A Comparison Framework for Interleaved Persistence Modules

Conley index approach to sampled dynamics

“…, p − 1} the topological group with the addition modulo p and the discrete topology. We define the space X := X × Z p , with the product topology, and dynamical systems f , f : X → X, by (17) f : X (x, i) −→ (f (x), i + 1) ∈ X and f : X (x, i) −→ (f (x), i) ∈ X, respectively. Consider the homeomorphism l : X (x, i) −→ (x, i + 1) ∈ X and observe that we have…”

“…Let f : R d → R d be a discrete dynamical system. Set R d := R d × I p and consider the dynamical system f on R d given by (17). Assume that K := p−1 i=0 (K σ i × {i}) ⊂ R d is an isolated invariant set with respect to f p , and M := p−1 i=0 (M σ i × {i}) is its isolating neighborhood.…”

“…Proof. Consider the space R d := R d ×I p , and the dynamical system f on R d , given by (17). By Proposition 8.2 we infer that S σ is an isolated invariant set with respect to f p .…”

“…). Moreover, definition (17) of f guaranties that the p-periodic trajectory of f through x passes through the components of Inv( p−1 i=0 N σ i , f ) in a proper order.…”

“…There are two challenges associated with this approach. The first is that persistent homology computations on large data sets can be prohibitively expensive (there is extensive work being done to address this problem [9,29,17]), and second that the development of a persistence theory of maps is in its early stages [10,11,4]. An alternative technique is to bin the data.…”