Related papers: Carnot metrics, Dynamics and Local Rigidity

In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism $L$ with simple real eigenvalues with…

This paper presents a theoretical overview of a Neural Contraction Metric (NCM): a neural network model of an optimal contraction metric and corresponding differential Lyapunov function, the existence of which is a necessary and sufficient…

We prove several superrigidity results for isometric actions on metric spaces satisfying some convexity properties. First, we extend some recent theorems of N. Monod on uniform and certain non-uniform irreducible lattices in products of…

We establish a rigidity theorem for Brendle and Hung's recent systolic inequality, which involves Gromov's notion of \(T^{\rtimes}\)-stabilized scalar curvature. Our primary technique is the construction of foliations by free boundary…

We show that the time-1 map of an Anosov flow, whose strong-unstable foliation is $C^2$ smooth and minimal, is $C^2$ close to a diffeomorphism having positive central Lyapunov exponent Lebesgue almost everywhere and a unique physical…

We prove the local well-posedness in Sobolev spaces of the free-boundary problem for compressible inviscid resistive isentropic MHD system under the Rayleigh-Taylor physical sign condition, which describes the motion of a free-boundary…

For a fixed topological Markov shift, we consider measure-preserving dynamical systems of Gibbs measures for 2-locally constant functions on the shift. We also consider isomorphisms between two such systems. We study the set of all…

We introduce the notion of a point on a locally closed subset of a symplectic manifold being "locally rigid" with respect to that subset, prove that this notion is invariant under symplectic homeomorphisms, and show that coisotropic…

We extend the Lyapunov function technique, a fundamental tool for investigating asymptotic stability and existence of attractors for ordinary differential equations, by introducing the notion of a {\it strong Lyapunov function} for an…

We extend some known results from smooth dynamical systems to the category of Lipschitz homeomorphisms of compact metric spaces. We consider dynamical properties as robust expansiveness and structural stability allowing Lipschitz…

We study a wide class of metrics in a Lebesgue space with a standard measure, the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the…

We prove a version of the Hopf-Rinow-theorem with respect to path metrics on discrete spaces. The novel aspect is that we do not a priori assume local finiteness but isolate a local finiteness type condition, called essential local…

We analyse the so-called Marginal Instability of linear switching systems, both in continuous and discrete time. This is a phenomenon of unboundedness of trajectories when the Lyapunov exponent is zero. We disprove two recent conjectures of…

We consider 3D free-boundary compressible elastodynamic system under the Rayleigh-Taylor sign condition. It describes the motion of an isentropic inviscid elastic medium with moving boundary. The deformation tensor satisfies the neo-Hookean…

Let $G$ be the group of orientation-preserving isometries of a rank-one symmetric space $X$ of non-compact type. We study local rigidity of certain actions of a solvable subgroup $\Gamma \subset G$ on the boundary of $X$, which is…

Inspired by the lotus effect, many studies in the last decade have focused on micro- and nano-patterned surfaces. They revealed that patterns at the micro-scale combined with high contact angles can significantly reduce skin drag. However,…

In the present work we obtain rigidity results analysing the set of regular points, in the sense of Oseledec's Theorem. It is presented a study on the possibility of an Anosov diffeomorphisms having all Lyapunov exponents defined…

This paper develops a family of Hofer-like metrics ($\dAnV{V}$) on the space of Anosov vector fields $\An(M)$, providing dynamically relevant distances based on the cost of deformation paths using $\Ck{k}$ or Sobolev $\SobolevHk{k}$ norms.…

While stability analysis is a mainstay for control science, especially computing regions of attraction of equilibrium points, until recently most stability analysis tools always required explicit knowledge of the model or a high-fidelity…

For Anosov diffeomorphisms on the $3$-torus which are strongly partially hyperbolic with expanding center, we construct systems of strong unstable and center stable Margulis measures which are holonomy-invariant. This allows us to obtain a…