Renato Feres

Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows

We consider in this note smooth dynamical systems equipped with smooth invariant affine connections and show that, under a pinching condition on the Lyapunov exponents, certain invariant tensor fields are parallel. We then apply this result to a problem of rigidity of geodesic flows for Riemannian manifolds with negative curvature.

Geodesic flows on manifolds of negative curvature with smooth horospheric foliations

We improve and extend a result due to M . Kanai about rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable horospheric foliation is smooth. More precisely, the main results proved here are: (1) Let M be a closed C ∞ Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow φ t : V → V on the unit tangent bundle V of M is C ∞ . Assume, moreover, that either (a) the sectional curvature of M satisfies −4 M is odd. Then the geodesic flow of M is C ∞ -isomorphic (i.e., conjugate under a C ∞ diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature. (2) For M as above, assume instead of (a) or (b) that dim M ≡ 2(mod 4). Then either the above conclusion holds or φ 1 , is C ∞ -isomorphic to the flow , on the quotient Γ\ , where Γ is a subgroup of a real Lie group ⊂ Diffeo ( ) with Lie algebra is the geodesic flow on the unit tangent bundle of the complex hyperbolic space ℂH m , m = ½ dim M.

Random Walks Derived from Billiards

We introduce a class of random dynamical systems derived from billiard maps, which we call random billiards, and study certain random walks on the real line obtained from them. The interplay between the billiard geometry and the stochastic properties of the random billiard is investigated. Our main results are concerned with the description of the spectrum of the random billiard’s Markov operator and the characteristics of diusion limits under appropriate scaling.

Random billiards with wall temperature and associated Markov chains

By a random billiard we mean a billiard system in which the standard rule of specular reflection is replaced with a Markov transition probabilities operator P that gives, at each collision of the billiard particle with the boundary of the billiard domain, the probability distribution of the post-collision velocity for a given pre-collision velocity. A random billiard with microstructure, or RBM for short, is a random billiard for which P is derived from a choice of geometric/mechanical structure on the boundary of the billiard domain, as explained in the text. Such systems provide simple and explicit mechanical models of particle–surface interaction that can incorporate thermal effects and permit a detailed study of thermostatic action from the perspective of the standard theory of Markov chains on general state spaces. The main focus of this paper is on the operator P itself and how it relates to the mechanical and geometric features of the microstructure, such as mass ratios, curvatures, and potentials. The main results are as follows: (1) we give a characterization of the stationary probabilities (equilibrium states) of P and show how standard equilibrium distributions studied in classical statistical mechanics such as the Maxwell–Boltzmann distribution and the Knudsen cosine law arise naturally as generalized invariant billiard measures; (2) we obtain some of the more basic functional theoretic properties of P, in particular that P is under very general conditions a self-adjoint operator of norm 1 on a Hilbert space to be defined below, and show in a simple but somewhat typical example that P is a compact (Hilbert–Schmidt) operator. This leads to the issue of relating the spectrum of eigenvalues of P to the geometric/mechanical features of the billiard microstructure; (3) we explore the latter issue, both analytically and numerically in a few representative examples. Additionally, (4) a general algorithm for simulating the Markov chains is given based on a geometric description of the invariant volumes of classical statistical mechanics. Our description of these volumes may have independent interest.

Leafwise holomorphic functions

It is a well-known and elementary fact that a holomorphic function on a compact complex manifold is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property holds in the setting of holomorphically foliated spaces.

From billiards to thermodynamics

Abstract We explore some beginning steps in stochastic thermodynamics of billiard-like mechanical systems by introducing extremely simple and explicit random mechanical processes capable of exhibiting steady-state irreversible thermodynamical behavior. In particular, we describe a Markov chain model of a minimalistic heat engine and numerically study its operation and efficiency.

Higher order approximation of isochrons

Phase reduction is a commonly used techinque for analysing stable oscillators, particularly in studies concerning synchronization and phase lock of a network of oscillators. In a widely used numerical approach for obtaining phase reduction of a single oscillator, one needs to obtain the gradient of the phase function, which essentially provides a linear approximation of the local isochrons. In this paper, we extend the method for obtaining partial derivatives of the phase function to arbitrary order, providing higher order approximations of the local isochrons. In particular, our method in order 2 can be applied to the study of dynamics of a stable oscillator subjected to stochastic perturbations, a topic that will be discussed in a future paper. We use the Stuart–Landau oscillator to illustrate the method in order 2.

Connection preserving actions of lattices inSLnℝ

We consider connection-preserving actions of lattice subgroups ofSLnℝ on compact Riemannian manifolds, forn≥3. The main result gives a description of such actions, when dimM=n+1, as follows: Either the action preserves a smooth invariant Riemannian metric onM or it can be described in terms of linear actions on then-torus.

Dynamics on the space of harmonic functions and the foliated Liouville problem

We study here the action of subgroups of PSL(2,R) on the space of harmonic functions on the unit disc bounded by a common constant, as well as the relationship this action has with the foliated Liouville problem. Given a foliation of a compact manifold by Riemannian leaves and a leafwise harmonic continuous function on the manifold, is the function leafwise constant? We give a number of positive results and also show a general class of examples for which the Liouville property does not hold. The connection between the Liouville property and the dynamics on the space of harmonic functions as well as general properties of this dynamical system are explored. It is shown among other properties that the Z-action generated by hyperbolic or parabolic elements of PSL(2,R) is chaotic.

Chapter 9 Ergodic theory and dynamics of G-spaces (with special emphasis on rigidity phenomena)

Publisher Summary This chapter discusses the case of group, as opposed to semi-group, actions. There are various technical complications that arise in passing from invertible to noninvertible actions, and in general, the nonamenable noninvertible case has not been given sufficient attention as yet to warrant inclusion into a general survey. The primary focus of the survey is on those aspects of the ergodic theory and differentiable dynamics of group actions that are most distinct from the theory for R and ℤ. As ergodic theory for actions of general amenable groups share with R and ℤ many key properties, the survey discussed in the chapter is concerned in large part with the actions of nonamenable groups. Orbit equivalence provides a particularly compelling example: all finite measure-preserving ergodic actions of discrete amenable groups are orbit equivalent. For certain groups that are both “sufficiently large” and “rigid,” orbit equivalence essentially implies isomorphism.