**Vitae**

**Name:** Jacob Palis Jr.

**Born:** March 15, 1940, Uberaba, MG, Brazil

**Parents:** Jacob Palis and Sames Palis

**Status:** Married, three children

**Degrees**

- Bachelor (Engineering) Federal University of Rio de Janeiro, 1962

- Master University of California - Berkeley, 1966

- Ph.D. University of California - Berkeley, 1967

**Fellowships**

- National Research Council of Brazil, 1965-1967

- Guggenheim Foundation, Post-Doctoral Fellowship - University of California Berkeley, 1973

**PhD Students** - 40 theses concluded to date

**PhD Descendants** - about 100 to date

**Research Areas**

- Global stability of dynamical systems

- Bifurcations and fractional dimensions

- Global scenario for chaotic systems

**Present Positions**

- Professor - Instituto Nacional de Matemática Pura e Aplicada - IMPA, since 1971

- President, Academy of Sciences for the Developing World - TWAS, 2007-2009

- President - Brazilian Academy of Sciences - ABC,
2007-2010

- Director, Brazilian Academy of Sciences - ABC, 1977-1979; 1979-1981; 2001-2004.

- Director, Instituto Nacional de Matemática Pura e Aplicada - IMPA, 1993-2003.

- Vice-President, Brazilian Academy of Sciences - ABC, 1999-2001.

- Member of the Scientific Committee, National Research Council of Brazil - CNPq,
1988-1992; 1994-1998; 2001-2005.

- President Elected, Academy of Sciences for the Developing World - TWAS, 2007-2009.

- Member of the Scientific and Strategic Council - COSS, Collège de France, 2003-2008.

- Secretary General, Academy of Sciences for the Developing World - TWAS, 2000-2006.

- Member of the Scientific Advisory Committee of the ETH, Zürich, 1990-2006.

- Chair, International Center for Theoretical Council - ICTP, Trieste-Italy, 2003-2005;
Member of the Scientific Committee, 1993-2005.

- Founding Member of the Latin American and Caribean Mathematical Union - UMALCA, 1995;
Chair of its first Scientific Committee.

- President, International Mathematical Union - IMU, 1999-2002; Secretary, 1991-1999;
Member of Executive Board, 1982-1991.

- Vice-President, International Council for Science - ICSU, 1996-1999.
Member of Executive Board, 1993-1996.

- Member of the Scientific Committee of Conferences, Workshops, Schools at IMPA,
ICTP and several main Universities/Institutes in Europe, South and North Americas.

- One of the two Coordinators of the Study Panel responsible for the InterAcademy
Council's Report: "Inventing a Better Future: A Strategy for Building Worldwide Capacities
in Science and Technology", launched at the United Nations in February 2004 by the Secretary
General Kofi Annan. The Report has been translated from English to French, Portuguese, Chinese
and Arab.

**Prizes**

- Trieste Science Prize, in Mathematics, 2006.

- Prize Mexico for Science and Technology, 2001.

- InterAmerican Prize for Science, Organization of the American States, 1995.

- National Prize for Science and Technology, Brazil, 1990.

- Prize The Academy of Sciences for the Developing World - TWAS, l988 - Mathematics.

- Prize Moinho Santista (highest Brazilian prize for Science at the time), 1976.

- Prize Universidade do Brasil, best graduating student, 1962.

**Decorations**

- Chevalier de la Légion d'Honneur - França, 2005.

- Brazilian Scientist honoured by UNESCO in its 60th Anniversary, 2006.

- Medal of Honour - Brazilian Ministry of Education, Coordenation of the Advancement of Higher Education Personnel - CAPES, 50 th Anniversary, 2001.

- Medal of Scientific Merit Carlos Chagas Filho - State of Rio de Janeiro, 2000.

- Medal Lecture - Academy of Sciences for the Developing World - TWAS, 1998.

- Grand-Croix National Order of Scientific Merit, Brazil, 1994.

**Doctor Honoris Causa**

- Doctor Honoris Causa, Universidad de Ingenieria, Peru, 2003.

