# On the complexity function for sequences which are not uniformly recurrent

## TL;DR

This paper investigates the complexity functions of non-minimal transitive subshifts, establishing growth bounds and structural restrictions, and extends known results on minimal systems to more general transitive cases.

## Contribution

It proves new growth bounds for complexity functions of non-minimal transitive subshifts and extends Boshernitzan's results on minimal systems to transitive systems.

## Key findings

- Non-minimal transitive subshifts have complexity growth exceeding 1.5n infinitely often.
- Certain transitive subshifts with specific complexity bounds must be minimal.
- Restrictions imply unique ergodicity for some non-minimal transitive subshifts.

## Abstract

We prove that every non-minimal transitive subshift $X$ satisfying a mild aperiodicity condition satisfies $\limsup c_n(X) - 1.5n = \infty$, and give a class of examples which shows that the threshold of $1.5n$ cannot be increased. As a corollary, we show that any transitive $X$ satisfying $\limsup c_n(X) - n = \infty$ and $\limsup c_n(X) - 1.5n < \infty$ must be minimal. We also prove some restrictions on the structure of transitive non-minimal $X$ satisfying $\liminf c_n(X) - 2n = -\infty$, which imply unique ergodicity (for a periodic measure) as a corollary, which extends a result of Boshernitzan from the minimal case to the more general transitive case.