A dichotomy between discrete and continuous spectrum for a class of special flows over rotations
Bassam Fayad A. Windsor
Journal of Modern Dynamics 2007, 1(1): 107-122 doi: 10.3934/jmd.2007.1.107
We provide sufficient conditions on a positive function so that the associated special flow over any irrational rotation is either weak mixing or $L^2$-conjugate to a suspension flow. This gives the first such complete classification within the class of Liouville dynamics. This rigidity coexists with a plethora of pathological behaviors.
keywords: time change of linear flows cohomological theory of dynamical systems weak mixing.
An application of topological multiple recurrence to tiling
Rafael De La Llave A. Windsor
Discrete & Continuous Dynamical Systems - S 2009, 2(2): 315-324 doi: 10.3934/dcdss.2009.2.315
We show that given any tiling of Euclidean space, any geometric pattern 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.
keywords: multiple topological recurrence. Pattern recurrence Tiling
Smooth dependence on parameters of solutions to cohomology equations over Anosov systems with applications to cohomology equations on diffeomorphism groups
Rafael de la Llave A. Windsor
Discrete & Continuous Dynamical Systems - A 2011, 29(3): 1141-1154 doi: 10.3934/dcds.2011.29.1141
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 cohomology 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 $\eta \in C$k+α$(M,$Diff$^r(N))$ for a compact manifold $N$, $k,r \in \N$, $r \geq 1$, and $0 < \alpha \leq \Lip$. We show that if there exists a $\varphi\in C$k+α$(M,$Diff$^1(N))$ solving

$ \varphi_{f(x)} = \eta_x \circ \varphi_x$

then in fact $\varphi \in C$k+α$(M,$Diff$^r(N))$. The existence of this solutions for some range of regularities is studied in the literature.

keywords: Anosov diffeomorphisms Cohomology equations rigidity. diffeomorphism groups Livšic theory

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