Symmetric product measures: Binomial processes and invariant manifold intersections in dynamical systems

This article proposes the concept of Symmetric Product Measures (SPM) for addressing the geometric characterization of the distribution of the intersections of the stable and unstable manifolds in (hyperbolic) dynamical systems. Prototypical examples of SPM associated with multiplicative binomial processes are addressed, and the application to manifold characterization in non-linear area-preserving systems (Henon-map) is developed. Moreover, the application of SPM to the characterization of measure-theoretical properties of the periodic orbit distribution in hyperbolic dynamical systems is addressed.