- Doctor Honoris Causa, Universidad de la Habana, 2001.

- Doctor Honoris Causa, University of Santiago de Chile, 2000.

- Doctor Honoris Causa, University of Warwick, 2000.

- Doctor Honoris Causa, University of Chile, 1996.

- Doctor Honoris Causa, State University of Rio de Janeiro, 1993.

**Members of Academies of Sciences**

- Member of the Russian Academy of Sciences, 2006.

- Member of Norwegian Academy of Sciences, 2005.

- Member of European Academy of Sciences, 2004.

- Foreign Member of French Academy of Sciences, 2002.

- Foreign Member, United States National Academy of Sciences, 2002.

- Foreign Member, Mexican Academy of Sciences, 2001.

- Honorary Member of The Peruvian Mathematical Society, 1999.

- Member of the Chilean Academy of Sicences, 1997.

- Member of the Indian Academy of Sciences, 1995.

- Member of The Academy of Sciences for the Developing World - TWAS, 1991.

- Member of the Brazilian Academy of Sciences, 1970.

**Especial Invited Lectures and Distinctions**

1. Honoured on the occasion of the 60th Anniversary: Geometry Methods in Dynamics,
Vols. I and II, Astérisque Journal, 286-287, 2003.

2. Newton's Distinguished Lecture, Jawaharlar Nehru Centre for Advanced Scientific Research, 2001.

3. Hallim Distinguished Lecture - The Korean Academy of Science and Technology, 1999.

4. Invited as a main speaker to major international conferences:

- Conference in honour of Kolmogorov - (Moscow University, 2003).

- Conference in Honour of F. Takens (University of Groningen, Holand, 2001).

- Conference in Honour of I. G. Petrovsky (Moscow University, 2001).

- Conference in Honour of J. Moser (ETH-Zurich, 2001).

- Conference in Honour of Adrien Douady (Univ. Paris, 1995).

- Conference in Honour of A. N. Kolmogorov (Euler Institute, St. Petersbourg, 1992).

- Conference in Honour of Stephen Smale (The Contribution of Smale to Dynamical Systems, University of California - Berkeley, l990).

- Conference in Honour of René Thom (Institut Henri Poincaré, 1988).

- Topological Methods in Analysis (Celebration of 600th years of the Univ.of Heidelberg, l986).

- Lefschetz Centennial Conference (IPN-Mexico and Princeton University, l984).

5. Invited speaker, International Congress of Mathematicians, Helsinki, 1978.

6. Editorial Board of Journals:

- Ergodic Theory and Dynamical Systems.

- Nonlinearity, London Mathematical Society.

- Annales de la Facultè de Sciences de Toulouse.

Currently:

- Bull. of the Brazilian Mathematical Society, Chief Editor;

- Annales de l'Institut Henri Poincaré;

- Acta Applicandae Mathematicae;

- Chaos, Nonlinear Science

- Chinese Annals of Math.

- Communications in Contemporary Mathematics

**List of Publications
**

**2. On Morse-Smale Dynamical Systems**

*Topology, vol.19, (385-405), 1969.
*

**3. Structural Stability Theorems**

*with S.Smale, Proceedings of the Institute on Global Analysis,
American Math. Society, vol.XIV, (223-232), 1970.
*

**4. A Note on Omega-Stability**

*Proceedings of the Institute on Global Analysis, American
Mathematical Society, vol.XIV, (220-222), 1970.
*

**5. Local Structure of Hyperbolic Fixed Points in Banach Space**

*Anais da Academia Brasileira de Ciencias, vol.40, (263-266), 1968.
*

**6. Neighborhoods of Hyperbolic Sets**

*with M.Hirsch, C. Pugh and M. Shub, Inventiones Mathematicae, vol.9, (212-234),1970.
*

