2022/03/07 by Snir Ben Ovadia, Ovadia, Snir Ben
Computer Science · Mathematics · #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2203.03507
openalex publication_date 2022/03/07 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
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 C1+ diffeomorphism of a closed manifold of dimension ≥2). 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.