Shadowing for codimension one sectional-Anosov flows

TL;DR

This paper extends the understanding of pseudo-orbit tracing properties in sectional-hyperbolic dynamics, showing that certain codimension one attractors with Lorenz-like singularities lack the finite pseudo-orbit tracing property.

Contribution

It generalizes Komuro's result by proving that all codimension one sectional-hyperbolic attractors with a boundary-type Lorenz-like singularity do not possess FPOTP.

Findings

Codimension one sectional-hyperbolic attractors lack FPOTP.

Generalization of Komuro's negative result to broader class of attractors.

Identification of boundary-type Lorenz-like singularities as key factor.

Abstract

In hyperbolic dynamics, a well-known result is that every hyperbolic attracting set, have a finite pseudo-orbit tracing property (FPOTP). It's natural to wonder if this result is maintained in the sectional-hyperbolic dynamics; Komuro in [Lorenz attractors do not have the pseudo-orbit tracing property], provides a negative answer for this question, by proving that the geometric Lorenz Attractor doesn't have a FPOTP. In this paper, we generalized the result of Komuro, we prove that every codimension one sectional-hyperbolic attractor set with a unique singularity Lorenz-like, which is of boundary-type, does not have FPOTP.