arXiv subjects

Martin Leguil

Publications and source records attributed to Martin Leguil.

20 records · Page 2Linked to original sources

arXiv2017

Weak mixing properties of interval exchange transformations and translation flows

Let $d >1$. In this paper we show that for an irreducible permutation $π$ which is not a rotation, the set of $[λ]\in \mathbb{P}_+^{d-1}$ such that the interval exchange transformation $f([λ],π)$ is not weakly mixing does not have full Hausdorff dimension. We also obtain an analogous statement for translation flows. In particular, it strengthens the result of almost sure weak mixing proved by G. Forni and the first author. We adapt here the probabilistic argument developed in their paper in order to get some large deviation results. We then show how the latter can be converted into estimates on the Hausdorff dimension of the set of "bad" parameters in the context of fast decaying cocycles, following a strategy developed by V. Delecroix and the first author.

math.DS↗

arXiv2016

Dynamics of a family of polynomial automorphisms of $\mathbb{C}^3$, a phase transition

The polynomial automorphisms of the affine plane have been studied a lot: if $f$ is such an automorphism, then either $f$ preserves a rational fibration, has an uncountable centralizer and its first dynamical degree equals $1$, or $f$ preserves no rational curves, has a countable centralizer and its first dynamical degree is $>1$. In higher dimensions there is no such description. In this article we study a family $(Ψ_α)_α$ of polynomial automorphisms of $\mathbb{C}^3$. We show that the first dynamical degree of $Ψ_α$ is $>1$, that $Ψ_α$ preserves a unique rational fibration and has an uncountable centralizer. We then describe the dynamics of the family $(Ψ_α)_α$, in particular the speed of points escaping to infinity. We also observe different behaviors according to the value of the parameter $α$.

math.DS↗