**7. Omega-Explosions**

*Bulletin of the Brazillian Mathematical Society, vol.1, (55-57), 1970.
*

**8. Omega-Stability and Explosions**

*Lecture Notes in Mathematics, Springer-Verlag, vol.1206, (40-42), 1971.
*

**9. Omega-Explosions for Flows**

*Proceedings American Math. Society, vol.27, (85-90), 1971.
*

**10. Sistemas Dinamicos**

*Seminar Notes, IMPA, 1971.
*

**11. Hyperbolic Nonwandering Sets on Two-Dimensional Manifolds**

*with S. Newhouse, in Dynamical Systems, Academic Press, (293-302), 1973.
*

**12. Bifurcations of Morse-Smale Dynamical Systems**

*with S. Newhouse, in Dynamical Systems, Academic Press, (303-366), 1973.*

**13. Vector Fields Generate Few Diffeomorphisms**

*Bulletin American Mathematical Society, vol.80, (503-505), 1974.
*

**14. Non Differentiability of Invariant Foliations**

*with C. Pugh and C. Robinson, Lecture Notes in Mathematics, Springer-Verlag, vol.468,
(234-241), 1975.
*

**15. Genericity Theorems in Topological Dynamics**

*with C. Pugh, M. Shub and D. Sullivan, Lecture Notes in Mathematics, Springer-Verlag,
vol.468, (241-251), 1975.
*

**16. Fifty Problems in Dynamical Systems**

*with C. Pugh, Lecture Notes in Mathematics, Springer-Verlag, vol.468, (345-353), 1975.
*

