We find robust obstructions to representing a Hamiltonian diffeomorphism as a full k-th power, k ≥ 2, and in particular, to including it into a one-parameter subgroup. The robustness is understood in the sense of Hofer's metric. Our approach is based on the theory of persistence modules applied in the context of filtered Floer homology. We present applications to geometry and dynamics of Hamiltonian diffeomorphisms.

We show that the L ∞ -norm of the contact Hamiltonian induces a nondegenerate right-invariant metric on the group of contactomorphisms of any closed contact manifold. This contact Hofer metric is not left-invariant, but rather depends naturally on the choice of a contact form α, whence its restriction to the subgroup of α-strict contactomorphisms is bi-invariant. The non-degeneracy of this metric follows from an analogue of the energy-capacity inequality. We show furthermore that this metric has infinite diameter in a number of cases by investigating its relations to previously defined metrics on the group of contact diffeomorphisms. We study its relation to Hofer's metric on the group of Hamiltonian diffeomorphisms, in the case of prequantization spaces. We further consider the distance in this metric to the Reeb one-parameter subgroup, which yields an intrinsic formulation of a small-energy case of Sandon's conjecture on the translated points of a contactomorphism. We prove this Chekanov-type statement for contact manifolds admitting a strong exact filling.2010 Mathematics Subject Classification. 53D10, 37J55, 57R17, 57S05.

We establish a C 0 a priori bound on the solutions of the quaternionic Calabi-Yau equation (of Monge-Ampère type) on compact HKT manifolds with a locally flat hypercomplex structure. As an intermediate step, we prove a quaternionic version of the Gauduchon theorem.

We introduce a new notion of persistence modules endowed with operators. It encapsulates the additional structure on Floer-type persistence modules coming from the intersection product with classes in the ambient (quantum) homology, along with a few other geometric situations. We provide sample applications to the C 0 -geometry of Morse functions and to Hofer's geometry of Hamiltonian diffeomorphisms, that go beyond spectral invariants and traditional persistent homology.

In this note we prove that if a closed monotone symplectic manifold M of dimension 2n, satisfying a homological condition that holds in particular when the minimal Chern number is N > n, admits a Hamiltonian pseudo-rotation, then the quantum Steenrod square of the point class must be deformed. This gives restrictions on the existence of pseudo-rotations. Our methods rest on previous work of the author, Zhao, and Wilkins, going back to the equivariant pair-of-pants product-isomorphism of Seidel.