A periodicity criterion and the section problem on the Mapping Class Group

Some years ago, V. Markovic proved that there is no section of the Mapping Class Group for a closed surface of genus g larger than 5 (in the case of homeomorphims) and more recently generalized this result with D. Saric to the case where g is larger than 1. We will state a periodicity criterion and will use it to simplify some of the arguments given by Markovic and Saric in the proof of their theorem. The periodicity criterion tells us that a homeomorphism of a connected surface must be periodic if the set of connected periodic open sets generates the topology of the surface.Comment: 40 page

A kinetic eikonal equation

We analyse the linear kinetic transport equation with a BGK relaxation operator. We study the large scale hyperbolic limit (t,x)\to (t/\eps,x/\eps). We derive a new type of limiting Hamilton-Jacobi equation, which is analogous to the classical eikonal equation derived from the heat equation with small diffusivity. We prove well-posedness of the phase problem and convergence towards the viscosity solution of the Hamilton-Jacobi equation. This is a preliminary work before analysing the propagation of reaction fronts in kinetic equations

Fast algorithmic Nielsen-Thurston classification of four-strand braids

We give an algorithm which decides the Nielsen-Thurston type of a given four-strand braid. The complexity of our algorithm is quadratic with respect to word length. The proof of its validity is based on a result which states that for a reducible 4-braid which is as short as possible within its conjugacy class (short in the sense of Garside), reducing curves surrounding three punctures must be round or almost round.Comment: One minor error corrected (Example 4.2 was wrong

Optimal growth for linear processes with affine control

We analyse an optimal control with the following features: the dynamical system is linear, and the dependence upon the control parameter is affine. More precisely we consider $\dot x_\alpha(t) = (G + \alpha(t) F)x_\alpha(t)$, where $G$ and $F$ are $3\times 3$ matrices with some prescribed structure. In the case of constant control $\alpha(t)\equiv \alpha$, we show the existence of an optimal Perron eigenvalue with respect to varying $\alpha$ under some assumptions. Next we investigate the Floquet eigenvalue problem associated to time-periodic controls $\alpha(t)$. Finally we prove the existence of an eigenvalue (in the generalized sense) for the optimal control problem. The proof is based on the results by [Arisawa 1998, Ann. Institut Henri Poincar\'e] concerning the ergodic problem for Hamilton-Jacobi equations. We discuss the relations between the three eigenvalues. Surprisingly enough, the three eigenvalues appear to be numerically the same

A one-dimensional Keller-Segel equation with a drift issued from the boundary

We investigate in this note the dynamics of a one-dimensional Keller-Segel type model on the half-line. On the contrary to the classical configuration, the chemical production term is located on the boundary. We prove, under suitable assumptions, the following dichotomy which is reminiscent of the two-dimensional Keller-Segel system. Solutions are global if the mass is below the critical mass, they blow-up in finite time above the critical mass, and they converge to some equilibrium at the critical mass. Entropy techniques are presented which aim at providing quantitative convergence results for the subcritical case. This note is completed with a brief introduction to a more realistic model (still one-dimensional).Comment: short version, 8 page

Blow-up dynamics of self-attracting diffusive particles driven by competing convexities

In this paper, we analyze the dynamics of an $N$ particles system evolving according the gradient flow of an energy functional. The particle system is a consistent approximation of the Lagrangian formulation of a one parameter family of non-local drift-diffusion equations in one spatial dimension. We shall prove the global in time existence of the trajectories of the particles (under a sufficient condition on the initial distribution) and give two blow-up criteria. All these results are consequences of the competition between the discrete entropy and the discrete interaction energy. They are also consistent with the continuous setting, that in turn is a one dimension reformulation of the parabolic-elliptic Keller-Segel in high dimensions