Summable orbits

Snir Ben Ovadia

doi:10.3934/jmd.2023017

Article Contents

2023, Volume 19: 625-715. Doi: 10.3934/jmd.2023017

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Summable orbits

Snir Ben Ovadia

Department of Mathematics, The Pennsylvania State University, University Park, McAllister Building, State College, PA 16802, USA

Received: February 3, 2022

Revised: August 26, 2022

Published online: August 10, 2023

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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 diffeomorphsims where we are able to simultaneously code all invariant measures in a finite-to-one almost everywhere fashion.

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Mathematics Subject Classification: Primary: 37D05; Secondary: 37B10.

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