Contractibility of a persistence map preimage

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4 Scopus citations


This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of snapshots of solutions, what conclusions can be drawn about solutions of the original dynamical system? We address this challenge in the context of an N dimensional system of ordinary differential equation defined in RN. To each point in RN (e.g. an initial condition) we associate a persistence diagram. The main result of this paper is that under this association the preimage of every persistence diagram is contractible. As an application we provide conditions under which multiple time series of persistence diagrams can be used to conclude the existence of a fixed point of the differential equation that generates the time series.

Original languageEnglish (US)
Pages (from-to)509-523
Number of pages15
JournalJournal of Applied and Computational Topology
Issue number4
StatePublished - Dec 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Computational Mathematics
  • Geometry and Topology


  • Dynamical systems
  • Fixed point theorem
  • Persistent homology
  • Topological data analysis


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