### A mountain pass theorem for actions of compact Lie groups.

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For symmetric classical field theories on principal bundles there are two methods of symmetry reduction: covariant and dynamic. Assume that the classical field theory is given by a symmetric covariant Lagrangian density defined on the first jet bundle of a principal bundle. It is shown that covariant and dynamic reduction lead to equivalent equations of motion. This is achieved by constructing a new Lagrangian defined on an infinite dimensional space which turns out to be gauge group invariant.

The main purpose of this paper is to present in a unified approach to different results concerning group actions and integrable systems in symplectic, Poisson and contact manifolds. Rigidity problems for integrable systems in these manifolds will be explored from this perspective.

A G-shift of finite type (G-SFT) is a shift of finite type which commutes with the continuous action of a finite group G. For irreducible G-SFTs we classify right closing almost conjugacy, answering a question of Bill Parry.

For a G-covering Y → Y/G = X induced by a properly discontinuous action of a group G on a topological space Y, there is a natural action of π(X,x) on the set F of points in Y with nontrivial stabilizers in G. We study the covering of X obtained from the universal covering of X and the left action of π(X,x) on F. We find a formula for the number of fixed points of an element g ∈ G which is a generalization of Macbeath's formula applied to an automorphism of a Riemann surface. We give a new method...