  We develop a variational method for constructing positive entropy invariant
measures of Lagrangian systems without assuming transversal intersections of
stable and unstable manifolds, and without restrictions to the size of
non-integrable perturbations. We apply it to a family of two and a half degrees
of freedom a-priori unstable Lagrangians, and show that if we assume that there
is no topological obstruction to diffusion (precisely formulated in terms of
topological non-degeneracy of minima of the Peierl's barrier function), then
there exists a vast family of "horsheshoes", such as "shadowing" ergodic
positive entropy measures having precisely any closed set of invariant tori in
its support. Furthermore, we give bounds on the topological entropy and the
"drift acceleration" in any part of a region of instability in terms of a
certain extremal value of the Fr\'{e}chet derivative of the action functional,
generalizing the angle of splitting of separatrices. The method of construction
is new, and relies on study of formally gradient dynamics of the action
(coupled parabolic semilinear partial differential equations on unbounded
domains). We apply recently developed techniques of precise control of the
local evolution of energy (in this case the Lagrangian action), energy
dissipation and flux. In Part II of the paper we will apply the theory to
obtain sharp bounds for topological entropy and drift acceleration for the same
class of equations in the case of small perturbations.
