Non-dense orbits on topological dynamical systems

TL;DR

This paper investigates the size and complexity of non-dense orbit sets in topological dynamical systems with the specification property, revealing they are either empty or have full topological pressure.

Contribution

It establishes a dichotomy for non-dense orbit sets in systems with the specification property, showing they are either empty or possess full topological pressure.

Findings

Non-dense orbit sets are either empty or have full topological pressure.

The result applies to any non-transitive point in the system.

The study advances understanding of orbit distribution in topological dynamics.

Abstract

Let $(X,d,T )$ be a topological dynamical system with the specification property. We consider the non-dense orbit set $E(z_0)$ and show that for any non-transitive point $z_0\in X$, this set $E(z_0)$ is empty or carries full topological pressure.