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So here's my question: Does there exist a minimal diffeomorphism of class at least $\mathcal{C^2}$ of a compact manifold X which is minimal uniquely ergodic with unique probability measure $\mu$ n …

The question that I have in mind is the following: Which kind of closed sets can arise as the $\omega$-limit of a point for a $1$-dimensional dynamical system? It is probably somewhat naive, but now …

So I have this question which is somewhat directed to people knowing a bit about translation surfaces. I am sure it is only a technical issue. I consider $f : (\Sigma, \omega) \longrightarrow (\Sigma …

Consider an interval exchange transformation that is, a bijective piecewise continuous map $[0,1] \rightarrow [0,1] $ whose restriction to its continuity intervals are translations. Assume that it is …

I am looking for a reference here. Consider a two-dimensional torus $\mathrm{T}^2 =S^1 \times S^1$ together with an affine structure, that is a $(Aff(\mathbb{R}^2), \mathbb{R}^2)$-structure. Such a st …

Let $P$ be a polygon in the plane. One can define the billiard flow on the unit tangent bundle of $P$, just following the trajectories of the billiard at speed one. A standard conjecture is that a t …

I'm coming with a very basic question for which I can't find an answer. Please forgive me if I didn't search efficiently enough. What can the closure of an orbit of an element $X$ of $\mathbb{R}^2$ u …

I imagine the dynamic of $SL(2,\mathbb{Z}$) on $\mathbb{C}^2$ has been studied. Does one know if it is recurrent or ergodic (with respect to the Lebesgue measure) ? Is there any explicit description o …