We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations.

Symbolic dynamics play a central role in the description of the evolution of nonlinear systems. Yet there are few methods for determining symbolic dynamics of chaotic data. One difficulty is that the data contains random fluctuations associated with the experimental process. Using data obtained from a magnetoelastic ribbon experiment we show how a topological approach that allows for experimental error and bounded noise can be used to obtain a description of the dynamics in terms of subshift dynamics on a finite set of symbols.