Non-dense orbits of systems with approximate product property

TL;DR

This paper demonstrates that in systems with the approximate product property, the set of points whose orbits avoid accumulation in large sets has full topological pressure, highlighting complex orbit behavior.

Contribution

It establishes a new result linking approximate product property to the full topological pressure of non-accumulating orbits.

Findings

Non-dense orbits can have full topological pressure.

The approximate product property influences orbit accumulation behavior.

The result applies broadly to systems with this property.

Abstract

We show that for any topological dynamical system with approximate product property, the set of points whose forward orbits do not accumulate to any point in a large set carries full topological pressure.