Jacob Palis

Geometric theory of dynamical systems : an introduction

1 Differentiable Manifolds and Vector Fields.- 0 Calculus in ?n and Differentiable Manifolds.- 1 Vector Fields on Manifolds.- 2 The Topology of the Space of Cr Maps.- 3 Transversality.- 4 Structural Stability.- 2 Local Stability.- 1 The Tubular Flow Theorem.- 2 Linear Vector Fields.- 3 Singularities and Hyperbolic Fixed Points.- 4 Local Stability.- 5 Local Classification.- 6 Invariant Manifolds.- 7 The ?-lemma (Inclination Lemma). Geometrical Proof of Local Stability.- 3 The Kupka-Smale Theorem.- 1 The Poincare Map.- 2 Genericity of Vector Fields Whose Closed Orbits Are Hyperbolic.- 3 Transversality of the Invariant Manifolds.- 4 Genericity and Stability of Morse-Smale Vector Fields.- 1 Morse-Smale Vector Fields Structural Stability.- 2 Density of Morse-Smale Vector Fields on Orientable Surfaces.- 3 Generalizations.- 4 General Comments on Structural Stability. Other Topics.- Appendix: Rotation Number and Cherry Flows.- References.

BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS

We consider one parameter families or arcs of diffeomorphisms. For families starting with Morse-Smale diffeomorphisms we characterize various types of (structural) stability at or near the first bifurcation point. We also give a complete description of the stable arcs of diffeomorphisms whose limit sets consist of finitely many orbits. Universal models for the local unfoldings of the bifurcating periodic orbits (especially saddle-nodes) are established, as well as several results on the global dynamical structure of the bifurcating diffeomorphisms. Moduli of stability related to saddle-connections are introduced.

Neighborhoods of hyperbolic sets

Inventiones math. 9, 121 - 134 (1970) Neighborhoods of Hyperbolic Sets M. HIRSCH, J. PALIS, C. PUGH, and M. SHUB (Univ. of Warwick) w 1. Introduction In this paper we study the asymptotic behavior of points near a compact hyperbolic set of a C r diffeomorphism (r__>l)f: M - ~ M , M being a compact manifold. The purpose of our study is to complete the proof of Smales O-stability Theorem by demonstrating (2.1), (2.4) of [6]. O denotes the set of non-wandering points for f Smales Axiom A requires [5]: (a) O has a hyperbolic structure, (b) the periodic points are dense in O. Hyperbolic structure, the stable manifold of O, and fundamental neighborhoods are discussed in ~j 2 and 5. The result of [-6] proved here is: I f f obeys Axiom A then there exists a proper fundamental neighbor- hood V for the stable manifold of f2 such that the union of the unstable manifold oft2 and the forward orbit of V contains a neighborhood of 0 in M. As a consequence we have: l f f obeys Axiom A then any point whose orbit stays near 0 is asymp- totic with a point of 0. Section 8 of the mimeographed version of [ 1 ] contains a generalization of the above results with an incorrect proof. A correct generalization is: (1.1) Theorem. I f A is a compact hyperbolic set then WU(A)uO+ V contains a neighborhood U of A, where V is any fundamental neighbor- hood for WS(A) and 0+ V= U f ( v ) . i f A has local product structure n>O then a proper fundamental neighborhood may be found and any point whose forward orbit lies in U is asymptotic with some point of A. Theorem (1.1) is proved in w 5, local product structure is discussed in [-5] and in w In w we prove the analogous theorems for flows. Here is an example, due to Bowen, of a compact hyperbolic set A which does not have local product structure, has no proper fundamental neighborhood and for which there are points asymptotic to A without being asymptotic with any point of A.

Hyperbolicity and the creation of homoclinic orbits

We consider one-parameter families {sp,; t E R} of diffeomorphisms on surfaces which display a homoclinic tangency for yt = 0 and are hyperbolic for yt < 0 (i.e., (pf has a hyperbolic nonwandering set); the tangency unfolds into transversal homoclinic orbits for yt positive. For many of these families, we prove that (p is also hyperbolic for most small positive values of yt (which implies much regularity of the dynamical structure). A main assumption concerns the limit capacities of the basic set corresponding to the homoclinic tangency.

