We introduce a natural principle Strong Chang Reflection strengthening the classical Chang Conjectures. This principle is between a huge and a two huge cardinal in consistency strength. In this note we prove that it implies the existence of an inner model with a huge cardinal. The technique we explore for building inner models with huge cardinals adapts to show that decisive ideals imply the existence of inner models with supercompact cardinals. Proofs for all of these claims can be found in [10].1,2.

In 1932, von Neumann proposed classifying the statistical behavior of differentiable systems. Joint work of B. Weiss and the author proved that the classification problem is complete analytic. Based on techniques in that proof, one is able to show that the collection of recursive diffeomorphisms of the 2-torus that are isomorphic to their inverses is -hard via a computable 1-1 reduction. As a corollary there is a diffeomorphism that is isomorphic to its inverse if and only if the Riemann Hypothesis holds, a different one that is isomorphic to its inverse if and only if Goldbach's conjecture holds and so forth. Applying the reduction to the Π01 sentence expressing "ZFC"is consistent gives a diffeomorphism T of the 2-torus such that the question of whether T ≅ T-1 is independent of ZFC.

The main result of this paper is that two large collections of er-godic measure preserving systems, the Odometer Based and the Circular Systems have the same global structure with respect to joinings that preserve underlying timing factors. The classes are canonically isomorphic by a contin-uous map that takes synchronous and anti-synchronous factor maps to synchronous and anti-synchronous factor maps, synchronous and anti-synchro-nous measure-isomorphisms to synchronous and anti-synchronous measure-isomorphisms, weakly mixing extensions to weakly mixing extensions and compact extensions to compact extensions. The first class includes all finite entropy ergodic transformations that have an odometer factor. By results in [6], the second class contains all transformations realizable as dif-feomorphisms using the untwisted Anosov–Katok method. An application of the main result will appear in a forthcoming paper [7] that shows that the diffeomorphisms of the torus are inherently unclassifiable up to measure-isomorphism. Other consequences include the existence of measure distal diffeomorphisms of arbitrary countable distal height.

The paper considers the equivalence relation of conjugacy-by-homeomorphism on diffeomorphisms of smooth manifolds. In dimension 2 and above it is shown that there is no Borel method of attaching complete numerical invariants. In dimension 5 and above it is shown that the equivalence relation is not Borel, and in fact is complete analytic.

Construction sequences are a general method of building symbolic shifts that capture cut-and-stack constructions and are general enough to give symbolic representations of Anosov-Katok diffeomorphisms. We show here that any finite entropy system that has an odometer factor can be represented as the limit of a special class of construction sequences, the odometer based construction sequences. These naturally correspond to those cut-and-stack constructions that do not use spacers. The odometer based construction sequences can be constructed to have the small word property and every Choquet simplex can be realized as the simplex of invariant measures of the limit of an odometer based construction sequence.

This paper is the first of a series of papers culminating in the result that measure preserving diffeomorphisms of the disc or 2-torus are unclassifiable. It addresses another classical problem: which abstract measure preserving systems are realizable as smooth diffeomorphisms of a compact manifold? The main result gives symbolic representations of Anosov–Katok diffeomorphisms.

In 1932 von Neumann proposed classifying the statistical behavior of differentiable systems. In modern language this is interpreted as classifying diffeomorphisms of compact manifolds up to measure isomorphism. This paper proves that this is impossible in a rigorous sense.

This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call Welch games. Player II having a winning strategy in the Welch game of length ω on κ is equivalent to weak compactness. Winning the game of length 2κ is equivalent to κ being measurable. We show that for games of intermediate length γ, II winning implies the existence of precipitous ideals with γ-closed, γ-dense trees. The second part shows the first is not vacuous. For each γ between ω and κ+, it gives a model where II wins the games of length γ, but not γ+. The technique also gives models where for all ω1 < γ ≤ κ there are κ-complete, normal, κ+-distributive ideals having dense sets that are γ-closed, but not γ+-closed.