Dynamic of generalized transvections

Abstract

Given an increasing odd homeomorphism $\sigma$ : R $\rightarrow$ R, the two bijective maps h $\sigma$ , v $\sigma$ : R 2 $\rightarrow$ R 2 dened by h $\sigma$ (x, y) = (x + $\sigma$ --1 (y), y) and v $\sigma$ (x, y) = (x, $\sigma$(x) + y). are called generalized transvections. We study the action on the plane of the group $\Gamma$($\sigma$) generated by these two maps. Particularly interesting cases arise when $\sigma$(x) = sgn(x)|x| $\alpha$. We prove that most points have dense orbits and that every nonzero point has a dense orbit when $\sigma$(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 $\sigma$ is linear in a neighborhood of the origin.