discuss various results concerning the relationship between measure and category. We are mostly interested in set-theoretic properties of these ideals, particularly, their cardinal characteristics. This is a very large area, and it was necessary to make some choices. We decided to present several new results and new approaches to old problems. In most cases we ...

We analyze the dual Ramsey theorem for k partitions and ℓ colors (DRTℓk) in context of reverse math, effective analysis, strong reductions. Over RCA0, stated Baire colorings Baire-DRTℓk is equivalent to statement clopen ODRTℓk a purely combinatorial CDRTℓk. When Borel k≥3, resulting principles are essentially relativizations For each α, there computable code Δα0-coloring such that any partition...

We study and characterize stability, NIP and NSOP in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, ‘Talagrand’s stability’, and explain the relationship between this property and NIP in continuous logic. Using a result of Bourgain, Fremlin and Talagrand, we prove the ‘almost and Baire 1 definability’ of types assuming NIP...

Measure and category There are two natural ways of saying that a set of countable graphs is “large”. Choose a fixed countable vertex set, and enumerate the pairs of vertices: {x0, y0}, {x1, y1}, . . . There is a probability measure on the set of graphs, obtained by choosing independently with probability 1/2 whether xi and yi are joined, for all i. Now a set of graphs is “large” if it has proba...

For an arbitrary infinite additive group G and for an uncountable compact Hausdorff topological group H with card(H) = card(Hא0) = card(H), H-valued measurable Gprocesses are constructed on the group H and some set-theoretical characteristics of their various F∗(HG)-invariant extensions are calculated, where F∗(HG) denotes a group of transformations of H generated by the eventually neutral sequ...

We expand the ergodic theory developed by Furstenberg and Hochman on dynamical systems that are obtained from magnifications of measures. We prove that any fractal distribution in the sense of Hochman is generated by a uniformly scaling measure, which provides a converse to a regularity theorem on the structure of distributions generated by the scenery flow. We further show that the collection ...

We prove a Hindman-type partition theorem for Baire partitions of [0, 1].

We de…ne combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and category versions are derived from one topological theorem on interior points applied to the usual topology and the density topology on the line. Likewise we unify the subgroup theorem by reference to a Ramsey prope...

We present an infinite game that characterizes the Borel functions on Baire Space.