Each flow dominates a unique gradient flow (the gradient part) which in turn dominates any other gradient flow dominated by the original flow. The set of points carried to rest points of the gradient part (the chain recurrent set) is characterized in terms of a recurrence relation. Filtrations of the given flow correspond to those of the gradient part.

This paper concerns estimates for periodic solutions of a very general class of Hamiltonian systems of prescribed energy. The estimtes are a priori upper and lower bounds for the action integral in terms of the period.

Positive tilt maps of the annulus are studied, and a correspondence is developed between the rotation set of the map and certain of its Morse decompositions. The main tool used is a characterization of fixed point free lifts of positive tilt maps. As an application, some alternative hypotheses under which the conclusions of the Aubry-Mather theorem hold are given, and it is also shown that the rotation band of a chain transitive set is always in the rotation set of the map.

The existence of transversal ejection—collision orbits in the restricted three-body problem is shown to imply, via the KAM theorem, the existence, for certain intervals of (large) values of the Jacobi constant, of an uncountable number of invariant punctured tori in the corresponding (non-compact) energy surface. The proof is based on a comparison between Levi-Civita and McGehee regularizing variables. That these transversal ejection-collision orbits do actually exist was proved in [5] in the case where one of the primaries has a small mass and the zero-mass body revolves around the other (and for all values of the Jacobi constant compatible with the existence of three connected components for the Hill region); it is proved here without any restriction on the masses, well in the spirit of Conley's thesis [3].

Numerical-valued cohomological index theories for G-pairs (X, A) over B, where G is a compact Lie group, have proved useful in critical point theory and in proving Borsuk—Ulam and Bourgin—Yang theorems. More information (which is lost in taking numerical values) is obtained using an ideal-valued theory, and this theory is applied to estimating the size of the zero set of a G-map from certain G-manifolds to a G-module. Parametrized versions of these theorems are also obtained by a principle which applies quite generally.

An equivariant version of Conley's homotopy index theory for flows is described and used to find periodic solutions of a Hamiltonian system locally near an equilibrium point which is at resonance.

We prove that if f is a homeomorphism of the annulus which is homotopic to the identity and has a compact invariant chain transitive set L, then either f has a fixed point or every point of L moves uniformly in one direction: clockwise or counterclockwise. If f is area-preserving, then the annulus itself is a chain transitive set, so, in the presence of a boundary twist condition, one obtains a fixed point. The same techniques apply to homeomorphisms of the torus T2. In this setting we show that if f is homotopic to the identity, preserves Lebesgue measure and has mean translation 0, then it has at least one fixed point.

We give a new proof of the fact that an area-preserving monotone twist map of the annulus with one p/q-periodic orbit on which the map preserves the ordering on the angular coordinate (i.e. Birkhoff or monotone periodic orbits) actually has a second such orbit distinct from the first.

A theorem is proved giving a condition under which certain standing wave solutions of non-linear Schrödinger-type equations are linearly unstable. The eigenvalue equations for the linearized operator at the standing wave can be analysed by dynamical systems methods. A positive eigenvalue is then shown to exist by means of a shooting argument in the space of Lagrangian planes. The theorem is applied to a situation arising in optical waveguides.

Conformal equivalence theorem from complex analysis says that every Riemannian metric on a compact surface with negative Euler characteristics can be obtained by multiplying a metric of constant negative curvature by a scalar function. This fact is used to produce information about the topological and metric entropies of the geodesic flow associated with a Riemannian metric, geodesic length spectrum, geodesic and harmonic measures of infinity and Cheeger asymptotic isoperimetric constant. The method is rather uniform and is based on a comparison of extremals for variational problems for conformally equivalent metrics.

The single pendulum is one of the fundamental model problems in the theory of dynamical systems; coupled pendula, or equivalently, two elastically coupled particles in a periodic potential on a line, are a natural extension of intrinsic interest. The system arises in various physical applications and it inherits some rudiments of the behaviour exhibited by its finite-dimensional parent, the sine-Gordon equation. Among these phenomena are the so-called caterpillar solutions, whose behaviour is reminiscent of solitons. These solutions turn out to have a transparent geometrical explanation. There is an interesting bifurcation picture associated with the system: the parameter region is broken up into the set of ‘pyramids’ parametrized by pairs of integers; these integers characterize the behaviour of the associated solutions.

The role in the Conley index of mappings between flows is considered. A class of maps is introduced which induce maps on the index level. With the addition of such maps to the theory, the homology Conley index becomes a homology theory. Using this structure, an analogue of the Lefschetz theorem is proved for the Conley index. This gives a new condition for detecting fixed points of flows, extending the classical Euler characteristic condition.

Let in be an exact, area-preserving, monotone twist diffeomorphism of the cylinder and let 10 be a Liouville number. We will show that arbitrarily close to in the C∞ topology there exists a C∞ diffeomorphism with no homotopically non-trivial invariant circle of rotation number ω.

The equilateral triangle family of relative equilibria of the 4-body problem consists of three particles of mass 1 at the vertices of an equilateral triangle and the fourth particle of arbitrary mass m at the centroid. For one value of the mass m this relative equilibrium is degenerate. We show that families of isosceles triangle relative equilibria bifurcate from the equilateral triangle family as m passes through the degenerate value.

The square family of relative equilibria of the 5-body problem consists of four particles of mass 1 at the vertices of a square and the fifth particle of arbitrary mass m at the centroid. For one value of the mass m this relative equilibrium is degenerate. We show that families of kite and isosceles trapezoidal relative equilibria bifurcate from the square family as m passes through the degenerate value.

This paper surveys the work of Charles Conley and his students on Morse decompositions for flows on compact metric spaces, as well as the more recent development of the connection matrix formalism for detecting connections between the Morse sets of a Morse decomposition.

This paper is concerned with minimal foliations; these are foliations whose leaves are extremals of a prescribed variational problem, as for example foliations consisting of minimal surfaces. Such a minimal foliation is called stable if for any small perturbation of the variational problem there exists a minimal foliation conjugate under a smooth diffeomorphism to the original foliation. In this paper the stability of special foliations of codimension 1 on a higher-dimensional torus is established. This result requires small divisor assumptions similar to those encountered in dynamical systems. This theorem can be viewed as a generalization of the perturbation theory of invariant tori for Hamiltonian systems to elliptic partial differential equations for which one obtains quasi-periodic solutions.

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.

We analyse isospectral sets of potentials associated to a given ‘generalized periodic’ boundary condition in SL(2, R) for the Sturm-Liouville equation on the unit interval. This is done by first studying the larger manifold M of all pairs of boundary conditions and potentials with a given spectrum and characterizing the critical points of the map from M to the trace a + d Isospectral sets appear as slices of M whose geometry is determined by the critical point structure of the trace function. This paper completes the classification of isospectral sets for all real self-adjoint boundary conditions.