The importance of this result is that it opens the possibility of applying standard algebraic topological tools, e.g. Lefschetz fixed point theorem, Conley index, to dynamics that is observed through the lens of persistent homology.

To state our goal precisely requires the introduction of notation. Throughout this paper ${{\mathcal{S}_N}}$ denotes a simplicial complex composed out of $N$ vertices and $N-1$ edges. More specifically, ${{\mathcal{S}_N}}= {{\mathcal{S}_N^{(0)}}}\cup {{\mathcal{S}_N^{(1)}}}$ where ${{\mathcal{S}_N^{(0)}}}$ denotes the $0$-skeleton of ${{\mathcal{S}_N}}$, i.e. the set of vertices $${{\mathcal{S}_N^{(0)}}}=\left\{ <i>\ |\ i=1,\dots,N \right\},$$ and ${{\mathcal{S}_N^{(1)}}}$ denotes the set of edges $${{\mathcal{S}_N^{(1)}}} = \left\{ <i,i+1>\ |\ i=1,\dots,N-1\right\}.$$ Observe that ${{\mathcal{S}_N}}$ can be viewed as a simplicial decomposition of a bounded interval $I\subset{\mathbb{R}}$. We are interested in sublevel set filtrations associated with piecewise monotone continuous functions on $I$, and thus we are interested in the following filtration on ${{\mathcal{S}_N}}$.

\[def:Sfiltration\] Let $z\in {\mathbb{R}}^N$. For $\theta\in {\mathbb{R}}$, define $${{\mathcal{S}_N}}(z,\theta) := {\left\{ {\sigma\in{{\mathcal{S}_N}}: f(z,\sigma)\leq \theta}\right\}}$$ where $f\colon {\mathbb{R}}^N\times{{\mathcal{S}_N}}\to {\mathbb{R}}$, is given by $$f(z,\sigma) := \begin{cases}
    z_j &\text{if $\sigma=<j>$,}\\
    \max{\left\{ {z_j, z_{j+1}}\right\}}&\text{if $\sigma=<j,j+1>$.}
    \end{cases}$$

\[defn:sublevelset\] Given $z=(z_1,\ldots,z_N)\in {\mathbb{R}}^N$, we can reorder the coordinates of $z$ such that $$z_{j_1} \leq z_{j_2} \leq \cdots \leq z_{j_N}.$$ The *sublevel-set filtration of ${{\mathcal{S}_N}}$ at $z$* is given by $${{\mathcal{S}_N}}(z,z_{j_1}) 