Sheldon E. Newhouse

The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms

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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.

Lectures on Dynamical Systems

A basic question in the theory of dynamical systems is to study the asymptotic behaviour of orbits. This has led to the development of many different subjects in mathematics. To name a few, we have ergodic theory, hamiltonian mechanics, and the qualitative theory of differential equations.

Quasi-Elliptic Periodic Points in Conservative Dynamical Systems

1. One of the most important and beautiful subjects in the theory of dynamical systems concerns the orbit structure near an elliptic periodic point of an area preserving diffeomorphism f of the two dimensional disk D2. Recall that a periodic point p of such an f is a point for which f ( p) = p for some integer n >0. Assuming n is the least such integer, p is called elliptic if the derivative of fn at p, Tpfn, has non-real eigenvalues of norm one. If the eigenvalues of Tpfn have norm different from one, p is called hyperbolic. It has been known for a long time that elliptic periodic orbits occur in many problems in mechanics, in particular, the restricted three body problems [3, 8]. When f is real analytic, Birkhoff established a normal form for f near an elliptic fixed point provided the eigenvalues of Tf are not roots of unity. If this normal form is not linear, he showed that the fixed point is a limit of infinitely many periodic points, and that among these accumulating periodic points both elliptic and hyperbolic types appear [28]. A theorem due to Kolmogorov, Arnold, and Moser asserts that many f-invariant circles enclose a general elliptic fixed point p, and that on each of these circles f behaves like a rotation through an angle 9 with 0/27r strongly irrational [8, 9]. This result implies that general elliptic orbits are Liapounov stable. By contrast, according to a theorem of Hartman and Grobman [12, 16], the local structure near a hyperbolic fixed point p is not complicated. The diffeomorphism f behaves topologically like its derivative. The points asymptotic to p under forward and backward iterates form smooth curves (the stable and unstable manifolds of p) meeting transversely at p, and the other nearby orbits lie on continuous curves which are easily described. Recently, E. Zehnder has shown that, generically, many hyperbolic periodic orbits near an elliptic periodic orbit have homoclinic points (non-periodic intersections of the stable and unstable manifolds of the same hyperbolic periodic orbit) [28]. Thus the rather intricate picture in Figure 1 (taken from [3]) generically occurs near any elliptic periodic point p. Each circle is invariant under a power

Entropy and volume

An inequality is given relating the topological entropy of a smooth map to growth rates of the volumes of iterates of smooth submanifolds. Applications to the entropy of algebraic maps are given.

On the estimation of topological entropy

We study a method for estimating the topological entropy of a smooth dynamical system. Our method is based on estimating the logarithmic growth rates of suitably chosen curves in the system. We present two algorithms for this purpose and we analyze each according to its strengths and pitfalls. We also contrast these with a method based on the definition of topological entropy, using(n, ɛ)-spanning sets.

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.

Pre-image entropy

We define and study new invariants called pre-image entropies which are similar to the standard notions of topological and measure-theoretic entropies. These new invariants are only non-zero for non-invertible maps, and they give a quantitative measurement of how far a given map is from being invertible. We obtain analogs of many known results for topological and measure-theoretic entropies. In particular, we obtain product rules, power rules, analogs of the Shannon–Breiman–McMillan theorem, and a variational principle.

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.

Rigorous and accurate enclosure of invariant manifolds on surfaces

Knowledge about stable and unstable manifolds of hyperbolic fixed points of certain maps is desirable in many fields of research, both in pure mathematics as well as in applications, ranging from forced oscillations to celestial mechanics and space mission design. We present a technique to find highly accurate polynomial approximations of local invariant manifolds for sufficiently smooth planar maps and rigorously enclose them with sharp interval remainder bounds using Taylor model techniques. Iteratively, significant portions of the global manifold tangle can be enclosed with high accuracy. Numerical examples are provided.