2017/03/06 by Sophie Grivaux, Grivaux, Sophie, Étienne Matheron +3 · 2 citations
Mathematics · #Holomorphic and Operator Theory #Mathematical Dynamics and Fractals #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.1703.01854
We solve a number of questions pertaining to the dynamics of linear operators\non Hilbert spaces, sometimes by using Baire category arguments and sometimes by\nconstructing explicit examples. In particular, we prove the following results.\n - A typical hypercyclic operator is not topologically mixing, has no\neigenvalues and admits no non-trivial invariant measure, but is densely\ndistributionally chaotic.\n - A typical upper-triangular operator is ergodic in the Gaussian sense,\nwhereas a typical operator of the form "diagonal plus backward unilateral\nweighted shift" is ergodic but has only countably many unimodular eigenvalues,\nin particular, it is ergodic but not ergodic in the Gaussian sense.\n - There exist Hilbert space operators which are chaotic and mathcal\nU-frequently hypercyclic but not frequently hypercyclic, Hilbert space\noperators which are chaotic and frequently hypercyclic but not ergodic, and\nHilbert space operators which are chaotic and topologically mixing but not\n mathcal U-frequently hypercyclic.\n We complement our results by investigating the descriptive complexity of some\nnatural classes of operators defined by dynamical properties.\n