**17. Arcs of Dynamical Systems: Bifurcations and Stability**

*Lecture Notes in Mathematics, Springer-Verlag, vol.468, (48-53), 1975.
*

**18. Cycles and Bifurcations Theory**

*with S. Newhouse, Astérisque, vol.31, (44-140), 1976.
*

**19. Stable Arcs of Diffeomorphisms
**

**20. La Topologie du Feuilletage d'un Champ de Vecteurs Holomorphe près d'une
Singularitè
**

**21. The Topology of Holomorphic Flows near a Singularity
**

**22. Topological Equivalence of Normally Hyperbolic Vector Fields
**

**23. Some Developments on Stability and Bifurcations of Dynamical Systems
**

**24. Geometry and Topology
**

**25. Introdução aos Sistemas Dinamicos
**

**26. Centralizeres of Diffeomorphisms and Stability of Suspended Foliations
**

**27. Invariantes de Conjugação e Módulos de Estabilidade dos Sistemas Dinâmicos
**

**28. A Differentiable Invariant of Topological Conjugacies and Moduli of Stability
**

**29. Moduli of Stability and Bifurcation Theory
**

**30. Moduli of Stability for Diffeomorphisms
**

**31. Characterization of the Modulus of Stability for a Class of Diffeomorphisms
**

**32. Families of Vector Fields with Finite Moduli of Stability
**

**33. Geometric Theory of Dynamical Systems
**

**34. Bifurcations and Stability of Families of Diffeomorphisms
**

**35. Geometric Dynamics
**

**36. Stability of Parameterized Families of Gradient Vector Fields
**

**37. A Note on the Inclination Lemma and Feigenbaum's Rate of Approach
**

**38. The Dynamics of a Diffeomorphism and Rigidity of its Centralizer
**

**39. Topological Invariants as Translation Number
**

**40. Cycles and Measure of Bifurcation Sets for Two-Dimensional Diffeomorphisms
**

**41. Homoclinic Orbits, Hyperbolic Dynamics and Fractional Dimension of Cantor Sets
**

**42. Dimensões Fracionárias de Conjuntos de Cantor e Dinâmica Hiperbólica
**

**43. Hyperbolicity and Creation of Homoclinic Orbits
**

**44. On the Solution of the C1 Stability Conjecture (Mañé) and the -Stability Conjecture
**

**45. On the Continuity of Hausdorff Dimension and Limit Capacity for Horseshoes
**

**46. Homoclinic Bifurcations and Hyperbolic Dynamics
**

**47. Topics in Dynamical Systems
**

**48. On the C ^{1} Omega-Stability Conjecture
**

**49. On the Solution of the Stability Conjecture and the Omega-Stability Conjecture
**

**50. Rigidity of Centralizeres of Diffeomorphisms
**

**51. Centralizers of Anosov Diffeomorphisms on Tori
**

**52. Homoclinic Bifurcations and Fractional Dimensions
**

**53. Gradient Flow, Stability Theory and Related Topics in Dynamical Systems
**

**54. Bifurcations and Global Stability Families of Gradient
**

**55. Centralizers of Diffeormorphisms
**

**56. Chaotic or Turbulent Systems, Attractors and Homoclinic Bifurcations
**

**57. Differentiable Conjugacies of Morse-Smale Diffeomorphisms
**

**58. Homoclinic Bifurcations, Sensitive-Chaotic Dynamics and Strange Attractors
**

**59. A Glimpse at Dynamical Systems: the Long Trajectory from the Sixties to Present Developments
**

**60. New Developments in Dynamics: Hyperbolicity and Chaotic Dynamics
**

**61. Dynamical Systems
**

**62. Hyperbolicity and Sensitive-Chaotic Dynamics at Homoclinic Bifurcations, Fractal Dimensions and Infinitely Many Attractors
**

**63. On the Contribution of Smale to Dynamical Systems, From Topology to Computation
**

**64. Homoclinic Tangencies for Hyperbolic Sets of Large Hausdorff Dimension
**

**65. High Dimension Diffeomorphisms Displaying Infinitely Many Sinks
**

**66. A View on Chaotic Dynamical Systems
**

**67. Chaotic and Complex Systems
**

**68. A Global View and Conjectures on Chaotic Dynamical Systems
**

**69. From Dynamical Stability and Hyperbolicity to Finitude of Ergodic Attractors.
**

**70. On the Arithmetic Sum of Regular Cantor Sets
**

**71. Chaotic and Complex Systems, Caos e Complexidade
**

**72. Uncertainty-Chaos in Dynamics. A Global view
**

**73. A Global View of Dynamics and a Conjecture on the Denseness of Finitude of Attractors
**

**74. Homoclinic bifurcations: from Poincaré to present time
**

**75. Nonuniformily Hyperbolic Horseshoes Unleashed by Homoclinic Bifurcations and Zero Density of Attractors
**

**76. Homoclinic tangencies and fractal invariants in arbitrary dimension
**

**77. Implicit formalism for affine-like map and parabolic composition.
**

**78. On the codimension of gradients with umbilic singularity.
**

**79. Wonders and Frontiers of Sciences – CNPq 45 Years.
**

Editors: Jacob Palis and José Galizia Tundisi

**80. Chaotic and Complex Systems
**

**81. A Global Perspective for Non-Conservative Dynamics
**

**82. Non-Uniformly Hyperbolic Horseshoes Arising from Bifurcations of Poincaré
Heteroclinic Cycles
**

**PhD. Students: 40
**

from eleven different countries

Ph.D. supervisor (theses concluded) of:

Welington de Melo, Ricardo Mañé, Pedro Mendes,
Geovan Tavares dos Santos, Paulo Sad, Artur Oscar Lopes, Luiz Fernando Carvalho da Rocha, Genésio Lima
dos Reis, Iaci Pereira Malta, Maria Izabel Camacho, Jorge Beloqui, Rafael Labarca, Sergio Plaza, Jaques
Gheiner, Roberto Markarian, Jaime Vera, Jorge da Rocha, Lorenzo Diaz, Leonardo Mora, Marcelo Viana,
J. Martin-Riva, Neptali Romero, Pedro Duarte, Raul Ures, Carlos Gustavo Moreira, Carlos Morales,
Elenora Catsigeras, Bernardo San Martin, Enrique Pujals, E. Luzzatto, E. Colli, M. Sambarino,
F. Sanchez-Sala, R. Metzger, V. Pinheiro, F. Rodriguez-Hertz, Ali Tahzib, Aubin Arroyo,
Carlos Vasquez, Bladismir Leal.