The topology of holomorphic flows with singularity

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Bifurcations of Morse–Smale Dynamical Systems

Publisher Summary This chapter describes a phenomenon that occurs in the bifurcation theory of one-parameter families of diffeomorphisms. This is the appearance of infinitely many different topological conjugacy classes of structurally stable diffeomorphisms, each class containing a diffeomorphism with an infinite nonwandering set, in every neighborhood of certain diffeomorphisms in the boundary of the Morse–Smale diffeomorphisms. This phenomenon occurs quite frequently in the following sense: any Morse–Smale diffeomorphism, on a manifold of dimension greater than one, can be moved through a one-parameter family which first exhibits some simple phase portrait changes to a new Morse–Smale diffeomorphism. The natural place where the phenomenon appears is in the construction of a cycle, and it occurs in that situation generically. On the other hand, under a reasonably general condition if one approaches the boundary of the Morse–Smale diffeomorphisms without creating a cycle, then the only new structurally stable diffeomorphisms one can find nearby will also be Morse–Smale. In this case as well, one can encounter an infinite number of topologically different Morse–Smale diffeomorphisms. It is shown that all of the conditions assumed for these results are true for open subsets of the space of one-parameter families of diffeomorphisms.

Vector fields generate few diffeomorphisms

Let M be a compact C manifold without boundary. Let Diff(M) be the group of C diffeomorphisms of M with the C topology. If r = 0 , this corresponds to the group of homeomorphisms of M. A C flow on M is a continuous group homomorphism (p:R->Diff(M), O^s^oo. In a natural way, C vector fields generate C flows and Lipschitz vector fields generate C° (topological) flows. We say that feDiff(M) embeds in a C flow, s^.r, iff is the map at time one of such a flow. Our main purpose is to announce results showing that few diffeomorphisms, in the sense of Baire category, embed in flows or are generated by vector fields with some mild differentiability or Lipschitz condition. Here we will prove only one of these results concerning flows generated by vector fields. Several authors have treated similar questions. For C diffeomorphisms of the circle, our last theorem follows from stronger results of Kopell [2] and it was also proved in [4], where more references can be found. The embedding of diffeomorphisms in topological flows was also considered in [5]. The author acknowledges very useful conversations with C. Pugh and M. Shub and several people at IMPA. We now show that with a mild assumption on the vector fields, the diffeomorphisms they possibly generate form a subset of first category in DiflP(Af)Fix a riemannian metric on M. Let x be a singularity for a vector field X. X is said to be Lipschitz at x if there exists a constant K>0 such that \X(y)\^Kd(x,y) for every y eM, where d(x,y) is the distance between x and y. Let % denote the set of C° vector fields on M that generate topological flows and are Lipschitz at the singularities.

Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms

SummaryWe describe a large class of one-parameter families φ, {ϕμ}, μ∈ℝ, of two-dimensional diffeomorphisms which arestable for μ<0, exhibit acycle for μ=0, and thereafter have a bifurcation set of positive but arbitrarily smallrelative measure for μ in small intervals [0, σ]. A main assumption is that the basic sets involved in the cycle havelimit capacities that are not too large.

Hyperbolic Nonwandering Sets on Two-Dimensional Manifolds†

Publisher Summary This chapter discusses the diffeomorphisms of a compact manifold whose nonwandering sets are hyperbolic. Considering f as such a diffeomorphism and Ω its nonwandering set, it considers two questions: (1) whether the periodic orbits of f dense in Ω and (2) whether f can be approximated by an Ω-stable diffeomorphism. The chapter discusses that both questions have a positive answer when M is a closed two-dimensional manifold. The first question was suggested by Smale and related to it is Anosovs closing lemma.

Stability of parametrized families of gradient vector fields

It is shown that the stable one-parameter families of gradient vector fields are open and dense.