Valentín Mendoza

The complexity of a dynamical system exhibiting a homoclinic orbit is given by its dynamical core which, due to Cantwell, Conlon and Fenley, is a set uniquely determined in the isotopy class, up to a topological conjugacy, of the end-periodic map relative to that orbit. In this work we prove that a sufficient condition to determine the dynamical core of a homoclinic orbit of a Smale diffeomorphism on the 2-disk is the non-existence of bigons relative to this orbit. Moreover, we propose a pruning method for eliminating bigons that can be used to find a Smale map without bigons and hence for finding the dynamical core.

The dynamics that necessarily coexists with a homoclinic orbit is captured by its dynamical core. In this work we characterize the dynamical core of a broad class of homoclinic orbits in the Smale horseshoe, specifically those with decorations of three types: maximal, P-lists and star decorations. For each of these families, we construct an explicit pruning region whose survival set — consisting of all symbolic sequences whose orbits avoid the region under the shift — coincides with the dynamical core. This provides a unified symbolic description of the forced dynamics and establishes a framework for computing dynamical invariants such as topological entropy.