Subshifts of very low complexity

Abstract: The word complexity function $p(n)$ of a subshift $X$ measures the number of $n$-letter words appearing in sequences in $X$, and $X$ is said to have linear complexity if $p(n)/n$ is bounded. It's been known since work of Ferenczi that subshifts X with linear word complexity function (i.e. $\limsup p(n)/n$ finite) have highly constrained/structured behavior. I'll discuss recent work with Darren Creutz, where we show that if $\limsup p(n)/n < 4/3$, then the subshift $X$ must in fact have measurably discrete spectrum, i.e. it is isomorphic to a compact abelian group rotation. Our proof uses a substitutive/S-adic decomposition for such shifts, and I'll touch on connections to the so-called S-adic Pisot conjecture.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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One World Numeration seminar

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