Available online at www.sciencedirect.com Indagationes Mathematicae 22 (2011) 144–146 www.elsevier.com/locate/indag On Floris Takens and our joint mathematical work Jacob Palis∗ Instituto Matematica´ Pura e Aplicada (IMPA), Estrada Dona Castorina, 110 Jardim Botanico, 22460-320 Rio de Janeiro, RJ, Brazil A Remarkable Personality Floris was one of the finest mathematicians of his generation: very creative and elegant inhis writings and expositions. His presence in mathematical meetings and, in general, mathematical circles, was always to be much noticeable due to his intellectual brilliance and assertive personality. At the same time, he was a sensitive human being and a wonderful friend for life. He certainly influenced many mathematicians worldwide, especially in Europe and Latin America, and, in this last case, Brazil. His frequent visits to IMPA, The National Institute for Pure and Applied Mathematics, certainly represented an invaluable contribution to create and to consolidate at the institute a magical mathematical research ambiance, particularly for young talents. Several other Brazilian centers shared the privilege of his presence. In recognition, the Brazilian Academy of Sciences has elected him as a Foreign Member in 1981, when he was just forty years old. He was also a Member of the Royal Dutch Academy of Arts and Sciences. A Glimpse on Our Joint Work It is no surprise that I first heard about Floris and actually met him at the IHES — Institut des Hautes Etudes´ Scientifique, the scientific home of Rene´ Thom and David Ruelle. I was invited to visit IHES by Thom, a good friend of Mauricio Peixoto and IMPA, but I was introduced to Floris by Ruelle, that later would become another special friend of IMPA. ∗ Tel.: +55 21 511 1749, +55 21 529 5270; fax: +55 21 512 4112, +55 21 512 4115. E-mail addresses: [email protected], [email protected]. 0019-3577/$ - see front matter ⃝c 2011 Published by Elsevier B.V. on behalf of Royal Netherlands Academy of Arts and Sciences. doi:10.1016/j.indag.2011.09.004 J. Palis / Indagationes Mathematicae 22 (2011) 144–146 145 Floris and Ruelle were working on no less than the nature of turbulence. I was taken by surprise with their rather simple but powerful idea that turbulence fundamentally concerns the presence of strange (chaotic) attractors. Globally, my joint work with Floris was partially done with Sheldon Newhouse, who had visited IMPA for about two years in the early seventies. It dealt with two very much interrelated topics in dynamics. The first one has to do in various ways with topological orbit equivalence of dynamics: a homeomorphism on the phase space sending trajectories of a dynamical system onto the trajectories of another one. Dynamical systems here are Cr ; r ≥ 1, flows or diffeomorphisms of a compact boundaryless C1 manifold. In particular, a dynamical system is structurally stable or just stable if it is topologically equivalent to all Cr , r ≥ 1, nearby dynamics. Similar concepts apply to parametrized families of dynamics. In such a case, two families of dynamics are topologically equivalent if there is a homeomorphism on the parameter space such that for corresponding parameter points the corresponding dynamical systems in the phase space are topologically equivalent. In one of our papers [1] published in the Annals of Math., we proved that there is an open and dense subset of C1 one-parameter families of gradient vector fields such that its elements are structurally stable. I have enjoyed very much to have done this work with Floris. One reason was that I had proved the stability result for gradient fields in low dimensions and with Steven Smale, my Ph.D. adviser, in all dimensions. As pointed out in the paper with Floris “the key difficulty is the fact that the stable manifolds (the attracting set) of the singularities vary in dimension”. For this reason the concept of stable and unstable tubular families or foliations was introduced already in low dimensions. Fitted together, they provide globally compatible coordinate systems, from which flow conjugacies are naturally constructed. In [1] we joined forces so that these ideas were extended to the parametrized families of gradients. But now we also had do deal quite deeply with the bifurcation theory of (gradient) fields, the second main topic of my work with Floris. A considerable part of the paper is dedicated to understanding one-parameter unfolding of tangencies between stable and unstable manifolds in the global context of gradient dynamics. Before that, we published a paper together in Topology, our first, on the