On synchronized non-sofic subshifts

TL;DR

This paper characterizes when synchronized coded systems are intrinsically ergodic with full support, linking this property to entropy conditions and extending results to systems with specification-like properties, including entropy calculations.

Contribution

It provides a characterization of intrinsic ergodicity for synchronized coded systems based on entropy comparisons and extends the understanding of systems with specification properties.

Findings

Synchronized coded systems are intrinsically ergodic of full support if entropy conditions are met.

SVGL systems, as non-mixing extensions of systems with specification, are also intrinsically ergodic.

The entropy of the Fischer cover's underlying graph is explicitly computed.

Abstract

We show that a synchronized coded system $X$ is intrinsically ergodic of full support if and only if $ h(Y)$, the topological entropy of $Y$, is less than $h(X) $ whenever $Y$ is a proper subsystem of $ X $. We also show that like systems with specification property, SVGL's the non-mixing extension of systems with specification property, are intrinsically ergodic of full support. Moreover, we compute the entropy of the underlying graph of the Fischer cover of a synchronized system.