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Sobhan Seyfaddini

Publications and source records attributed to Sobhan Seyfaddini.

26 records · Page 2Linked to original sources
arXiv2015

Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces

Inspired by Le Calvez' theory of transverse foliations for dynamical systems of surfaces, we introduce a dynamical invariant, denoted by N, for Hamiltonians of any surface other than the sphere. When the surface is the plane or is closed and aspherical, we prove that on the set of autonomous Hamiltonians this invariant coincides with the spectral invariants constructed by Viterbo on the plane and Schwarz on closed and aspherical surfaces. Along the way, we obtain several results of independent interest: We show that a formal spectral invariant, satisfying a minimal set of axioms, must coincide with N on autonomous Hamiltonians thus establishing a certain uniqueness result for spectral invariants, we obtain a "Max Formula" for spectral invariants on aspherical manifolds, give a very simple description of the Entov-Polterovich quasi-state on aspherical surfaces and characterize the heavy and super-heavy subsets of such surfaces.

math.SG↗
arXiv2015

Coisotropic rigidity and C^0-symplectic geometry

We prove that symplectic homeomorphisms, in the sense of the celebrated Gromov-Eliashberg Theorem, preserve coisotropic submanifolds and their characteristic foliations. This result generalizes the Gromov-Eliashberg Theorem and demonstrates that previous rigidity results (on Lagrangians by Laudenbach-Sikorav, and on characteristics of hypersurfaces by Opshtein) are manifestations of a single rigidity phenomenon. To prove the above, we establish a C^0-dynamical property of coisotropic submanifolds which generalizes a foundational theorem in C^0-Hamiltonian dynamics: Uniqueness of generators for continuous analogs of Hamiltonian flows.

math.SG↗
arXiv2013

The displaced disks problem via symplectic topology

We prove that a $C^0$--small area preserving homeomorphism of a closed surface with vanishing mass flow can not displace a topological disk of large area. This resolves the displaced disks problem posed by F. Béguin, S. Crovisier, and F. Le Roux.

math.DS↗
arXiv2012

A note on C^0 rigidity of Hamiltonian isotopies

We show that a symplectic isotopy that is a $C^0$ limit of Hamiltonian isotopies is itself Hamiltonian, if the corresponding sequence of generating Hamiltonians converge in $L^{(1, \infty)}$ topology.

math.SG↗
arXiv2012

C^0-limits of Hamiltonian paths and the Oh-Schwarz spectral invariants

In this article we study the behavior of the Oh-Schwarz spectral invariants under C^0-small perturbations of the Hamiltonian flow. We obtain an estimate, which under certain assumptions, relates the spectral invariants of a Hamiltonian to the C^0-distance of its flow from the identity. Using the mentioned estimate we show that, unlike the Hofer norm, the spectral norm is C^0-continuous on surfaces. We also present applications of the above results to the theory of Calabi quasimorphisms and improve a result of Entov, Polterovich and Py. In the final section of the paper we use our results to answer a question of Y.-G. Oh about spectral Hamiltonian homeomorphisms.

math.SG↗
arXiv2012

Descent and C^0-rigidity of spectral invariants on monotone symplectic manifolds

We obtain estimates showing that on monotone symplectic manifolds (asymptotic) spectral invariants of Hamiltonians which vanish on a non-empty open set, U, descend to Ham_c(M\setminus U) from its universal cover. Furthermore, we show these invariants and are continuous with respect to the C^0-topology on Ham_c(M\setminus U). We apply these results to Hofer geometry and establish unboundedness of the Hofer diameter of $Ham_c(M\setminus U)$ for stably displaceable $U$. We also answer a question of F. Le Roux about $C^0$-continuity properties of the Hofer metric.

math.SG↗