Symbolic Extensions and dominated splittings for Generic C^1-Diffeomorphisms

TL;DR

This paper proves that for generic C^1-diffeomorphisms on compact manifolds, homoclinic classes either admit a specific dominated splitting or lack symbolic extensions, revealing structural dichotomies in dynamical systems.

Contribution

It establishes a residual subset of C^1-diffeomorphisms where homoclinic classes exhibit a clear dichotomy between dominated splittings and absence of symbolic extensions.

Findings

Homoclinic classes either have a dominated splitting or no symbolic extensions.

The result applies to a residual subset of C^1-diffeomorphisms.

Provides a structural classification of homoclinic classes in generic dynamics.

Abstract

Let Diff^1(M) be the set of all C^1-diffeomorphisms f : M \rightarrow M, where M is a compact boundaryless d-dimensional manifold, d \geq 2. We prove that there is a residual subset R of Diff^1(M) such that if f \in R and if H(p) is the homoclinic class associated with a hyperbolic periodic point p, then either H(p) admits a dominated splitting of the form E\oplusF1 \oplus...\oplusFk \oplusG, where Fi is not hyperbolic and one-dimensional, or f|H(p) has no symbolic extensions.