. This article deals with relative barcode entropy, a notion recently introduced by & Ccedil;ineli, Ginzburg, and G & uuml;rel. It is a non-negative number assigned to a Hamiltonian diffeomorphism and a pair of Lagrangian submanifolds, and defined through Lagrangian Floer complexes associated to them. We exhibit several situations in which the relative barcode entropy is positive. In analogy to a result in the absolute case by the above authors, we obtain that the topological entropy of any horseshoe K is a lower bound if the two Lagrangians contain a local unstable resp. stable manifold in K. In dimension 2, we also estimate the relative barcode entropy of a pair of closed curves that lie in special homotopy classes in the complement of a certain set of periodic orbits in K. Furthermore, we define a variant of relative barcode entropy and present first examples for which it is positive. As applications, certain robustness features of the volume growth and the topological entropy are discussed.
. This paper establishes robust obstructions to representing Hamiltonian diffeomorphisms as k-th powers (k >= 2) or embedding them in Hamiltonian flows for certain higher-dimensional symplectic manifolds (M, w), including surface bundles. We prove that in the Hamiltonian group (Ham(M, w), dH) equipped with the Hofer metric, there exist arbitrarily large balls that are disjoint from the set of k-th powers. Furthermore, we demonstrate that the free group of two generators embeds into every asymptotic cone of (Ham(M,w),dH), revealing the large-scale geometric complexity of the Hamiltonian group.
. We consider a perturbation f of a hyperbolic toral automorphism L. We study rigidity related to exceptional properties of the strong and weak stable foliations for f. If the strong foliation is mapped to the linear one by the conjugacy h between f and L, we obtain smoothness of h along the weak foliation and regularity of the joint foliation of the strong and unstable foliations. We also establish a similar global result. If the weak foliation is sufficiently regular, we obtain smoothness of the conjugacy along the strong foliation and regularity of the joint foliation of the weak and unstable foliations. If both conditions hold, then we get smoothness of h along the stable foliation. We also deduce a rigidity result for the symplectic case. The main theorems are obtained in a unified way using our new result on the relation between holonomies and normal forms.
We show that the second iteration T2 of the outer symplectic billiard map with respect to a convex domain M in a symplectic vector space is approximated by an explicit Hamiltonian flow for points far away from M. More precisely, denote by N the symplectic polar dual of the symmetrization M circle minus M of M. If we write N as the unit level set of a 1-homogeneous function H, then the difference between T2 and the time-2-Hamiltonian flow of H applied to a point x is smaller than c/|x| for some constant c depending only on M. Moreover, we show that if an orbit escapes to infinity, then its distance to the origin grows no faster than k in the number of iterations. Finally, we prove that a k-periodic orbit needs to be close, in terms of k, to M.
Arnol'd flows are a class of area-preserving flows on surfaces. In this paper, we prove that typical Arnol'd flows have the minimal self-joinings property. Consequently, we can classify centralizers and factors of typical Arnol'd flows.
