Barcodes and area-preserving homeomorphisms

Abstract

In this paper we use the theory of barcodes as a new tool for studying dynamics of area-preserving homeomorphisms. We will show that the barcode of a Hamiltonian diffeomorphism of a surface depends continuously on the diffeomorphism, and furthermore define barcodes for Hamiltonian homeomorphisms.Our main dynamical application concerns the notion of weak conjugacy, an equivalence relation which arises naturally in connection to C 0 continuous conjugacy invariants of Hamiltonian homeomorphisms. We show that for a large class of Hamiltonian homeomorphisms with a finite number of fixed points, the number of fixed points, counted with multiplicity, is a weak conjugacy invariant. The proof relies, in addition to the theory of barcodes, on techniques from surface dynamics such as Le Calvez's theory of transverse foliations.In our exposition of barcodes and persistence modules, we present a proof of the Isometry Theorem which incorporates Barannikov's theory of simple Morse complexes.

Viterbo conjecture for Zoll symmetric spaces

“…In the case of surfaces, a similar statement was proven in [66] using different tools, while the same statement was proven in [60,Remark 8] using [60,Corollary 6]. It was also shown in [18] for closed symplectically aspherical manifolds, using [60,Corollary 6].…”

“…This map is Lipschitz with respect to the L 1,∞ -distance on H = H M , and the bottleneck distance on the space barcodes of barcodes. This observation was used in [76], in [3,40,78,92,99,105] and more recently in [18,31,60,66,93,95] to produce various quantitative results in symplectic topology. Set barcodes ′ for the quotient space of barcodes with respect to the isometric R-action by shifts.…”

“…Following [66], we use Corollary 6, and the conjugation invariance property (15) of B ′ , to establish that B ′ : Ham(CP n , ω F S ) → barcodes ′ is constant on weak conjugacy classes. Two elements φ 0 , φ 1 of a topological group G are called weakly conjugate if θ(φ 0 ) = θ(φ 1 ) for all continuous conjugacy-invariant maps θ : G → Y, to a Hausdorff topological space Y.…”

Lagrangian configurations and Hamiltonian maps

“…Indeed, as was described in [17, Section 5.3], by Gromov's result [26] that Symp c (W ) = Ham c (W ) for each topological ball W ⊂ R 4 , fragmentation results for Hamiltonian maps apply in dimension four. Specifically, the argument proving [33, Lemma 4.4, Proposition 4.2] extends to our case and the proofs of analogues of [36,Lemma 4.7] and hence of [12,Lemma 4.6], [13,Lemma 3.11] go through. Furthermore it is easy to see that in this case c k,B , c k ′ ,B ′ are defined on G = G by the obvious map G → Ham(M a ).…”

Completeness of derived interleaving distances and sheaf quantization of non-smooth objects

“…However, one still obtains an Arnold-type theorem for a Hamiltonian homeomorphism if one reformulates the conjecture with the notion of spectral invariants, as in [BHS21,Kaw19,BHS19]. In these studies, the C 0 -continuity of persistence modules associated with Hamiltonian diffeomorphisms was effectively used (see also [RSV18]). Guillermou [Gui13] (see also [Gui19,Part 7]) applied the microlocal theory of sheaves to C 0 -symplectic geometry.…”