Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

Abstract

We study topological entropy of compactly supported Hamiltonian diffeomorphisms from a perspective of persistence homology and Floer theory. We introduce barcode entropy, a Floer-theoretic invariant of a Hamiltonian diffeomorphism, measuring exponential growth under iterations of the number of not-too-short bars in the barcode of the Floer complex. We prove that the barcode entropy is bounded from above by the topological entropy and, conversely, that the barcode entropy is bounded from below by the topological entropy of any hyperbolic locally maximal invariant set, e.g., a hyperbolic horseshoe. As a consequence, we conclude that for Hamiltonian diffeomorphisms of surfaces the barcode entropy is equal to the topological entropy.

Hofer's geometry and topological entropy

“…Very recently the authors in [15] show that the topological entropy of a Hamiltonian diffeomorphism ϕ on a closed surface coincides with its barcode entropy (ϕ) which they introduce, and which measures the growth of the number of certain bars in the barcode of the iterates of ϕ. Hence, together with the results in this paper, this shows that the results obtained in Theorems 1.4, 1.6 and Corollary 1.5 hold additionally for .…”

“…Note first that this is the case if the expression in ( 16) is considered as one in terms of the free group in a, b, c. Therefore, if the genus is g ≥ 3, no cancellation of terms in (16) follows from the fact that (i Σg ) * : π(C g , s 0 ) → π(Σ g , s 0 ) is injective. (i Σ 2 ) * : π(C 2 , s 0 ) → π(Σ 2 , s 0 ) is injective up to (15), so clearly one only has to h top > C; note that the union of these neighborhoods V is an open and dense set with respect to d Hofer .…”

“…, so Γ(S ∪ {a m b n }) ≥ log(#S + 1). (i Σ 2 ) * is injective up to relation (15), but for each l, only 4 products could appear in those equations and hence also in this case Γ(S ∪ {a m b n }) ≥ log(#S + 1). Note that in the case m = n = 1 clearly T ∞ ([ab]) ≤ log(2), and hence in fact T ∞ ([ab]) = log(2).…”

On the growth of the Floer barcode

“…In fact, we are not aware of any example where the diameter would be shown to grow faster than linearly. When M is not aspherical or non-contractible periodic orbits are included, the notion of the diameter is more involved and less unambiguous, but even then the effective diameter grows polynomially in most cases; see [ÇGG21,Rmk. 3.4].…”

“…With this in mind, one useful way to measure the growth of the Floer complex of ϕ k is by counting the number of bars b ǫ ϕ k of length greater than ǫ > 0 in its barcode. The limit (ϕ) as ǫ ց 0 of the exponential growth rate of b ǫ ϕ k is called the barcode entropy and closely related to the topological entropy h top (ϕ) of ϕ; [ÇGG21]. In particular, (ϕ) ≤ h top (ϕ) and hence b ǫ ϕ k grows at most exponentially, and (ϕ) = h top (ϕ) in dimension two.…”