Robust vanishing of all Lyapunov exponents for iterated function systems

Given any compact connected manifold $M$ , we describe $C^2$ -open sets of iterated functions systems (IFS’s) admitting fully-supported ergodic measures whose Lyapunov exponents along $M$ are all zero. Moreover, these measures are approximated by measures supported on periodic orbits. We also describe $C^1$ -open sets of IFS’s admitting ergodic measures of positive entropy whose Lyapunov exponents along $M$ are all zero. The proofs involve the construction of non-hyperbolic measures for the induced IFS’s on the flag manifold.     

Recurrence and genericity

We prove a C¹-connecting lemma for pseudo-orbits of diffeomorphisms on compact manifolds. We explore some consequences for C¹-generic diffeomorphisms. For instance, C¹-generic conservative diffeomorphisms are transitive. To cite this article: C. Bonatti, S. Crovisier, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A spectral-like decomposition for transitive Anosov flows in dimension three

Given a (transitive or non-transitive) Anosov vector field X on a closed three dimensional manifold M, one may try to decompose (M, X) by cutting M along tori and Klein bottles transverse to X. We prove that one can find a finite collection \(\{S_1,\dots ,S_n\}\) of pairwise disjoint, pairwise non-parallel tori and Klein bottles transverse to X, such that the maximal invariant sets \(\Lambda _1,\dots ,\Lambda _m\) of the connected components \(V_1,\dots ,V_m\) of \(M-(S_1\cup \dots \cup S_n)\) satisfy the following properties:

• each \(\Lambda _i\) is a compact invariant locally maximal transitive set for X;
• the collection \(\{\Lambda _1,\dots ,\Lambda _m\}\) is canonically attached to the pair (M, X) (i.e. it can be defined independently of the collection of tori and Klein bottles \(\{S_1,\dots ,S_n\}\));
• the \(\Lambda _i\)’s are the smallest possible: for every (possibly infinite) collection \(\{S_i\}_{i\in I}\) of tori and Klein bottles transverse to X, the \(\Lambda _i\)’s are contained in the maximal invariant set of \(M-\cup _i S_i\).

To a certain extent, the sets \(\Lambda _1,\dots ,\Lambda _m\) are analogs (for Anosov vector field in dimension 3) of the basic pieces which appear in the spectral decomposition of a non-transitive axiom A vector field. Then we discuss the uniqueness of such a decomposition: we prove that the pieces of the decomposition \(V_1,\dots ,V_m\), equipped with the restriction of the Anosov vector field X, are “almost unique up to topological equivalence”.

     

The <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> generic diffeomorphism has trivial centralizer

Answering a question of Smale, we prove that the space of C ¹ diffeomorphisms of a compact manifold contains a residual subset of diffeomorphisms whose centralizers are trivial.     

Guaranteed approximation of Markov chains with applications to multiplexer engineering in ATM networks

Discrete Markov chains are applied widely for analysis and design of high speed ATM networks due to their essentially discrete nature. Unfortunately, their use is precluded for many important problems due to explosion of the state space cardinality. In this paper we propose a new method for approximation of a discrete Markov chain by a chain of considerably smaller dimension which is based on the duality theory of optimization. A novel feature of our approach is that it provides guaranteed upper and lower bounds for the performance indices defined on the steady state distribution of the original system. We apply our method to the problem of engineering multiplexers for ATM networks.     

On maximal transitive sets of generic diffeomorphisms

Abstract. – We construct locally generic C¹-diffeomorphisms of 3-manifolds with maximal transitive Cantor sets without periodic points. The locally generic diffeomorphisms constructed also exhibit strongly pathological features generalizing the Newhouse phenomenon (coexistence of infinitely many sinks or sources). Two of these features are: coexistence of infinitely many nontrivial (hyperbolic and nonhyperbolic) attractors and repellors, and coexistence of infinitely many nontrivial (nonhyperbolic) homoclinic classes.?We prove that these phenomena are associated to the existence of a homoclinic class H(P,f) with two specific properties:?– in a C¹-robust way, the homoclinic class H(P,f) does not admit any dominated splitting,?– there is a periodic point P^′ homoclinically related to P such that the Jacobians of P^′ and P are greater than and less than one, respectively. Manuscrit reĉu le 13 décembre 2000. RID="*" ID="*"This paper was partially supported by CNPq, Faperj, and Pronex Dynamical Systems (Brazil), PICS-CNRS and the Agreement Brazil-France in Mathematics. The authors acknowledge to IMPA and Laboratoire de Topologie, Université de Bourgogne, for the warm hospitality during their visits while preparing this paper. We also acknowledge M.-C. Arnaud, F. Béguin and the referees for their comments on the first version of this paper.     

SRB measures for partially hyperbolic systems whose central direction is mostly expanding

We construct Sinai-Ruelle-Bowen (SRB) measures supported on partially hyperbolic sets of diffeomorphisms – the tangent bundle splits into two invariant subbundles, one of which is uniformly contracting – under the assumption that the complementary subbundle is non-uniformly expanding. If the rate of expansion (Lyapunov exponents) is bounded away from zero, then there are only finitely many SRB measures. Our techniques extend to other situations, including certain maps with singularities or critical points, as well as diffeomorphisms having only a dominated splitting (and no uniformly hyperbolic subbundle). Oblatum 16-IV-1999 & 29-X-1999?Published online: 21 February 2000