# Dynamique de transvections généralisées

IRIMAS, Université de Haute-Alsace

Date(s) : 12/11/2019   iCal
11 h 00 min - 12 h 00 min

Given an increasing odd homeomorphism σ : {{R}} → {{R}}, the two bijective maps {h}σ , {v}σ : {{R}}2 → {{R}}2 defined by
{h}σ({x}, {y}) = ({x} + σ-1({y}), {y})
and
{v}σ({x}, {y}) = ({x}, σ({x}) + {y})
are called generalized transvections. We study the action on the plane of the group Γ(σ) generated by these two maps. Particularly interesting cases arise when σ({x}) = sgn({x})|{x}|α. We prove that most points have dense orbits and that every nonzero point has a dense orbit when σ({x}) = sgn({x})|{x}|2. We also look at invariant measures and thanks to Nogueira’s work about SL(2, {{Z}})-invariant measure, we can determine these measures when σ is linear in a neighborhood of the origin.

Joint work with Nicolas Chevallier.