For any discrete time dynamical system with a rational evolution, we define an entropy, which is a global index of complexity for the evolution map. We analyze its basic properties and its relations to the singularities and the irreversibility of the map. We indicate how it can be exactly calculated. Postal address: Laboratoire de Physique Théorique et des Hautes EnergiesUniversité Pierre et Marie Curie, boîte postale 126.

We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but which nevertheless are nonintegrable. As a more sensitive integrability test, we propose the analysis of the complexity ("algebraic entropy") of the map using the growth of the degree of its iterates: integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems.

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In an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations (in Yang-Mills theory) is a C 00 principal fibration. The underlying quotient space, the gauge orbit space, is seen explicitly to be a C 00 manifold modelled on a Hubert space.

We compute all possible anomalous terms in quantum gauge theory in the natural class of polynomials of differential forms. By using the appropriate cohomological and algebraic methods, we do it for all dimensions of spacetime and all structure groups with reductive Lie algebras.

The appropriate language for describing the pure Yang-Mills theories is introduced, . An elementary but precise presentation of the mathematical tools which are necessary for a geometrical description of gauge fields is given. After recalling basic notions of diA'erential geometry, it is shown in what sense a gauge potential is a connection in some fiber bundle, and the corresponding gauge field the associated curvature. It is also shown how the global aspects of the theory (e.g., boundary conditions) are coded into the structure of the bundle. Gauge transformations and equations of motion, as well as the selfduality equations, acquire then a global character, once they are defined in terms of operations in the bundle space. Finally the orbit space, that is to say, the set of gauge inequivalent potentials, is defined, and its is shown why there is no continuous gauge fixing in the non-Abelian case. CONTENTS

We produce the general solution of the Wess-Zumino consistency condition for gauge theories of the Yang-mills type, for any ghost number and form degree. We resolve the problem of the cohomological independence of these solutions. In other words we fully describe the local version of the cohomology of the BRS operator, modulo the differential on space-time. This in particular includes the presence of external fields and non-trivial topologies of space-time.

We state some new results about the configuration space of pure Yang-Mills theory. These results come from the study of the kinetic energy term of the Lagrangian of the theory. This term defines a riemannian metric on the space of non-equivaίent gauge potentials. We develop a riemannian calculus on the configuration space, compute the riemannian connection, the curvature tensor, and solve for the geodesies, etc. We show that the Gribov ambiguity is more than an artefact of the choice of a gauge condition, and is related to the existence of conjugate points on the geodesies, and is thus an intrinsic feature of the theory.