Name: Jacob Palis Jr.
Born: March 15, 1940, Uberaba, MG, Brazil
Parents: Jacob Palis and Sames Palis
Status: Married, three children
PhD Students - 40 theses concluded to date
PhD Descendants - about 100 to date
Doctor Honoris Causa
Members of Academies of Sciences
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:
5. Invited speaker, International Congress of Mathematicians, Helsinki, 1978.
6. Editorial Board of Journals:
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.
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
with S. Newhouse and F. Takens, Bulletin American Mathematical Society, vol.82, (499-502), 1976.
20. La Topologie du Feuilletage d'un Champ de Vecteurs Holomorphe près d'une
with C. Camacho and N. Kuiper, C.R.A.S. Paris, vol.282, (959-961), 1976.
21. The Topology of Holomorphic Flows near a Singularity
with C. Camacho and N. Kuiper, Publications Math. Institut Hautes Études Scientifiques, vol.48, (5-38), 1978.
22. Topological Equivalence of Normally Hyperbolic Vector Fields
with F. Takens, Topology, vol.16, (335-345), 1977.
23. Some Developments on Stability and Bifurcations of Dynamical Systems
Lecture Notes in Mathematics, Springer-Verlag, vol.597, (495-509), 1977.
24. Geometry and Topology
editor, with M. do Carmo, Proc. of the III Latin American Mathematical School, Lecture Notes in Mathematics, Springer-Verlag, vol.597, 1977.
25. Introdução aos Sistemas Dinamicos
with W.de Melo, Notes of the X Bazilian Mathematical Colloquium, 1975. Book in Projeto Euclides, IMPA-CNPq, 1978.
26. Centralizeres of Diffeomorphisms and Stability of Suspended Foliations
Lecture Notes in Mathematics, Springer-Verlag, vol.652, (114-121), 1978.
27. Invariantes de Conjugação e Módulos de Estabilidade dos Sistemas Dinâmicos
Proceedings of the XI Brazilian Mathematical Colloquium,1978.
28. A Differentiable Invariant of Topological Conjugacies and Moduli of Stability
Astérisque, vol.51, (335-346), 1978.
29. Moduli of Stability and Bifurcation Theory
Proceedings of the International Congress of Mathematicians, Helsinki, (835-839), 1978.
30. Moduli of Stability for Diffeomorphisms
with W. de Melo, Proc. Symp. Dyn. Systems, Lecture Notes in Mathematics, Springer-Verlag, vol.819, (318-339), 1980.
31. Characterization of the Modulus of Stability for a Class of Diffeomorphisms
with W. de Melo and S.Van Strien, Lecture Notes in Mathematics, Springer-Verlag, vol.898, (266-285), 1981.
32. Families of Vector Fields with Finite Moduli of Stability
with I. P. Malta, Lecture Notes in Mathematics, Springer-Verlag, vol.898, (212-229), 1981.
33. Geometric Theory of Dynamical Systems
with W. de Melo, Springer-Verlag, 1982. Translated into Russian and Chinese.
34. Bifurcations and Stability of Families of Diffeomorphisms
with S. Newhouse and F. Takens, Publications Math. Institut Hautes Études Scientifiques, vol.57, (5-72), 1983.
35. Geometric Dynamics
editor, Proc. Int. Symposium Dynamical Systems, IMPA-1981. Lecture Notes in Mathematics, Springer-Verlag, vol.1007, 1983.
36. Stability of Parameterized Families of Gradient Vector Fields
with F. Takens, Annals of Mathematics, vol.118, (383-421), 1983.
37. A Note on the Inclination Lemma and Feigenbaum's Rate of Approach
Lecture Notes in Mathematics, Springer-Verlag, vol.1007, (630-636), 1983.
38. The Dynamics of a Diffeomorphism and Rigidity of its Centralizer
Singularities and Dynamical Systems, North Holland, (15-21), 1985.
39. Topological Invariants as Translation Number
with R. Roussarie, in Dynamical Systems and Bifurcations, Lecture Notes in Mathematics, vol.1125, (64-86), 1985.
40. Cycles and Measure of Bifurcation Sets for Two-Dimensional Diffeomorphisms
with F. Takens, Inventiones Mathematicae, vol.82, (397-422), 1985.
41. Homoclinic Orbits, Hyperbolic Dynamics and Fractional Dimension of Cantor Sets
Contemporary Mathematics, vol.58, (203-216), 1987.
42. Dimensões Fracionárias de Conjuntos de Cantor e Dinâmica Hiperbólica
Proceedings of the XV Brazillian Mathematical Colloquium, (341-353), 1987.
43. Hyperbolicity and Creation of Homoclinic Orbits
with F.Takens, Annals of Mathematics, vol.125, (337-374), 1987.
44. On the Solution of the C1 Stability Conjecture (Mañé) and the -Stability Conjecture
Proceedings of the XVI Brazillian Mathematical Colloquium, (599-606), 1988.
45. On the Continuity of Hausdorff Dimension and Limit Capacity for Horseshoes
with M.Viana, Dynamical Systems, Lecture Notes in Mathematics, Springer-Verlag, vol.1331, (150-160), 1988.
