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Let $v$ be continuous and nowhere-vanishing vector field tangent to the $n$-sphere $\mathbb{S}^n$ (hence $n$ is odd, w.r.t the Hairy-Ball Theorem). Let $x$ be a trajectory on $\mathbb{S}^n$, defined f …

Let $K$ be a compact set of $\mathbb{R}^n$ $(n\geq 1)$ and $v\in\mathcal{C}^1(K,\mathbb{R}^n)$ be a speed field on $K$ such that for any initial condition $x_0\in K$, the following dynamical system \b …

Let $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ $(n\geq 1)$ a velocity field such that every solution $(x_t)_{t\geq 0}$ of $(d/dt)x_t=v(x_t)$ is periodic. Denote, for a non-stationary point $x\in\m …

Let $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ $(n\geq 2)$ a vector field, such that the set $E=\{v=0\}$ is a manifold of dimension $n-2$. Assume that for every $x\in\mathbb{R}^n-E$, the trajector …

I assume that there exist a family of probability measures $(d\mu_{t})_{t\geq 0}$ over the circle $\mathbb{R}_{|2\pi\mathbb{Z}}$ satisfying the following equation of evolution: for every continuous te …