Computing the Dynamical Core of Horseshoe Homoclinic Orbits

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.