Specification property for step skew products

TL;DR

This paper establishes that certain step skew products with specific properties in the fibre maps and base subshift exhibit the specification property, linking dynamical features of the base and fibre maps.

Contribution

It proves that step skew products with continuous, piecewise monotone, expanding, and surjective fibre maps, combined with a subshift having the specification property and a mixing periodic orbit, also possess the specification property.

Findings

Step skew products with specified fibre map conditions have the specification property.

The specification property in the base subshift influences the skew product's dynamics.

Mixing and periodic orbits in the base are key to the main result.

Abstract

Step skew products with interval fibres and a subshift as a base are considered. It is proved that if the fibre maps are continuous, piecewise monotone, expanding and surjective and the subshift has the specification property and a periodic orbit such that the composition of the fibre maps along this orbit is mixing, then the corresponding step skew product has the specification property.