Journal of Modern Dynamics
2021 , Volume 17
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We consider Bourgain's ergodic theorem regarding arithmetic averages in the cases where quantitative mixing is present in the dynamical system. Focusing on the case of the horocyclic flow, those estimates allow us to bound from above the Hausdorff dimension of the exceptional set, providing evidence towards conjectures by Margulis, Shah, and Sarnak regarding equidistribution of arithmetic averages in homogeneous spaces. We also prove the existence of a uniform upper bound for the Hausdorff dimension of the exceptional set which is independent of the spectral gap.
We study the regularity of a conjugacy between an Anosov automorphism
Given a countable amenable group
We generalize the notion of cusp excursion of geodesic rays by introducing for any
We construct a rigid, rank 1, prime transformation that is not quasi-simple and whose self-joinings form a Poulsen simplex. This seems to be the first example of a prime system whose self-joinings form a Poulsen simplex.
We construct some positive entropy automorphisms of rational surfaces with no periodic curves. The surfaces in question, which we term tri-Coble surfaces, are blow-ups of
We address the problem of computing the critical regularity of groups of homeomorphisms of the interval. Our main result is that if
Consider a symplectic surface
We introduce numerical invariants of contact forms in dimension three and use asymptotic cycles to estimate them. As a consequence, we prove a version for Anosov Reeb flows of results due to Hutchings and Weiler on mean actions of periodic points. The main tool is the Action-Linking Lemma, expressing the contact area of a surface bounded by periodic orbits as the Liouville average of the asymptotic intersection number of most trajectories with the surface.
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