##### Department of Mathematics,

University of California San Diego

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### Math 211B - Group Actions Seminar

## Prof. Darren Creutz

#### U.S. Naval Academy

## Word complexity cutoffs for mixing properties of subshifts

##### Abstract:

In the setting of zero-entropy transformations, the class of subshifts--closed shift-invariant subsets $X$ of $\mathcal{A}^{\mathbb{Z}}$ for a finite alphabet $\mathcal{A}$--possesses a quantitative measure of complexity: the number of distinct `words' of a given length $p(q) = |\{ w \in \mathcal{A}^{q} : \exists x \in X \text{ s.t. w is a substring of x}\}|$.

I will discuss my work, some joint with R. Pavlov, pinning down the relationship between this quantitative notion of complexity with the qualitative dynamical complexity properties of probability-preserving systems known as strong and weak mixing.

Specifically, I will present results that strong mixing can occur with word complexity arbitrarily close to linear but cannot occur when $\liminf p(q)/q < \infty$ and that weak mixing can occur when $\limsup p(q)/q = 1.5$ but cannot occur when $\limsup p(q)/q < 1/5$.

The condition that $\limsup p(q)/q < 1.5$ is a (much) stronger version of zero entropy. A corollary of our work is that the celebrated Sarnak conjecture holds for all such systems.

Host: Brandon Seward

### February 8, 2024

### 10:00 AM

APM 7321

Research Areas

Ergodic Theory and Dynamical Systems****************************