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.
We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids
We show the existence, over an arbitrary infinite ergodic
Very little is currently known about the dynamics of non-polynomial entire maps in several complex variables. The family of transcendental Hénon maps offers the potential of combining ideas from transcendental dynamics in one variable and the dynamics of polynomial Hénon maps in two. Here we show that these maps all have infinite topological and measure theoretic entropy. The proof also implies the existence of infinitely many periodic orbits of any order greater than two.
We extend the notion of Rauzy induction of interval exchange transformations to the case of toral
We focus on one example,
It is shown that in a class of counterexamples to Elliott's conjecture by Matomäki, Radziwiłł, and Tao [
We prove that for any partially hyperbolic diffeomorphism having neutral center behavior on a 3-manifold, the center stable and center unstable foliations are complete; moreover, each leaf of center stable and center unstable foliations is a cylinder, a Möbius band or a plane.
Further properties of the Bonatti–Parwani–Potrie type of examples of of partially hyperbolic diffeomorphisms are studied. These are obtained by composing the time
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