Research Topics

My research domain is topological dynamics. A topological dynamical system is a continuous transformation T from X to X, the space X being most of the time metric compact. The evolution of the system is given by iterating successively the map, Tⁿ being the transformation T composed n times (Tⁿ = T o T o ... o T). One wants to study the behaviour of the system when the time tends to infinity.

I am mainly, but not only, interested in one-dimensional dynamical systems: continuous transformations on the interval or topological graphs. I also worked on dynamical systems on compact metric spaces and on topological Markov chains, which are symbolic systems on infinite graphs. Interval maps can, under certain condition, be represented by topological Markov chains, and this representation is a key tool to study the existence of measures of maximal entropy.

Asymptotic pairs in positive entropy systems

Historically, the term of chaos was introduced by Li and Yorke in 1975 to describe the behaviour of some systems on the interval. In the sequel, others definitions of chaos were proposed; they do not coincide in general, and none can be considered as the only ``good'' definition of chaos. Indeed, the idea of chaos relies on a body of properties. It is important to study the relations between these properties and also to known if some regular behaviours can be found together with a given chaotic property.

Let T be a continuous transformation from X to X, where X is a compact metric space; d is the distance. If x and y are two points in X, (x,y) is called a Li-Yorke pair if

$\begin{displaymath} \limsup_{n\to+\infty}d(T^n x, T^n y)>0\quad\mbox{et}\quad \liminf_{n\to+\infty}d(T^n x, T^n y)=0. \end{displaymath}$

The system (X,T) is said chaotic in the sense of Li-Yorke if there exists an uncountable set S such that two distinct points of S form a Li-Yorke pair. On the other hand, (x,y) is called an asymptotic pair if

$\displaystyle\lim_{n\to +\infty} d(T^n x,T^n y)=0$ ;

the pair is proper if x is not equal to y.

I showed with François Blanchard and Bernard Host that a dynamical system (X,T) of positive entropy has necessarily proper asymptotic pairs [2]. This results negatively answers an question of Huang and Ye who studied systems in which all pairs of distinct points are Li-Yorke and wondered if such systems can have positive entropy. Almost at the same time, Blanchard, Glasner, Kolyada and Maass showed that positive entropy implies chaos in the sense of Li-Yorke. Consequently there exist in a positive entropy system both "chaotic" (Li-Yorke) and "non chaotic" (asymptotic) pairs of points.

More precisely we showed that, for every ergodic measure of positive entropy, almost every point belongs to a proper asymptotic pair. If in addition the transformation is invertible then for almost every point x there exists an uncountable set of points y such that the pair (x,y) is asymptotic for T and Li-Yorke for T^-1, which recalls stable and unstable manifolds in Anosov. These results rely almost entirely on ergodic proofs.

Chaos for interval maps

An interval map is a dynamical system defined by a continuous transformation from a compact interval to itself. I investigated in order to do a survey of the known results concerning chaos for interval maps [book]. The situation is quite different from the general case because the various notions of chaos mostly coincide. For example, transitivity, which is often considered as a basic assumption to obtain some uniformity, leads to a strongly chaotic behaviour. Indeed, it implies sensitivity to initial conditions and density of periodic points, thus chaos in the sense of Devaney, as well as chaos in the sense of Li-Yorke and a positive entropy. Moreover, it is almost equivalent to topological mixing, which in turn implies other properties (specification, uniformly positive entropy).

At this stage, the following question is raised: what imply the various properties if one does not assume transitivity? For some of them, as sensibility, generic chaos or density of periodic points, the reverse implication is partially true, that is, there exists a transitive component composed of one or several subintervals. On the other hand, the various periods of the periodic points that can coexist are ruled by Sharkovskii's order, and the kind of periodic points is linked with topological entropy: positive entropy is equivalent to the existence of a periodic point whose period is not a power of 2. Moreover, an interval map has a positive entropy if and only there is a subsystem which is chaotic in the sense of Devaney.

Concerning dense chaos (density of Li-Yorke pairs in the product space), I showed that it implies that the entropy is greater that or equal to log 2/2 and that there exists a periodic point of period 6 [6].

I was also interested in the existence of transitive sensitive subsystems [5]. It is known that an interval map of positive entropy has a transitive sensitive subsystem; I showed that the reciprocal does not hold. Moreover I proved that for an interval map the existence of such a subsystem implies chaos in the sense of Li-Yorke and I built an counter-example showing that the reciprocal is false.

Topological Markov chains and graphs classification

A topological Markov chain (or Markov shift) is a symbolic system defined by the set of two-sided infinite paths on a countable oriented graph, endowed with the shift transformation. Contrary to probabilistic Markov chains there is no a priori probability. Not only these systems are interesting by themselves but they are a tool in the study of probability measures of maximal entropy (or maximal measures) because we know necessary and sufficient conditions for existence and uniqueness of such measures; we will come back on this point in the next section, dedicated to maximal measures for interval maps. If the graph is finite the situation is simple; in particular there always exists a maximal measure, that can be computed with matrix calculus. We are only interested in infinite graphs.

Vere-Jones classified connected oriented graphs in three groups (transient, null recurrent, positive recurrent) with respect to In 1970, Gurevich showed that this classification is strongly related to the existence of maximal measures. Indeed, if the graph is connected, the Markov chain on this graph admits a maximal measure if and only if it is positive recurrent; in this case the maximal measure is unique and it is a Markov measure. Let us indicate that for null recurrent graphs, there exists an infinite measure which plays the same role as a maximal measure, but it this situation the entropy has to be defined in a different way.

