"The secret of change is to focus all of your energy not on fighting the old, but on building the new" - Socrates

Selected Publications (by topic): 

Thermodynamic formalism

Equilibrium States, Linear Response and Phase Transitions 

A firt aim of the thermodynamic formalism in dynamical systems is the construction and study of ergodic properties of invariant measures - equilibrium states - that attain the topological complexity of the dynamics. In the case of a constant potential, equilibrium states are maximal entropy measures.  A special interest is given to the dependence of equilibrium states on the potential and the dynamical system, and to the existence of phase transitions. After pioneering work of Sinai, Ruelle and Bowen in the sixties and many contributions in the last decades, it is still a challenge is to understand the thermodynamic formalism of dynamical systems with some non-uniform hyperbolicity. 

Statistical properties of weak Gibbs measures

Gibbs measures play a key role in the thermodynamic formalism of uniformly hyperbolic dynamical systems, as equilibrium states for regular potentials arise are invariant Gibbs measures. The concept of weak Gibbs measure appeared naturally in dealing with non-uniformly hyperbolic dynamical systems, where the Gibbs property can be obtained at the instants of hyperbolicity. The geometric aspects of weak Gibbs measures allow one to study the velocity of convergence in the Shannon-McMillan-Breiman, their multifractal analysis and relation with large deviations  (even in  non-additive contexts).

Decay of Correlations and Limit Theorems for Flows

A major problem in ergodic theory is to describe the speed of mixing of dynamical systems, which measures an asymptotic independence of invariant probability measures. This problem has shown to be harder for flows than for diffeomorphisms. The time-1 map of a flow is at most partially hyperbolic, with an invariant foliation given by the orbits of the flow. Even though the neutral behavior of the foliation by orbits, the topological and ergodic mixing properties of hyperbolic and non-uniformly hyperbolic flows have much to do with a shear along the orbits. For instance, constant suspension flows over topologically mixing Anosov diffeomorphisms define transitive but not topologically mixing Anosov flows.  We gave first examples of flows (singular and non-singular) with robust exponential decay of correlations. The global understanding of the decay of correlations for hyperbolic and non-uniformly hyperbolic flows, including a solution to Bowen-Ruelle's conjecture in full generality, is still  incomplete.


Topological dynamics

Specification, gluing orbit properties and their consequences

Besides from its intrinsic interest, the specification property introduced by Bowen is a very useful tool in ergodic theory, in the study of large deviations for continuous observables and multifractal analysis, just to mention two applications. However, this property and the gluing orbit property turned out to be rare beyond the uniformly hyperbolic context, even though it is generic for volume preserving homeomorphisms. 

Birkhoff and Lyapunov non-typical points 

The ergodic theorems ensure that Birkhoff averages are convergent in a full probability measure set, and similarly whenever the Birhoff averages are replaced by quotients of Birkhoff averages or by sub-additive sequences of maps. In the context of smooth conformal expanding maps it is known that set of points which are non-typical is very rich: it is Baire generic, has full topological pressure and has full Hausdorff dimension. As the specification property, used in the classical argument, is a rather strong requirement,  other approaches have been used to describe the set of non-typical points for large classes of dynamical systems. 

Metric mean dimension

The metric mean dimension, introduced by Lindenstrauss and Weiss, is an entropy-like invariant that describes the complexity of dynamical systems with infinite topological entropy, including generic homeomorphisms or shifts with a compact and uncountable alphabet.


Rotation Theory

The concept of rotation number introduced by Poincaré to study circle homeomorphisms has been much fruitful as a mechanism to detect periodic points or minimal behavior in this one-dimensional context, among other properties. In the case of continuous maps on the torus homotopic to the identity, Misiurewicz and Ziemian introduced several notions of rotation sets, which are convex and seem to replicate the one-dimensional toolbox in the special case of the 2-torus. The study of totation sets for maps on higher dimensional tori and flows requires the use of different techniques and have been much less studied.

Algebraic aspects in dynamical systems

Centralizers in Dynamical Systems

Smale conjecured that typical smooth diffeomorphisms should have trivial symmetries ie, every symmetry is an iterate of the diffeomorphism. The conjecture was proved in special contexts, including C^r circle diffeomorphisms, hyperbolic basic sets for C^r-diffeomorphisms, C^1-generic diffeomorphisms  and C^infty uniformly hyperbolic dynamics on tori.  The problem has a negative answer for less regular maps.  The symmetries of flows and R^d actions need to contemplate their reparameterizations, and their triviality is associated to the existence of non-trivial first integrals.  A complete solution to Smale's conjecture is still unavailable.

Group and Semigroup Actions

Continuous maps acting on compact metric spaces have invariant probability measures, and satisfy a variational principle for the topological pressure. In the context of group actions by continuous maps on a compact metric space, there exist probability measures invariant by all elements in a group action if and only if the group is amenable. Moreover, while stationary measures play a crucial role in the context of group actions, these need not describe the thermodynamic formalism of semigroup actions. For the development of ergodic theory of non-amenable (semi)group actions it seems necessary to understand the suitable notions of invariance and entropy.

Some other topics
Uniform and non-uniform hyperbolicity 

In the context of linear cocycles, Oseledets celebrated theorem ensures that almost every point has well defined Lyapunov exponents and invariant subspaces. A major question in dynamics is to understand if non-uniform hyperbolicity is abundant. In a recent work we prove that typical linear cocycles on semisimple Lie groups have at least one positive Lyapunov exponent. 

Quantitative Poincaré recurrence

Poincaré recurrence theorem implies the recurrence of almost every point (with respect to any invariant probability measure).  Kac's formula provides a first quantitative estimate on the average expected return time to positive measures subsets. A step further is to provide sharper quantitative estimates for recurrence, e.g. proving limit laws for return and/or hitting times. 

Ergodic optimization

Ergodic optimization in dynamical systems concerns the description of the non-emtpy set of invariant measures which have largest (or smallest) time averages for Birkhoff or Kingman ergodic theorems, and study of their ergodic properties. The ergodic optimization for non-uniformly hyperbolic dynamical systems is still not completely understood.

(Updated 2023)