New York Combinatorics Seminar

Given a time series, we can extract a collection of permutations of a fixed length by sliding a window across our data and recording the relative order of the values in the window.  It is known that the distribution of permutations that appear conveys information about the complexity of the time series; one commonly studied measure is permutation entropy.  Interestingly, in the specific case that the time series is defined by iterating a piecewise monotone map of the interval, there are forbidden permutations.  Moreover, the number of allowed permutations encodes the system’s topological entropy, a measure of complexity that is itself closely related to the permutation structure of periodic points.

Using permutation-based measures of entropy in time series analysis as motivation, I will talk about some interesting problems that arise at this intersection of combinatorics, dynamics and time series analysis.

Patterns in Negative Beta Shifts

Abstract: The β-shift is the transformation from the unit interval to itself that maps x to the fractional part of βx. Permutations realized by the relative order of the elements in the orbits of these maps have been studied for positive integer values of β and for real values β>1. In both cases, a combinatorial description of the smallest positive value of β needed to realize a permutation is provided. In this paper we extend these results to the case of negative β, both in the integer and in the real case. Negative β-shifts are related to digital expansions with negative real bases, studied by Ito and Sadahiro, and Liao and Steiner.

Follow this link to the arXiv preprint.