Citation/Abstract

The variational method has shown many advantages over the geometric method in proving the existence of various connecting orbits since it requires much weaker hyperbolicity and less smoothness. Many results known to be difficult to obtain by the geometric method can now be obtained through a variational principle with relative ease. In particular, a variational principle provides a constructive approach to the existence of heteroclinic orbits and chaotic invariant sets.

The monotone twist map f¯ on [Special characters omitted.] and its lift f on [Special characters omitted.] considered in this thesis always have an associated variational principle h : [Special characters omitted.] through its generating function, and we mainly work with such a variational principle h and h¯ : [Special characters omitted.] defined by h¯ ( u ) = h ( u , u ). We first construct a locally minimal heteroclinic connection between an adjacent minimal pair u ¹ < u ² of critical points of h¯ , i.e., h¯ ( u ) > h¯ ( u ¹ ) = h¯ ( u ² ) for u ∈ ( u ¹ , u ² ). This makes it possible to define a bump for any adjacent minimal pair of critical points. For an adjacent minimal chain of critical points, if there exists a bump for each of the adjacent minimal pairs in the chain, we construct a heteroclinic connection between the two end minimal points of the chain. The results on h are carried over to f and f¯ wherever relevant.

Finally, assuming that there is a bump for any two neighboring globally minimal critical points of h¯ , we construct an invariant set Λ ^m such that when m is sufficiently large, the shift map of the n -symbol space is a factor of [Special characters omitted.] , where n is the total number of the globally minimal fixed points of f¯ .