topological equivalence of normally hyperbolic dynamical systems [2]. In this paper, we considered C1 vector fields X and X 0 on smooth manifolds M and M0, leaving invariant normally hyperbolic compact submanifolds N in M and N 0 in M0. We then proved that a topological equivalence between X=N and X 0=N 0, which can be lifted to bundle isomorphisms between the stable and unstable bundles, can be extended to neighborhoods of N in M and of N 0 in M0. Anticipating what we would pursue in a much more complex situation in [1], we made use of invariant foliations to obtain the desired extension of the initial topological equivalence between X=N and X 0=N 0. I now come to a joint work with Floris and Sheldon [7]. It was a hard, very interesting piece of work and it took us years to complete. It was published in Publications Math. IHES in 1983 [8]. Several of the results were announced in the Bulletin of the AMS in 1976. There we described how one-parameter families of diffeomorphisms starting at Morse–Smale ones first bifurcate or 146 J. Palis / Indagationes Mathematicae 22 (2011) 144–146 cease to be stable and how stable such a process can be up to and including the first bifurcation parameter value. In this paper we discussed stability for families of diffeomorphisms, up to and including the bifurcation parameter value, according to whether or not the family of conjugacies are continuous with respect to the parameter. The results are fairly complete. The most beautiful and surprising case is that of what we have called a saddle–node bifurcation, displaying unexpected rigidity for continuous conjugacies at its center manifold. Also, the strong stable and unstable foliations must be preserved by a continuous conjugacy between two families going through saddle–nodes. The last part of my joint work with Floris concerns one-parameter families of hyperbolic surface diffeomorphisms. It consists of two papers published in Inventiones Math. [3] and the Annals of Math. [4], as well as a book published by Cambridge University Press [5] that substantially expanded a previous version of a text for a course delivered at the 16th Brazilian Math. Colloquium and published by IMPA [6]. It was an intense and very fruitful collaboration. In the papers we discuss when for such families we lose hyperbolicity through the creation of a homoclinic tangency or the creation of a cycle. We then show that if the Hausdorff dimension of the hyperbolic set is smaller than one (or if this holds for their sum, when more than one set is involved in a cycle), we have total density of points in the parameter line, at the first bifurcating one, that correspond to hyperbolic diffeomorphisms. This sharpened substantially a previous result obtained with Newhouse stated only in terms of limsup and in a more limited context. All these facts and much more were discussed in our joint book ‘Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations’ [5]. Particular attention was given to Smale’s horseshoe, to the relations between limit capacities and Hausdorff dimension and the beautiful results of Newhouse on the creation of infinitely many periodic attractors (sinks) at unfoldings of homoclinic tangencies for surface diffeomorphisms. On Floris I still read our book and papers. I miss his flute concerts. At one time with Sheldon at the guitar. References [1] Palis Jacob, Takens Floris, Stability of parametrized families of gradient vector fields, Ann. of Math. (2) 118 (3) (1983) 383–421. [2] Palis Jacob, Takens Floris, Topological equivalence of normally hyperbolic dynamical systems, Topology 16 (4) (1977) 335–345. [3] Palis Jacob, Takens Floris, Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms, Invent. Math. 82 (3) (1985) 397–422. [4] Palis Jacob, Takens Floris, Hyperbolicity and the creation of homoclinic orbits, Ann. of Math. (2) 125 (2) (1987) 337–374. [5] Palis Jacob, Takens Floris, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, in: Fractal dimensions and infinitely many attractors, in: Cambridge Studies in Advanced Mathematics, vol. 35, Cambridge University Press, Cambridge, 1993, p. x C 234pp. [6] Palis Jacob, Takens Floris, Homoclinic bifurcations and hyperbolic dynamics. [16th Brazilian Mathematics Colloquium] Instituto de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, 1987, iv C 134 pp. [7] Newhouse Sheldon, Palis Jacob, Takens Floris, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Etudes´ Sci. Publ. Math. (57) (1983) 5–71. [8] Newhouse Sheldon, Palis Jacob, Takens Floris, Stable arcs of diffeomorphisms, Bull. Amer. Math. Soc. 82 (3) (1976) 499–502..