**1. W. de Melo**

Structural Stability on 2-Manifolds, Inventiones Mathematicae, l973.

**2. P. Mendes**

Stability on Open Manifolds,Journal of Differential Equations, l974.

**3. R. Mañé**

Persistent Submanifolds, Bulletin American Mathematical Society, 1974.

Transactions American Mathematical Society, l977.

**4. Geovan Tavares dos Santos**

Polynomial Vector Fields in the Plane, Proc.of the III Latin American School of Mathematics

Lecture Notes in Mathematics, Springer Verlag, l977.

**5. P. Sad**

Centralizers of Vector Fields, P.Sad, Topology, l979.

**6. A. O. Lopes**

Structural Stability and Hyperbolic Attractors,Transactions American Mathematical Society, l979.

**7. I. P. Malta**

Hyperbolic Birkhoff Center, Transactions American Mathematical Society, l980.

**8. M. I. T. Camacho**

Generic Properties of Homogeneous Vector Fields in R3, Transactions American Mathematical Society, l981.

**9. G. L. dos Reis**

Stability of Equivariant Vector Fields, Transactions American Mathematical Society, l983.

**10. L. F. da Rocha**

Characterization of Isotopy Classes of Morse-Smale Diffeomorphisms on Surfaces, Ergodic

Theory and Dynamical Systems, l985.

**11. J. A. Beloqui**

Moduli of Stability for Vector Fields on 3-Manifolds, Journal of Differential Equations, l985.

**12. R. Labarca**

Stability of Parametrized Families of Vector Fields, Anais da Academia Brasileira de Ciencias, l985.

Dynamical Systems and Bifurcation Theory, Longman, Pitman Research Notes in Mathematics Series, vol.160, 1987.

**13. M. Viana**

Abundance of Strange Attractors in Higher Dimensions, Bull. Braz. Math. Soc., vol.241, (13-62), 1993.

*Other work done at the time:*

- Continuity of Hausdorff dimensions and limit capacity for horseshoes, with J.Palis,
Topics in Dynamics.

Lecture Notes in Mathematics, Springer-Verlag, vol. l33l, (l50-l60), 1988.

- Discontinuity of Hausdorff dimension and limit capacity on arcs of diffeomorphisms,
with L.J.Diaz,

Ergodic Theory and Dynamical Systems, vol.9, (403-425), 1989.

- Abundance of Strange Attractors, with L. Mora, Acta Mathematica, 1993.

**14. J. da Rocha**

Rigidity of Centralizers of Analytic Diffeomorphisms, Ergodic Theory and Dynamical Systems, 1993.

**15. L. Mora**

Birkhoff-Henon Attractors for Dissipative Perturbations of Area-Preserving Twist Maps,

Ergodic Theory and Dynamical Systems, 1994.

*Other work done at the time:*

- Abundance of Strange Attractors, with M. Viana, Acta Mathematica, 1993.

**16. S. Plaza**

Bifurcation and Global Stability of Saddle-Nodes in Higher Dimensions, Anais da Academia
Brasileira de Ciências,

l988. Annales de la Facultè des Sciences de Toulouse, 1994.

**17. J. Gheiner**

Bifurcations of Codimension Two for Diffeomorphisms, Anais da Academia Brasileira de Ciências, 1989.

Nonlinearity, vol.7 (1), 1994.

**18. R. Markarian**

Non Uniform Hyperbolic Billiards, Longman, Pitman Research Notes in Mathematics, vol.221, 1990.

Annales de la Facultè des Sciences de Toulouse, 1994.

*Other work done at the time:*

- Billiards with Pesin Region of Measure One, Communications in
Mathematical Physics, vol.ll8, l988.

**19. P. Duarte**

There are Many Elliptic Orbits in the Standard Family of Area Preserving Maps,
Annales de L'Institut Henri Poincaré,

Analyse Non Lineaire, 1994.

