No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

We prove we can build (transitive or nontransitive) Anosov flows on closed three-dimensional manifolds by gluing together filtrating neighborhoods of hyperbolic sets. We give several applications of this result; for example: 1.We build a closed three-dimensional manifold supporting both a transitive Anosov vector field and a nontransitive Anosov vector field. 2.For any n, we build a closed three-dimensional manifold M supporting at least n pairwise different Anosov vector fields. 3.We build transitive hyperbolic attractors with prescribed entrance foliation; in particular, we construct some incoherent transitive hyperbolic attractors. 4.We build a transitive Anosov vector field admitting infinitely many pairwise nonisotopic transverse tori.

We study orthogonal polynomials with respect to self-similar measures, focusing on the class of infinite Bernoulli convolutions, which are defined by iterated function systems with overlaps, especially those defined by the Pisot, Garsia, and Salem numbers. By using an algorithm of Mantica, we obtain graphs of the coefficients of the 3-term recursion relation defining the orthogonal polynomials. We use these graphs to predict whether the singular infinite Bernoulli convolutions belong to the Nevai class. Based on our numerical results, we conjecture that all infinite Bernoulli Convolutions with contraction ratios greater than or equal to 1/2 belong to Nevai's class, regardless of the probability weights assigned to the self-similar measures.