Minimal topological chaos coexisting with a finite set of homoclinic and periodic orbits

• The pruning method can be applied to certain physical models.
• The combinatorics of the pruning map is found uncrossing invariant manifolds.
• Infinite pruning regions are related to singularities without rotation.

In this note we explain how to find the minimal topological chaos relative to finite set of homoclinic and periodic orbits. The main tool is the pruning method, which is used for finding a hyperbolic map, obtained uncrossing pieces of the invariant manifolds, whose basic set contains all orbits forced by the finite set under consideration. Then we will show applications related to transport phenomena and to the problem of determining the orbits structure coexisting with a finite number of periodic orbits arising from the bouncing ball model.