**20. L. Diaz**

Robust Nonhyperbolic Dynamics and Heterodimensional Cycles, Ergodic Theory and Dynamical Systems, 1995.

Nonconnected Heterodimensional Cycles: Bifurcation and Stability, Nonlinearity, 1992.

*Other work done at the time:*

- Descontinuity of Hausdorff Dimension and Limit Capacity on Arcs of Diffeomorphisms,

Ergodic Theory and Dynamical Systems, vol.9, 1989.

**21. J. Vera**

Stability and Bifurcations of a Large Class of 3-Dimensional Vector Fields, Nonlinearity, 1996.

**22. N. Romero**

Persistence of Homoclinic Tangencies in Higher Dimensions, Ergodic Theory and Dynamical Systems, 1995.

**23. R. Ures**

On the Approximation of Henon-like Attractors by Homoclinic Tangencies,
Ergodic Theory and Dynamical Systems, 1995.

Abundance of Hyperbolicity in the C1 Topology, Annales Scientifiques de l'Ècole Normale Superieur, 1995.

**24. C. G. Moreira**

On the Arithmetic Difference of Cantor Sets and Bifurcations in Dynamics,

Annales de L'Institute Henri Poincaré, Analyse Non Lineaire, 1995.

**25. C. A. Morales**

Lorenz Attractor trough Saddle-Node Bifurcations, the Annales de l'Institute Henri Poincaré,
Analyse Non Lineaire, 1996.

**26. E. Pujals**

Singular Strange Attractors on the Boundary of Morse-Smale Systems, Annales Scient. École
Normale Superieure, 1997.

**27. E. Catsigeras**

Period Doubling Bifurcations and Homoclinic Tangencies, Annales de L'Institute Henri Poincaré,
Analyse Non Lineaire, 1998.

**28. B. San Martin**

Saddle-Focus Singular Cycles and Hiperbolicity, Annales de L'Institute Henri Poincaré,
Analyse Non Lineaire, 1998.

**29. E. Colli**

Infinitely Many Coexisting Strange Attractors, Annales de L'Institute Henri Poincaré,
Analyse Non Lineaire, 1998.

**30. E. Luzzatto**

Critical and Singular Dynamics in the Lorenz Equations, Astérisque, 1999.

**31. M. Sambarino**

Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Annals of Mathematics, 2000.

**32. R. Metzger**

On the Existence of Sinai-Ruelle-Bowen measures for contracting Lorenz Maps and Flows,

Annales de L'Institute Henri Poincaré, Analyse Non Lineaire, 2000.

**33. F. Sanchez-Salas**

Some geometric properties of ergodic attractors, Divulg. Math., vol.9, 2001.

**34. J. Martins-Rivas**

Homoclinic and Period-Doubling Bifurcations for Higher Codimensions.

**35. V. Pinheiro**

Combinatorial properties and distortion control for unimodal maps, to appear.

**36. F. Rodriguez-Hertz**

Stable ergodicity of certain linear automorphisms of the torus, Annals of Mathematics, 2004.

**37. A. Tahzib**

Stably ergodic systems which are not partially hyperboli, to appear.

**38. A. Arroyo**

Homoclinic Bifurcations and Uniform Hyperbolicity for Three-Dimensional Flows,

Annales de L'Institute Henri Poincaré, Analyse Non Lineaire, 2004.

**39. C. Vasquez**

Statistical Stability for Diffeomorphisms with Dominated Splitting,

Erg. Theory & Dynamics Systems, 2005.

**40. L. Bladismir Leal**

High Dimension Diffeomorphisms Exhibiting Infinitely Many Strange Attractors, to appear.

**Master Degree**

1. The concept of foliated tubular families and their compatibility (also called global stable and unstable foliations), that plays a key role on questions of global structural stability of dynamical systems (vector fields and diffeomorphisms) as well as of stability of parametrized families of dynamical systems. This idea was introduced in l966 in my Ph.D. thesis to prove the stability of Morse-Smale systems in dimensions less than or equal to three. The was done independently of the famous work of Anosov and, in fact, for Anosov systems the foliations are uniquely defined and their leaves (stable and unstable, respectively) have the same dimension everywhere. In general, the foliations have different dimensions and compatibility means that they should fit together: when interesting each other, a leaf of smaller dimension should be contained in a leaf of bigger dimension.

