VIANA, MARCELO / Mathématiques / Chercheurs

Centre International de Recherche Scientifique


Mathématiques / VIANA, MARCELO


Professor of Mathematics & Deputy Director
Instituto de Matematica Pura e Aplicada (IMPA)
Rio de Janeiro, Brazil

Thèmes de recherche

Marcelo Viana is an established mathematician worldwide in the field of dynamical systems where he has made major contributions.

- Dynamical Systems, Ergodic Theory, Bifurcation Theory.
- Mathematical theory of chaotic dynamics and strange attractors. Ergodic
properties of chaotic systems: physical (SRB) measures, correlation functions,
stochastic stability. H\'enon-like and Lorenz-like systems. Multidimensional strange
- Properties of weak uniform hyperbolicity and their consequences: partial
hyperbolicity, existence of dominated splittings. Geometric and ergodic properties
of weakly hyperbolic systems, especially existence and niteness of physical
- Non-uniformly hyperbolic systems. Lyapunov exponents of dynamical systems
and linear cocycles. Dependence of Lyapunov exponents on the dynamics.
Bifurcations leading to chaotic behaviour: homoclinic tangencies, heteroclinic
cycles, saddle-node cycles. Relative frequency of dynamical phenomena in the
unfolding of these bifurcations, in two and higher dimensions. Coexistence of
in nitely many attractors.
- Fractional invariants, such as Hausdor\' dimension, limit capacity, and thickness,
and their role in Dynamics.

Prix et récompenses

- Srinivasa Ramanujan Prize, 2005
Invited Professor, College de France, Paris, 2002.
- UMALCA (Uni~ao Matematica da America Latina e do Caribe) Award in
Mathematics, 2000.
- Grã Cruz da Ordem do Mérito Cientifico\" granted by the President of
Brazil, 2000.
- Member of the Third World Academy of Sciences, ellected in 2000.
- Plenary Speaker, Intern. Congr. of Mathematicians, ICM-Berlin, 1998.
- Third World Academy of Sciences Award in Mathematics, 1998.
- Member of the Brazilian Academy of Sciences, ellected in 1997.
- Invited Speaker, Intern. Congr. of Mathematicians, ICM-Zurich, 1994.
- Plenary Speaker, Intern. Congr. of Mathematical Physics, Paris, 1994.
- CNPq Research Grant, since 1991, level 1A
- Guggenheim Foundation Fellowship, 1993-1994
- International Mathematical Union Grant to attend the ICM90 - Kyoto
- Calouste Gulbenkian Foundation Fellowship, 1988-1990
- Prize \"Ciencias Exactas\", Fundacão Eng. Antonio Jose de Almeida, 1984.

- Brazilian Mathematical Society, since 1987.
- American Mathematical Society, since 1994.
- Brazilian Academy of Sciences, since 1997.
- Third World Academy of Sciences, since 2000.


Pisa lectures on Lyapunov exponents, with J. Bochi, in Dynamical Systems.
Part II: Topological, geometrical and ergodic properties of dynamics.
Pubblicazioni della Classe di Scienze, Scuola Normale Superiore, Pisa.
Centro di Ricerca Matematica \"Ennio De Giorgi\" : Proceedings.

Dynamics beyond uniform hyperbolicity: A global geometric and probabilistic perspective, with C. Bonatti, L. J. Daz. Encyc. Math. Sciences, Springer Verlag.

The Lyapunov exponents of generic volume preserving and symplectic systems, with J. Bochi, Annals of Mathematics 161 (2005).

Random perturbations and statistical properties of Henon-like maps, with M. Benedicks, Annales Inst. Henri Poincare - Analyse Non-Lineaire.

Almost all cocycles over any hyperbolic system have non-zero Lyapunov
exponents, preprint 2005.

Hausdorff dimension for non-hyperbolic repellers II: DA dieomorphisms,
with V. Horita, Discrete & Continuous Dynamical Systems 13(2005), 1125{1152.

Existence and uniqueness of maximizing measures for robust classes of local
dieomorphisms, with K. Oliveira, preprint 2005.

Dynamics in the moduli space of Abelian diferentials, with A. Avila,
preprint 2005.

Singular hyperbolic attractors are chaotic, with V. Araujo, M. J. Pacico, E.
Pujals, preprint 2005.

Geometrie et dynamique des surfaces plates, preprint 2005.

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