46. Homoclinic Bifurcations and Hyperbolic Dynamics
with F. Takens, Lecture Notes, vol.66. Proceedings of the XVI Brazillian Mathematical Colloquium, (10-15), 1988.
47. Topics in Dynamical Systems
Editor with R. Bamón e R. Labarca, Lecture Notes in Mathematics, Springer-Verlag, vol.1331, 1988.
48. On the C1 Omega-Stability Conjecture
Publications Math. Institut Hautes Etudes Scientifiques, vol.66, (210-215), 1988.
49. On the Solution of the Stability Conjecture and the Omega-Stability Conjecture
IX International Congress on Mathematical Physics, Adam Higher, (469-471), 1989.
50. Rigidity of Centralizeres of Diffeomorphisms
with J.C.Yoccoz, Ann. Scient. Ècole Normale Superièure, vol.22, 1989 (81-98), 1989.
51. Centralizers of Anosov Diffeomorphisms on Tori
with J.C.Yoccoz, Ann. Scient. Ecole Normale Superieure, vol.22, (99-108), 1989.
52. Homoclinic Bifurcations and Fractional Dimensions
Publicaciones Matematicas del Uruguay, vol.1, (55-66), 1989.
53. Gradient Flow, Stability Theory and Related Topics in Dynamical Systems
World Scientific, (142-145), 1989.
54. Bifurcations and Global Stability Families of Gradient
with M. J. Carneiro, Publications Mathematiques Institut Hautes Etudes Scientifiques, vol.70, (103-168), 1989.
55. Centralizers of Diffeormorphisms
Pitman Research Notes in Math., vol.221, (19-23), 1990.
56. Chaotic or Turbulent Systems, Attractors and Homoclinic Bifurcations
Revista Matematica Universitaria, Brazilian Mathematical Society, 1990.
57. Differentiable Conjugacies of Morse-Smale Diffeomorphisms
with J.C. Yoccoz, Bulletin Brazilian Mathematical Society, New Series, vol.2, (25-48), 1990.
58. Homoclinic Bifurcations, Sensitive-Chaotic Dynamics and Strange Attractors
Dyn. Systems and Related Topics, World Scientific, (466-473), 1991.
59. A Glimpse at Dynamical Systems: the Long Trajectory from the Sixties to Present Developments
Prize Talk at the Third World Academy of Sciences, Proceedings of the TWAS, 1992.
60. New Developments in Dynamics: Hyperbolicity and Chaotic Dynamics
Chaos, Resonance and Collective Dynamical Phenomena in the Solar Systems, International Astron. Union, Kluwer Acad. Publ., (363-369), 1992.
61. Dynamical Systems
with R. Bamon, R. Labarca e J. Lewowicz, Pitman Research Notes in Math., vol.285, 1993.
62. Hyperbolicity and Sensitive-Chaotic Dynamics at Homoclinic Bifurcations, Fractal Dimensions and Infinitely Many Attractors
with F. Takens, book, Cambridge University Press, 1993. Second edition: 1994
63. On the Contribution of Smale to Dynamical Systems, From Topology to Computation
volume in honour of Stephen Smale, Springer-Verlag, (165-178), 1993.
64. Homoclinic Tangencies for Hyperbolic Sets of Large Hausdorff Dimension
with J.C. Yoccoz, Acta Mathematica, vol.172, (91-136), 1994.
65. High Dimension Diffeomorphisms Displaying Infinitely Many Sinks
with M. Viana, Annals of Mathematics, vol.140, (207-250), 1994.
66. A View on Chaotic Dynamical Systems
Brazilian Journal of Physics, vol.24, (926-930), 1994.
67. Chaotic and Complex Systems
Science International, vol.58, November (27-31), 1995.
68. A Global View and Conjectures on Chaotic Dynamical Systems
Dynamical Systems and Chaos, World Scientific, vol.1, (217-225), 1995.
69. From Dynamical Stability and Hyperbolicity to Finitude of Ergodic Attractors.
Proceedings of the Third World Academy of Sciences, 11th General Conference, Italy, 1996.
70. On the Arithmetic Sum of Regular Cantor Sets
with J. C. Yoccoz, Annales de l'Inst. Henri Poincaré, Analyse Non Lineaire, vol.14, (439-456), 1997.
71. Chaotic and Complex Systems, Caos e Complexidade
Editora UFRJ, (27-38), 1999.
72. Uncertainty-Chaos in Dynamics. A Global view
Medal Lecture Third World Academy of Sciences, 1998. Proceedings of 10th General Meeting, (33-38), 1999.
73. A Global View of Dynamics and a Conjecture on the Denseness of Finitude of Attractors
Astérisque, vol.261, (339-351), 2000.
74. Homoclinic bifurcations: from Poincaré to present time
The Mathematical Sciences, After the Year 2000. World Scientific, (123-134), 2000.
75. Nonuniformily Hyperbolic Horseshoes Unleashed by Homoclinic Bifurcations and Zero Density of Attractors
with J.C.Yoccoz, C.R.A.S. Paris, vol.333, (1-5), 2001.