I showed that if the entropy of a topological Markov chain is greater than its local entropy then the graph is positive recurrent thus has a measure of maximal entropy [3]. Since there are links between topological Markov chains and interval maps, this result strengthens Buzzi's conjecture stating that the same result is true for interval maps, but this question is still open. Let us recall that for dynamical systems on a compact metric space it is known that null local entropy implies existence of a maximal measure.

Salama gave a more geometric approach of the classification transient/null recurrent/ positive recurrent in terms of existence of subgraphs or supergraphs of same entropy: a connected graph with no proper subgraph of same entropy is positive recurrent, and a connected graph is transient if and only if it is strictly included in a transient graph of same entropy. I completed this work by showing that a transient graph G can always be included in a recurrent graph of equal entropy, which is either null recurrent or positive recurrent depending on the properties of G [3].

Measures of maximal entropy for interval maps

A maximal measure is an invariant probability measure the entropy of which reaches the supremum of metric entropies; by the variational principle its entropy is equal to the topological entropy. Maximal measures are particularly interesting because they reflect the whole topological complexity and they enable to see where this complexity concentrates. They have a less obvious physical sense than absolutely continuous measures, however they are preserved by conjugacies.

One can associate to an interval map f an oriented, generally infinite, graph called Markov diagram. This construction, based on the dynamics of monotone subintervals, was first done by Hofbauer for piecewise monotone maps then generalised by Buzzi. Under some conditions, the topological Markov chain on this graph represents most of the dynamics of f. In particular Buzzi showed that if f is C¹ and if its entropy is greater than the topological entropy of critical points (i.e., points in a neighbourhood of which f is not monotone) then there is a bijection between ergodic maximal measures of f and those of its Markov diagram. The problem of existence of such measures is carried on the graph.

An interval map f which is, either piecewise monotone, or $C^{\infty}$ admits at least a maximal measure, and this measure is unique if f is transitive (results of Hofbauer for the piecewise monotone case, Newhouse and Buzzi for $C^{\infty}$ case). This result is not true if f is supposed to be only continuous, as shown by Gurevich and Zargaryan. The $C^{\infty}$ condition cannot be weaken either: for every integer n I built Cⁿ transitive interval maps with no maximal measure [1]. I used the geometric approach of Salama presented in the previous section to show that the Markov graph associated to these interval maps is transient; then the absence of maximal measure for the graph is carried on the interval.

I deduced for the results above that for every integer n there exist Cⁿ transitive interval map that are not Borel conjugate to any $C^{\infty}$ transformation [4].

On the other hand, Jérôme Buzzi and I showed that the smoothness of the map enables to give a sufficient condition for existence [7] by combining results related to differentiability and properties of topological Markov chains. Consider a C¹ interval mapf. Let C be the set of critical points, h_top(C,f)the entropy of the set C and h_loc(f) the local entropy of f. If the inequality h_top(f) > h_top(C,f) + h_loc(f) is satisfied then f has a finite number of ergodic maximal measures, with uniqueness in case of transitivity. Using a bound of the entropy of the zeros of the derivative and of local entropy we obtain a condition which can be checked more easily for C^r maps: if

$\begin{displaymath} h_{top}(f)> \frac{2}{r} \log \Vert f'\Vert _{\infty}, \end{displaymath}$

then the number of ergodic maximal measures if finite and non null.

Rotation set for transformations of degree 1 on some topological graphs

A connected set X is a topological graph if there exists a finite subset S such that every connected component of X\S is homeomorphic to an open interval. If in addition X has no subset homeomorphic to a circle, X is called a tree. Because of their dimension 1, dynamical systems on topological graphs share some properties with interval maps.

Sharkovskii's theorem determines the possible sets of periods of periodic points for interval maps. A similar determination was given for some graphs, in particular n-stars (n intervals glued at one endpoint) but it remains an open problem in the general case. For circle maps of degree 1 the set of periods is given by the theory of rotation numbers. The set of rotation numbers is a closed interval [a,b] and for every rational number p/q in this interval with p, q coprime there exists a periodic point of period q and of rotation number p/q. More precisely if p/q belongs to (a,b) the set of periods of periodic points of rotation number p/q is exactly the set of multiple of q; if a (resp. b) is a rational number the periods of periodic points of rotation number a (resp. b) can be determined using Sharkovskii's order.

I worked with Lluís Alsedà to generalise rotation numbers for maps of degree 1 on graphs with a unique loop, and more generally on the class T^o of graphs G with a loop S satisfying that for every connected component C of G\S the closure of C intersects S in a unique point [9]. Such a map f can be lifted to a map F on an infinite graph T included in the complex plane, 1-periodic, containing the real line (corresponding to the loop S) and such that F(x+1)=F(x)+1 for all x in T. The periodic points for f are the periodic points mod 1 for F. There is no difficulty to generalise the definition of rotation numbers to this context. The set of rotation numbers is not necessarily connected however the subset of rotation numbers of points x in IR, denoted by Rot_IR(F), is a non-empty compact interval, and if the union of Fⁿ(IR) is dense in T (in particular if F is transitive) then Rot_IR(F) is equal to the rotation set. A periodic point of period q has a rotation number equal to p/q for some relative integer p. Reciprocally if p/q belongs to Rot_IR(F) then there exists a periodic point of rotation number p/q, and if in addition p/q is in the interior of Rot_IR(F) then there exists N such that for all n>N there exists a periodic point in IR, of period nq and of rotation number p/q.

We conjecture that the rotation set is closed and has a finite number of connected components. I have obtained significative advances in this direction.

Last revised on October 1st, 2004