Summable Orbits

TL;DR

This paper introduces a new class of orbits with zero Lyapunov exponents that still exhibit sensitivity, develops a coding scheme for these orbits, and constructs stable/unstable foliations with applications to non-uniform hyperbolic systems.

Contribution

It constructs a countable Markov partition and coding for orbits with zero Lyapunov exponents, and develops shadowing and foliation theories for such orbits.

Findings

Constructed a countable Markov partition with finite-to-one coding.

Developed shadowing theory for orbits with zero Lyapunov exponents.

Established absolute continuity of stable and unstable foliations.

Abstract

We introduce a class of orbits which may have $0$ Lyapunov exponents, but still demonstrate some sensitivity to initial conditions. We construct a countable Markov partition with a finite-to-one almost everywhere induced coding, and which lifts the geometric potential with summable variations (for a $C^{1+}$ diffeomorphism of a closed manifold of dimension $\geq2$). An important tool we use is a shadowing theory for orbits which may have $0$ Lyapunov exponents. We construct (weak) stable and unstable leaves for such orbits using a Graph Transform method, and prove the absolute continuity of these foliations w.r.t holonomies. In particular, we discuss setups where these foliations exist, and are strictly weak -- i.e., do not demonstrate exponential contraction. One example is a family of non-uniformly hyperbolic diffeomorhpims where we are able to simultaneously code all invariant measures in a finite-to-one almost everyhwere fashion.