Observing expansive maps

TL;DR

This paper investigates the observability of positively expansive maps through continuous functions, establishing generic conditions and determining that two functions suffice for maps on tori.

Contribution

It extends previous work by proving a general generic observability result and precisely characterizes the minimal number of functions needed for positively expansive maps on tori.

Findings

Two functions are necessary and sufficient for observing positively expansive maps on tori.

A general result on the generic observability of locally injective maps is proved.

The minimal number of functions for observation is characterized for specific cases.

Abstract

We consider the problem of the observability of positively expansive maps by the time series associated to continuous real functions. For this purpose we prove a general result on the generic observability of a locally injective map of a compact metric space of finite topological dimension, extending earlier work by Gutman [6]. We apply this result to partially solve the problem of finding the minimal number of functions needed to observe a positively expansive map. We prove that two functions are necessary and sufficient for positively expansive maps on tori.