  The purpose of this paper is to generalize the variational principle, which
states that the topological entropy is equal to the supremum of the measure
theoretical entropies and also the minimum of the metric theoretical entropies,
to non-compact dynamical systems by utilizing the co-compact entropy. The
equivalence between positive co-compact entropy and the existence of
p-horseshoes over the real line is also proved. This implies a system over real
line with positive co-compact entropy contains a compact invariant subset.

Dynamical Systems