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.
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
I showed with François Blanchard and Bernard Host that a dynamical system (X,T) of positive entropy has necessarily proper asymptotic pairs . 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.
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 .
I was also interested in the existence of transitive sensitive subsystems . 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.
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 . 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 .
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 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 case). This result is not true if f is supposed to be only continuous, as shown by Gurevich and Zargaryan. The condition cannot be weaken either: for every integer n I built Cn transitive interval maps with no maximal measure . 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 transformation .
On the other hand, Jérôme Buzzi and I showed that the
smoothness of the map enables to give a sufficient condition for
existence  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,
htop(C,f)the entropy of the set C and
hloc(f) the local entropy of f. If the
inequality htop(f) > htop(C,f) +
hloc(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 Cr maps: if
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 To 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 . 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 RotIR(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 RotIR(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 RotIR(F) then there exists a periodic point of rotation number p/q, and if in addition p/q is in the interior of RotIR(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.