Guido AHUMADA - Dynamique de transvections généralisées
Guido AHUMADA (IRIMAS, Université de Haute-Alsace)
Given an increasing odd homeomorphism σ : R → R, the two bijective maps hσ , vσ : R2 → R2 defined by
hσ(x, y) = (x + σ-1(y), y)
vσ(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.