Journal of Modern Dynamics (JMD)

Smooth diffeomorphisms with homogeneous spectrum and disjointness of convolutions

Pages: 439 - 481, Volume 10, 2016      doi:10.3934/jmd.2016.10.439

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Philipp Kunde - Department of Mathematics, University of Hamburg, Bundesstra├če 55, 20146 Hamburg, Germany (email)

Abstract: On any smooth compact connected manifold $M$ of dimension $m\geq 2$ admitting a smooth non-trivial circle action $\mathcal S = \left\{S_t\right\}_{t\in \mathbb{S}^1}$ and for every Liouville number $\alpha \in \mathbb{S}^1$ we prove the existence of a $C^\infty$-diffeomorphism $f \in \mathcal{A}_{\alpha} = \overline{\left\{h \circ S_{\alpha} \circ h^{-1} \;:\;h \in \text{Diff}^{\,\,\infty}\left(M,\nu\right)\right\}}^{C^\infty}$ with a good approximation of type $\left(h,h+1\right)$, a maximal spectral type disjoint with its convolutions and a homogeneous spectrum of multiplicity two for the Cartesian square $f\times f$. This answers a question of Fayad and Katok (10,[Problem 7.11]). The proof is based on a quantitative version of the approximation by conjugation-method with explicitly defined conjugation maps and tower elements.

Keywords:  Smooth ergodic theory, homogeneous spectrum, disjointness of convolutions, periodic approximation.
Mathematics Subject Classification:  Primary: 37A05, 37A30, 37C40; Secondary: 37C05.

Received: March 2015;      Revised: July 2016;      Available Online: October 2016.