76. Homoclinic tangencies and fractal invariants in arbitrary dimension
with C. Moreira and M. Viana, in C.R.A.S. Paris, vol.333 (5), 2001.
77. Implicit formalism for affine-like map and parabolic composition.
with J.C. Yoccoz, Global Analysis of Dynamical systems, Inst. of Phys., IOP-London, (67-87), 2001.
78. On the codimension of gradients with umbilic singularity.
with M. J. Dias Carneiro, in prepration.
79. Wonders and Frontiers of Sciences CNPq 45 Years.
Publication of MCT-CNPq, 2001.
Editors: Jacob Palis and José Galizia Tundisi
80. Chaotic and Complex Systems
Current Science, vol.82 (4), 403-406, 2002.
81. A Global Perspective for Non-Conservative Dynamics
Annales de l'Inst. Henri Poincaré, Analyse Non Lineaire, vol.22, 485-507, 2005.
82. Non-Uniformly Hyperbolic Horseshoes Arising from Bifurcations of Poincaré
with J.C.Yoccoz, Publications Math. Institut Hautes Études Scientifiques, 194 pages, 2006. To appear
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:
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:
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:
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 Other work done at the time: 21. J. Vera 22. N. Romero 23. R. Ures 24. C. G. Moreira 25. C. A. Morales 26. E. Pujals 27. E. Catsigeras 28. B. San Martin 29. E. Colli 30. E. Luzzatto 31. M. Sambarino 32. R. Metzger 33. F. Sanchez-Salas 34. J. Martins-Rivas 35. V. Pinheiro 36. F. Rodriguez-Hertz 37. A. Tahzib 38. A. Arroyo 39. C. Vasquez 40. L. Bladismir Leal Master Degree
Robust Nonhyperbolic Dynamics and Heterodimensional Cycles, Ergodic Theory and Dynamical Systems, 1995.
Nonconnected Heterodimensional Cycles: Bifurcation and Stability, Nonlinearity, 1992.
Ergodic Theory and Dynamical Systems, vol.9, 1989.
Stability and Bifurcations of a Large Class of 3-Dimensional Vector Fields, Nonlinearity, 1996.
Persistence of Homoclinic Tangencies in Higher Dimensions, Ergodic Theory and Dynamical Systems, 1995.
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.
On the Arithmetic Difference of Cantor Sets and Bifurcations in Dynamics,
Annales de L'Institute Henri Poincaré, Analyse Non Lineaire, 1995.
Lorenz Attractor trough Saddle-Node Bifurcations, the Annales de l'Institute Henri Poincaré, Analyse Non Lineaire, 1996.
Singular Strange Attractors on the Boundary of Morse-Smale Systems, Annales Scient. École Normale Superieure, 1997.
Period Doubling Bifurcations and Homoclinic Tangencies, Annales de L'Institute Henri Poincaré, Analyse Non Lineaire, 1998.
Saddle-Focus Singular Cycles and Hiperbolicity, Annales de L'Institute Henri Poincaré, Analyse Non Lineaire, 1998.
Infinitely Many Coexisting Strange Attractors, Annales de L'Institute Henri Poincaré, Analyse Non Lineaire, 1998.
Critical and Singular Dynamics in the Lorenz Equations, Astérisque, 1999.
Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Annals of Mathematics, 2000.
On the Existence of Sinai-Ruelle-Bowen measures for contracting Lorenz Maps and Flows,
Annales de L'Institute Henri Poincaré, Analyse Non Lineaire, 2000.
Some geometric properties of ergodic attractors, Divulg. Math., vol.9, 2001.
Homoclinic and Period-Doubling Bifurcations for Higher Codimensions.
Combinatorial properties and distortion control for unimodal maps, to appear.
Stable ergodicity of certain linear automorphisms of the torus, Annals of Mathematics, 2004.
Stably ergodic systems which are not partially hyperboli, to appear.
Homoclinic Bifurcations and Uniform Hyperbolicity for Three-Dimensional Flows,
Annales de L'Institute Henri Poincaré, Analyse Non Lineaire, 2004.
Statistical Stability for Diffeomorphisms with Dominated Splitting,
Erg. Theory & Dynamics Systems, 2005.
High Dimension Diffeomorphisms Exhibiting Infinitely Many Strange Attractors, to appear.
Centralizers of Diffeomorphisms of the Circle
Dual Cantor Sets
Some of the Main Contributions
Other work done at the time:
21. J. Vera
22. N. Romero
23. R. Ures
24. C. G. Moreira
25. C. A. Morales
26. E. Pujals
27. E. Catsigeras
28. B. San Martin
29. E. Colli
30. E. Luzzatto
31. M. Sambarino
32. R. Metzger
33. F. Sanchez-Salas
34. J. Martins-Rivas
35. V. Pinheiro
36. F. Rodriguez-Hertz
37. A. Tahzib
38. A. Arroyo
39. C. Vasquez
40. L. Bladismir Leal
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 C1 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.