arXiv research

We present two proofs of the fact, originally due to Reiner Martin, that any fully irreducible hyperbolic element of $Out(F_N)$ acts on the projectivized space of geodesic currents $\mathbb{P}Curr(F_N)$ with uniform north-south dynamics. The first proof, using purely train-track methods, provides an elaborated and corr…

Unified proof of Nielsen-Thurston classification via Teichmüller's theorem.

This paper is devoted to rigidity of smooth bundles which are equipped with fiberwise geometric or dynamical structure. We show that the fiberwise associated sphere bundle to a bundle whose leaves are equipped with (continuously varying) metrics of negative curvature is a topologically trivial bundle when either the ba…

Polynomial inequalities lie at the heart of many mathematical disciplines. In this paper, we consider the fundamental computational task of automatically searching for proofs of polynomial inequalities. We adopt the framework of semi-algebraic proof systems that manipulate polynomial inequalities via elementary inferen…

Development of theory of dynamical systems admitting the normal shift in 1993-1999 is reviewed. Basics are given with complete proofs.

New method connects veering triangulations to dynamic pairs.

We give a sharp lower bound for the number of geometrically distinct contractible periodic orbits of dynamically convex Reeb flows on prequantizations of symplectic manifolds that are not aspherical. Several consequences of this result are obtained, like a new proof that every bumpy Finsler metric on $S^n$ carries at l…

We announce some results towards the classification of partially hyperbolic diffeomorphisms on 3-manifolds, and outline the proofs in the case when the diffeomorphism is dynamically coherent. Detailed proofs are long and technical and will appear later.

Stable actions of hyperbolic groups on their boundaries.

We present a new proof of the following theorem of Benoist-Quint: Let $G:=SO^\circ(d,1)$, $d\ge 2$ and $Δ<G$ a cocompact lattice. Any orbit of a Zariski dense subgroup $Γ$ of $G$ is either finite or dense in $Δ\backslash G$. While Benoist and Quint's proof is based on the classification of stationary measures, our proo…

Proves effective slope gaps for lattice surfaces.

New dynamic allocation methods for multi-armed bandit models.

Survey on strong closing lemmas in Hamiltonian dynamics.

Model analyzes Proof-of-Stake network dynamics and speculative capital effects on token prices.

The abstract discusses rational approximations for Hitchin representations on surfaces.

We provide conditions for the existence and the unicity of strictly stationary solutions of the usual Dynamic Conditional Correlation GARCH models (DCC-GARCH). The proof is based on Tweedie's (1988) criteria, after having rewritten DCC-GARCH models as nonlinear Markov chains. Moreover, we study the existence of their f…

In this note we make use of some properties of vector fields on a manifold to give an alternate proof to [3] for the equivalence between connections and parallel transport on vector bundles over manifolds. Out of the proof will emerge a new approach to connections on a bundle as a consistent way to lift the dynamics of…

The paper examines stability of shares in Proof of Stake protocol, identifying different investor behaviors and phase transitions.

Proposes a method to prove closing of periodic orbits in dynamical systems.

Formula derived for zeta functions of 3D foliated systems.

We give the simple proof of Bobkov's inequality using the arguments of dynamical programming principle. As a byproduct of the method we obtain a characterization of optimizers.

Simplified proof of stability for Ricci flow near ALE metrics.

Zeta functions extended to nonorientable surfaces, order of vanishing computed.

This paper studies dynamic stochastic optimization problems parametrized by a random variable. Such problems arise in many applications in operations research and mathematical finance. We give sufficient conditions for the existence of solutions and the absence of a duality gap. Our proof uses extended dynamic programm…

The paper studies invariant weighted Bergman metrics on domains.

Integrable dynamics explained via geometric maps and cluster algebras.

Probabilistic proof of smooth boundaries in optimal stopping problems.

New example solves topological dynamics problem.

We present an equivariant Liapunov stability criterion for dynamical systems with symmetry. This result yields a simple proof of the energy-momentum-Casimir stability analysis of relative equilibria of equivariant Hamiltonian systems.

The paper revisits and applies FTAP to life insurance and annuities pricing.

Mirzakhani's thesis counts geodesics on hyperbolic surfaces, finding a specific asymptotic formula.

We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain Conley pairs $(N,L)$, established in [2,3], as a dynamical thickening of the stable…

The present paper addresses the issue of choosing an optimal dynamic reinsurance policy, which is state-dependent, for an insurance company that operates under multiple insurance business lines. The optimal survival function is characterized as the unique nondecreasing viscosity solution of the associated Hamilton-Jaco…

We describe the polynomial time complexity algorithm for computing first coefficients of the skein (Homflypt) and Kauffman polynomial invariants of links, discovered by D.Vertigan in 1992 but never published.

The aim of this (mostly expository) article is twofold. We first explore a variety of length functions on the space of currents, and we survey recent work regarding applications of length functions to counting problems. Secondly, we use length functions to provide a proof of a folklore theorem which states that pseudo-…

We present a geometric proof of the averaging theorem for perturbed dynamical systems on a Riemannian manifold, in the case where the flow of the unperturbed vector field is periodic and the $\mathbb{S}^{1}$-action associated to this vector field is not necessarily trivial. We generalize the averaging procedure \cite{A…

We study the coarse geometry of the moduli space of dilation tori with two singularities and the dynamical properties of the action of the Teichmuller flow on this moduli space. This leads to a proof that the vertical foliation of a dilation torus is almost always Morse-Smale. As a corollary, we get that the generic pi…

Optimal trading strategy in Proof-of-Stake blockchain using continuous-time control.

Construct Hermitian-Einstein metrics on stable holomorphic vector bundles using dynamical methods.

A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow of any convex hypersurface in $\mathbb{R}^{2n}$ carries an elliptic closed orbit. Two important contributions toward its proof were given by Ekeland in 1986 and Dell'Antonio-D'Onofrio-Ekeland in 1995 proving this for convex hypersurfaces satis…

We analyze training dynamics in Gaussian mixture models using a comparison theorem.

Kyle's equilibrium model stability proven for 1-2 trading times, but not for 3 or more.

Adaptive algorithm improves convergence rate of Langevin dynamics.

Study the exponential map on surfaces using fluid dynamics.

This study explains gradient flow dynamics in neural networks for small initialisation.

We exhibit a pseudogroup of smooth local transformations of the real line which is compactly generated, but not realizable as the holonomy pseudogroup of a foliation of codimension 1 on a compact manifold. The proof relies on a description of all foliations with the same dynamic as the Reeb component.

We discuss some classification results for Ricci solitons, that is, self similar solutions of the Ricci Flow. Some simple proofs of known results will be presented. In detail, we will take the equation point of view, trying to avoid the tools provided by considering the dynamic properties of the Ricci flow.

The paper analyzes the mean field Langevin dynamics and its convergence rate.