Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics

We obtain a large deviation function for the stationary measures of twisted Brownian motions associated to the Lagrangians $L_{\lambda}(p,v)=\frac{1}{2}g_{p}(v,v)- \lambda\omega_{p}(v)$, where $g$ is a $C^{\infty}$ Riemannian metric in a compact surface $(M,g)$ with nonpositive curvature, $\omega$ is a closed 1-form such that the Aubry-Mather measure of the Lagrangian $L(p,v)=\frac{1}{2}g_{p}(v,v)-\omega_{p}(v)$ has support in a unique closed geodesic $\gamma$; and the curvature is negative at every point of $M$ but at the points of $\gamma$ where it is zero. We also assume that the Aubry set is equal to the Mather set. The large deviation function is of polynomial type, the power of the polynomial function depends on the way the curvature goes to zero in a neighborhood of $\gamma$. This results has interesting counterparts in one-dimensional dynamics with indifferent fixed points and convex billiards with flat points in the boundary of the billiard. A previous estimate by N. Anantharaman of the large deviation function in terms of the Peierl's barrier of the Aubry-Mather measure is crucial for our result

The stability conjecture for geodesic flows of compact manifolds without conjugate points and quasi-convex universal covering

Let $(M,g)$ be a $C^{\infty}$ compact, boudaryless connected manifold without conjugate points with quasi-convex universal covering and divergent geodesic rays. We show that the geodesic flow of $(M,g)$ is $C^{2}$-structurally stable from Ma\~{n}\'{e}'s viewpoint if and only if it is an Anosov flow, proving the so-called $C^{1}$-stability conjecture.Comment: 29 pages, 4 figure

Free-floating molecular clumps and gas mixing: hydrodynamic aftermaths of the intracluster−-interstellar medium interaction

The interaction of gas-rich galaxies with the intra-cluster medium (ICM) of galaxy clusters has a remarkable impact on their evolution, mainly due to the gas loss associated with this process. In this work, we use an idealised, high-resolution simulation of a Virgo-like cluster, run with RAMSES and with dynamics reproducing that of a zoom cosmological simulation, to investigate the interaction of infalling galaxies with the ICM. We find that the tails of ram pressure stripped galaxies give rise to a population of up to more than a hundred clumps of molecular gas lurking in the cluster. The number count of those clumps varies a lot over time -- they are preferably generated when a large galaxy crosses the cluster (M$_{200c} > 10^{12}$ M$_\odot$), and their lifetime ($\lesssim 300$ Myr) is small compared to the age of the cluster. We compute the intracluster luminosity associated with the star formation which takes place within those clumps, finding that the stars formed in all of the galaxy tails combined amount to an irrelevant contribution to the intracluster light. Surprisingly, we also find in our simulation that the ICM gas significantly changes the composition of the gaseous disks of the galaxies: after crossing the cluster once, typically 20% of the cold gas still in those disks comes from the ICM.Comment: 9 pages, 6 figures. Accepted for publication in MNRA

Noncommutativity in the analysis of piecewise discrete-time dynamical systems

In this paper, we present a new method for the analysis of piecewise dynamical systems that are similar to the Collatz conjecture in regard to certain properties of the commutator of their sub-functions. We use the fact that the commutator of polynomials $E(n)=n/2$ and $O(n)=(3n+1)/2$ is constant to study rearrangements of compositions of $E(n)$ and $O(n)$. Our main result is that for any positive rational number $n$, if $(E^{e_1} \circ O^{o_1} \circ E^{e_2} \circ \dotsb \circ O^{o_l} \circ E^{e_{l+1}})(n)=1$, then $(E^{e_1} \circ O^{o_1} \circ E^{e_2} \circ \dotsb \circ O^{o_l} \circ E^{e_{l+1}})(n) = \lceil(E^{e_1 + \dotsb + e_{l+1}} \circ O^{o_1 + \dotsb + o_{l}})(n)\rceil$, where exponentiation is used to denote repeated composition and $e_i$ and $o_i$ are positive integers. Composition sequences of this form have significance in the context of the Collatz conjecture. The techniques used to derive this result can be used to produce similar results for a wide variety of repeatedly composed piecewise functions.Comment: 7 page