class of coverings has the structure of a meet semi-lattice that we use extensively as a new form of bookkeeping.

Section \[secsubsets\] describes the preimage set in terms of star-shaped sets, generated by semi-lattice morphisms on the set of multi-indices. However, analyzing the topology of the nerve complex defined using those star-shaped sets directly turns out to be hard, and its intersection structure is unclear. Instead, we introduce a covering using more coarse sets.

Section \[secfirststep\] provides an argument for the initial step of the induction argument. To provide the reader with some intuition we illustrate the results with an example for a particular case.

Section \[secmotiv\] provides a motivating example for the contractibility inductive argument. We carry out formally the inductive argument for contractibility of the coarse covering in Section \[seccontr\].

Acknowledgements
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The work of JC and KM was partially supported by grants NSF-DMS-1125174, 1248071, 1521771 and a DARPA contracts HR0011-16-2-0033.

In addition KM was partially supported by DARPA contract FA8750-17-C-0054 and NIH grant R01 GM126555-01.

Preliminaries {#sec:background}
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#### Notation

Throughout this paper $n$ and $N$ denote natural numbers where $n\leq N$.

Sublevel-set filtrations and persistence diagrams
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Before we introduce finite dimensional dynamical systems we study 