2. Stability of Morse-Smale systems (with S.Smale).In particular, among gradient vector fields the stable ones are dense. This solved positively the basic question of whether there are structurally stable systems on every manifold. It also inspired the beautiful Stability Conjecture formulated by Smale and myself: a system is stable if and only if its limit set is hyperbolic and it satisfies the transversality condition. In the complex domain, our result implies rather immediately the local stability of holomorphic flows near a singularity in the Poincaré domain.

3. A theory on how Morse-Smale systems bifurcate or lose stability. This was done with S.Newhouse, and later with F.Takens. It turns out that much of the so called "chaotic" phenomena (complicated dynamics) already takes place just across some of these bifurcations. We introduced here the notion of rotation interval for endomorphisms of the circle (much studied afterwards). We also obtained (somewhat surprisingly) a sharp rigidity for the equivalence of unfolding of saddle-node bifurcations. The study was done in the context of one-parameter families of dynamical systems.

4. The discover of some surprising differentiable invariants of topological conjugacies of dynamical systems (ratio of eigenvalues or logarithm of eigenvalues), related to nontransversal saddle-connections. This lead to "moduli spaces" for topological conjugacies or equivalences in dynamical systems. Such line of research was much developed by several authors and is much in line with (3) above. Moduli space for holomorphic flows near a singularity in the Siegel domain in C3 in general and in the linear case in Cn was determined in a work with C. camacho and N. Kuiper : they are always non trivial. That is, the moduli space has always finite but positive dimension. Actually, due to a subsequent result of Chaperon, the same is also true in the nonlinear case in Cn (the linear case was independently studied by Ily'ashenko and Ladis.) We also provided globally stable holomorphic flows in CP2 (Poincaré-type singularities).

5. Stability of a generic 1-parameter family of gradient vector fields, done with F. Takens, and of 2-parameter families of gradients done with M.J.Carneiro. This work has published in the Annals of Math. and Publ. Math. Institut des Hautes Études Scientifiques, respectively. A well known question of Thom and Arnold that inspired our work is whether this is true, up to four parameters (we know that this is false for a larger number of parameters).

6. Solution of the Centralizer Problem (posed by Smale): most hyperbolic diffeomorphisms (hyperbolic limit set) have trivial centralizer. This means that a generic hyperbolic diffeomorphism commutes only with its own powers. The work was done with J.C.Yoccoz. The results can be interpreted in the context of the holonomy group of a foliation to conclude that there are many structurally stable foliations.

7. Poincaré manifested much amazement with his "discovery" of homoclinic orbits and how complicated is the dynamics of a transformation in the presence of such orbits. Earlier contributions to the understanding of these orbits were done notably by Birkhoff, Cartwright-Littlewood, Levinson and Smale. With S.Newhouse, we studied how homoclinic orbits are created for one-parameter families of diffeomorphisms starting at a dynamically simple one (Morse-Smale); see (3) above. A main hypothesis was that the homoclinic orbit corresponded to the first bifurcation point (in the so called Feigenbaum-Hènon case there is a cascade of different bifurcations beforehand). We analysed the unfolding of these homoclinic orbits and showed that rather "frequently" in the parameter space one gets hyperbolic dynamics (hyperbolic limit sets). Recently, Takens and myself improved this result substantially: the parameter value corresponding to the initial homoclinic orbit is indeed a point of density of hyperbolic dynamics. (Density points in a parameter space for certain kind of dynamics was first presented by Arnold for diffeormorphisms of the circle). Moreover, our families can start at hyperbolic systems (not necessarily Morse-Smale ones) if the Hausdorff dimension (or Kolmogorov's capacity) of certain stable and unstable foliation is not too big. We conjectured that the converse is also true (large fractional dimensions should imply much bifurcation).This was later proved to be true by myself and Yoccoz, as mentioned in (9) below.

