arXiv subjects
Publications and source records attributed to Alistair Windsor.
Let $T$ be any invertible, ergodic, aperiodic measure-preserving transformation of a Lebesgue probability space $(X, \calB, μ)$, and ¶\, any finite measurable partition of $X$. We show that a (finite) Alpern multitower may always be constructed whose base is independent of ¶.
We consider the dependence on parameters of the solutions of cohomology equations over Anosov diffeomorphisms. We show that the solutions depend on parameters as smoothly as the data. As a consequence we prove optimal regularity results for the solutions of equations taking value in diffeomorphism groups. These results are motivated by applications to rigidity theory, dynamical systems, and geometry. In particular, in the context of diffeomorphism groups we show: Let $f$ be a transitive Anosov diffeomorphism of a compact manifold $M$. Suppose that $η\in C^{\reg}(M,\Diff^r(N))$ for a compact manifold $N$, $k,r \in \N$, $r \geq 1$, and $0 < α\leq \Lip$. We show that if there exists a $φ\in C^{\reg}(M,\Diff^1(N))$ solving \begin{equation*} φ_{f(x)} = η_x \circ φ_x \end{equation*} then in fact $φ\in C^{\reg}(M,\Diff^r(N))$.
We show that given any tiling of Euclidean space, any geometric patterns of points, we can find a patch of tiles (of arbitrarily large size) so that copies of this patch appear in the tiling nearly centered on a scaled and translated version of the pattern. The rather simple proof uses Furstenberg's topological multiple recurrence theorem.
The celebrated Livsic theorem states that given M a manifold, a Lie group G, a transitive Anosov diffeomorphism f on M and a Holder function η: M \mapsto G whose range is sufficiently close to the identity, it is sufficient for the existence of ϕ:M \mapsto G satisfying η(x) = ϕ(f(x)) ϕ(x)^{-1} that a condition -- obviously necessary -- on the cocycle generated by ηrestricted to periodic orbits is satisfied. In this paper we present a new proof of the main result. These methods allow us to treat cocycles taking values in the group of diffeomorphisms of a compact manifold. This has applications to rigidity theory. The localization procedure we develop can be applied to obtain some new results on the existence of conformal structures for Anosov systems.
We augment the method of $C^\infty$ conjugation approximation with explicit estimates on the conjugacy map. This allows us to construct ergodic volume preserving diffeomorphisms measure-theoretically isomorphic to any apriori given Liouville rotation on a variety of manifolds. In the special case of tori the maps can be made uniquely ergodic.
We provide sufficient conditions on a positive function so that its associated special flow over any irrational rotation is either weak mixing or $L^2$-conjugate to a suspension flow.