Table of contents

The authors consider the steady group motions of a rigid body with a fixed point moving in a gravitational field. For an asymmetric top, rotation about the axis of gravity is the only permissible group motion; for a Lagrange top, simultaneous rotation about the axis of gravity and spin about the axis of symmetry of the top is permissible. The analysis of the heavy top follows the reduced energy momentum method of Simo et al. (1991), which is applicable to a wide range of conservative systems with symmetry. Steady group motions are characterized as solutions of a variational problem on the configuration space; local minima of the amended potential correspond to nonlinearly orbitally stable steady motions. The combination of a low-dimensional configuration space and a relatively large number of parameters that produce substantial qualitative changes in the dynamics makes possible a thorough, detailed analysis, which not only reproduces the classical results for this well known system, but leads to some results which we believe are new. They determine general equilibrium and nonlinear stability conditions for steady group motions of a heavy top with a fixed point. They rederive the classical equilibrium and stability conditions for sleeping tops and precessing Lagrange tops, analyse in detail the stability of a family of steady rotations of tilted tops which bifurcate from the branch of sleeping tops parametrized by angular velocity, and classify the possible stability transitions of an arbitrary top as its angular velocity is increased. They obtain a simple, general expression for the characteristic polynomial of the linearized equations of motion and analyse the linear stability of both sleeping tops and the family of tilted top motions previously mentioned. Finally, they demonstrate the coexistence of stable branches of steadily precessing tops that bifurcate from the branch of sleeping Lagrange tops throughout the range of angular velocities for which the sleeping top is stable.

The authors prove that the speed of convergence of two Markov families determines the smoothness of the conjugacy between them. One of the applications of this result that they give is that the attractors of any two quadratic foldings at the Feigenbaum accumulation point of period doubling are C^2+0.11 conjugate. The main result provides the basis for a complete unification of renormalization and smooth conjugacy results which includes both the classical theorems and more recent results about critical systems.

Invertible compositions of one-dimensional maps are studied which are assumed to include maps with a non-positive Schwarzian derivative and others whose sum of distortions is bounded. If the assumptions of the Koebe principle hold, the author shows that the joint distortion of the composition is bounded. On the other hand, if all maps with possibly non-negative Schwarzian derivatives are almost linear-fractional and their nonlinearities tend to cancel leaving only a small total, then they can all be replaced with affine maps with the same domains and images and the resulting composition is a very good approximation of the original one. These technical tools are then applied to prove a theorem about critical circle maps.

The author introduces and studies a 2-form which forms an antiplectic pair with the well known Adler-Gel'fand-Dikii bracket. He re-examines from this point of view the theorem that the composition of ordinary differential operators is a canonical transformation.

The finite fields of degree two are reinterpreted as discrete phase spaces on the two-dimensional torus. The authors study dynamical systems obtained by iterating linear maps over these fields, from a geometrical viewpoint. These maps can be regarded as the two-dimensional discrete equivalent of a Bernoulli shift. They yield irregular motions, which may coexist with spatial order. They find that the dynamics of orbits of long period can be characterized as a percolation process. The question of randomness in dynamical systems over finite sets is discussed.

The authors prove the existence of 'cantori' of all incommensurate rotation vectors, for symplectic maps of arbitrary dimension near enough to any non-degenerate anti-integrable limit, and derive an asymptotic form for them. Cantori are invariant Cantor sets which can be thought of as remnants of KAM tori.

Greene (1979) proposed a relationship between the existence of an invariant circle for an area-preserving map and the 'residues' of periodic orbits with nearby rotation number. It remains without a doubt the most effective practical criterion for calculating the breakup of invariant circles. In this work four results are proved which go a long way towards placing the criterion on a rigorous foundation, and the issues remaining to be resolved in order to formulate a rigorous version of the criterion are discussed.

N-mode truncations of the three-dimensional Navier-Stokes equations with periodic boundary conditions are considered. The authors show that the solution of the truncated model exhibits, under some assumptions concerning the external force, general properties of symmetry and invariance. A particular seven-mode truncation is then derived and investigated in detail by making use of numerical tools typical of dynamical systems. An intricate phenomenon is found with the significant presence of two- and three-dimensional tori. Among different transitions the one from quasiperiodicity to chaos in a three-torus appears of particular interest.

The degree of pseudo-randomness of several sequences and its relation to Anderson localization is investigated. It is rated with the help of the chi ² test and the spectral test, which are extensively used in computer science. The relationship between this rating and localization for tight-binding models with hopping to nearest neighbours and to next-nearest neighbours in one dimension is investigated. The results are relevant for localization in systems which are studied in 'quantum chaos', where the potentials are pseudo-random but not truly random.