8. In 1988, in a remarkable paper, Mañé proved the C1 Stability Conjecture, formulated more than 20 years before by Smale and myself. Based on Mañé 's work, I succeeded in proving the C^{1} *Omega*-Stability Conjecture, where *Omega* stands for the nonwandering set. Thus, we now fully understand that hyperbolicity and the no cycle property on the limit set (or nonwandering set) are necessary and sufficient conditions for stability of the main part of the dynamics, i.e. in the limit set. This settles a fundamental question concerning the structure of stable dynamical systems in general.

9. Beginning with the work mentioned in (7) and Newhouse's result about the existence of infinitely many sinks, we have been building up a theory of homoclinic bifurcations - chaotic systems. The scenario that we envisage is the following:

a) any diffeomorphism (endomorphim) can be approximated by a hyperbolic one or else by one exhibiting a homoclinic bifurcation,

b) the Hausdorff dimension and thickness of the hyperbolic set associated to the homoclinic tangency, can indicate what kind of dynamics is prevalent when we unfold this tangency: hyperbolicity, strange attractors, sinks or other persistent phenomena.

We already know that the following phenomena occur when we unfold a homoclinic tangency:

I- there are cascade of period doubling bifurcations (Yorke-Alligood);

II- there are strange attractors for a positive measure set in the parameter space (Benedicks - Carleson, Mora - Viana),

III- thee are intervals in the parameter space with residual subsets corresponding to infinitely many sinks (Newhouse),

IV- there is a prevalence of hyperbolicity in the presence of Hausdorff dimension smaller than one (Palis-Takens).

V- there is no prevalence of hyperbolicity in the presence of large Hausdorff dimension (Palis - Yoccoz).

Also, we conjecture that the set of values in the parameter space corresponding to infinitely many sinks has (Lebesgue) measure zero.

Thus, there should be a prevalence of hyperbolicity or some weak form of hyperbolicity and of strange attractors in main bifurcations like the homoclinic ones and critical saddle-mode cycles.

The program above, already partially fulfilled, conveys a main structural mechanism granting chaotic or turbulent regimes: the unfolding of homoclinic tangencies or critical sadde-mode cycles. Moreover, we conjecture that all the above complicated bifurcating phenomena are near each other: near a system exhibiting one of them, there is a system exhibiting any other one. Also, the following principle seems to emerge from this work: fractal dimensions of invariant sets associated to a bifurcation play a fundamental role in the measure of the bifurcation set in a parameter space. Some of the results have been extended to higher dimensions, like the simultaneous existence of infinitely many periodic attractors (Palis-Viana: Annals of Math.).

10. In 1995, we have set up a program to the following global conjecture for non-conservative dynamics in compact manifolds: A Global View on Dynamics: Conjectures and Results, invited to be presented in Astérisque:

Setting : diffeomorphisms, flows, transformations (real, complex) on compact manifolds with suitable topology : C1, C, C, ...

Global conjecture on the finitude of attractors and their metric stability:

I- Denseness of finitude of attractors - there is a dense set D of dynamics such that any element of D has finitely many attractors whose union of basins of attraction has total probability;

II- Existence of physical (SBR) measures - the attractors of the elements in D are either periodic or they support a physical measure;

III- Metric stability of basins of attraction - for any element in D and any of its attractors, for almost all small perturbations in generic k-parameter families of dynamics, k , there are finitely many attractors whose union of basins is nearly (Lebesque) equal to the basin of the initial attractor; such perturbed attractors are periodic or they support a physical measure;

IV- Stochastic stability of attractors - the attractors of elements in D are stochastically stable in their basins of attraction;

V- For generic families of one-dimensional dynamics, we conjecture that with total probability in parameter space, the attractors are either periodic or carry an absolutely continuous invariant measure.

The general view is to study persistent (positive probability) dynamical properties in general k-parameter families through elements of D near the initial system.