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Leonid Polterovich

Publications and source records attributed to Leonid Polterovich.

71 records · Page 4Linked to original sources
arXiv2003

Calabi quasimorphisms for the symplectic ball

We prove that the group of compactly supported symplectomorphisms of the standard symplectic ball admits a continuum of linearly independent real-valued homogeneous quasimorphisms. In addition these quasimorphisms are Lipschitz in the Hofer metric and have the following property: the value of each such quasimorphism on any symplectomorphism supported in any "sufficiently small" open subset of the ball equals the Calabi invariant of the symplectomorphism. By a "sufficiently small" open subset we mean that it can be displaced from itself by a symplectomorphism of the ball. As a byproduct we show that the (Lagrangian) Clifford torus in the complex projective space cannot be displaced from itself by a Hamiltonian isotopy.

math.SG↗
arXiv2003

Stable mixing for cat maps and quasi-morphisms of the modular group

It is well-known that the action of a hyperbolic element (``cat map'') of the modular group on the 2-torus has strong chaotic dynamical properties such as mixing and exponential decay of correlations. In this note we study stability of this behaviour with respect to kicks. Our approach is based on geometric group theory, and in particular on a new result on quasimorphisms of the modular group.

math.DS↗
arXiv2003

Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory

We consider Lagrangian submanifolds lying on a fiberwise strictly convex hypersurface in some cotangent bundle or, respectively, in the domain bounded by such a hypersurface. We establish a new boundary rigidity phenomenon, saying that certain Lagrangians on the hypersurface cannot be deformed (via Lagrangians having the same Liouville class) into the interior domain. Moreover, we study the "non-removable intersection set" between the Lagrangian and the hypersurface, and show that it contains a set with specific dynamical behavior, known as Aubry set in Aubry-Mather theory.

math.SG↗
arXiv2002

Calabi quasimorphism and quantum homology

We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on any diffeomorphism supported in a sufficiently small open subset of the sphere equals to the Calabi invariant of the diffeomorphism. This result extends to more general symplectic manifolds: If the symplectic manifold is monotone and its quantum homology algebra is semi-simple we construct a similar quasimorphism on the universal cover of the group of Hamiltonian diffeomorphisms.

math.SG↗
arXiv2001

Growth of maps, distortion in groups and symplectic geometry

In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism: the uniform norm of the differential of its n-th iteration and the word length of its n-th iteration. In the latter case we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms. We find lower bounds on the growth rates of these sequences in a number of situations. These bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism. They are obtained by using recent results of Schwarz on Floer homology. Applications to the Zimmer program are presented. We prove non-existence of certain non-linear symplectic representations for finitely generated groups including some lattices and Baumslag-Solitar groups.

math.DS↗
arXiv2001

Propagation in Hamiltonian dynamics and relative symplectic homology

The main result asserts the existence of noncontractible periodic orbits for compactly supported time dependent Hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating Hamiltonian is sufficiently large over the zero section. The proof is based on Floer homology and on the notion of a relative symplectic capacity. Applications include results about propagation properties of sequential Hamiltonian systems, periodic orbits on hypersurfaces, Hamiltonian circle actions, and smooth Lagrangian skeletons in Stein manifolds.

math.SG↗
arXiv2001

A linear isoperimetric inequality for the punctured Euclidean plane

It follows from a general theorem of Bonk and Eremenko that closed plane curves which are contractible in the complement to the integral lattice satisfy a linear isoperimetric inequality. We give an alternative proof of this fact. Our approach is based on a non-standard combinatorial isoperimetric inequality which requires a refinement of the small cancellation theory. We present an application of the isoperimetric inequality for the punctured plane to Hamiltonian dynamics. Combining it with methods of symplectic topology we show that every non-identical Hamiltonian diffeomorphism of the 2-torus has at least linear asymptotic growth of the differential.

math.GR↗
arXiv2000

An obstruction to conservation of volume in contact dynamics

A theorem of Moser guarantees that every diffeomorphism of a closed manifold can be isotoped to a volume preserving one. We show that this statement cannot be extended into contact category: some connected components of contactomorphism groups of certain contact manifolds contain no volume-preserving diffeomorphisms. This phenomenon can be considered from different viewpoints: geometric (isometric action of the contact mapping class group on the moduli space of contact forms), topological (action in symplectic homology) and dynamical (diffusion). We define a numerical invariant - a kind of contact Lyapunov exponent - which leads to a quantitive version of the abovementioned result.

math.SG↗
arXiv2000

Kick stability in groups and dynamical systems

We consider a general construction of ``kicked systems''. Let G be a group of measure preserving transformations of a probability space. Given its one-parameter/cyclic subgroup (the flow), and any sequence of elements (the kicks) we define the kicked dynamics on the space by alternately flowing with given period, then applying a kick. Our main finding is the following stability phenomenon: the kicked system often inherits recurrence properties of the original flow. We present three main examples. 1) G is the torus. We show that for generic linear flows, and any sequence of kicks, the trajectories of the kicked system are uniformly distributed for almost all periods. 2) G is a discrete subgroup of PSL(2,R) acting on the unit tangent bundle of a Riemann surface. The flow is generated by a single element of G, and we take any bounded sequence of elements of G as our kicks. We prove that the kicked system is mixing for all sufficiently large periods if and only if the generator is of infinite order and is not conjugate to its inverse in G. 3) G is the group of Hamiltonian diffeomorphisms of a closed symplectic manifold. We assume that the flow is rapidly growing in the sense of Hofer's norm, and the kicks are bounded. We prove that for a positive proportion of the periods the kicked system inherits a kind of energy conservation law and is thus superrecurrent. We use tools of geometric group theory and symplectic topology.

math.DS↗
arXiv1999

Partially ordered groups and geometry of contact transformations

We prove, for a class of contact manifolds, that the universal cover of the group of contact diffeomorphisms carries a natural partial order. It leads to a new viewpoint on geometry and dynamics of contactomorphisms. It gives rise to invariants of contactomorphisms which generalize the classical notion of the rotation number. Our approach is based on tools of Symplectic Topology.

math.SG↗
arXiv1999

On the asymptotic geometry of area-preserving maps

We study the asymptotic behaviour of 1-parameter subgroups with respect to Hofer's metric when the underlying symplectic manifold is an open surface of infinite area. We prove that, depending on the topology of the level sets of the Hamiltonian H, the distance either is bounded or behaves asymptotically linear. Moreover, the slope can be calculated explicitly as the difference of two distinguished critical values of H.

math.DG↗
arXiv1998

Hamiltonian loops from the ergodic point of view

The paper provides a link between ergodic theory and symplectic topology. A classical notion of ergodic theory is a skew product map associated with a loop in a group of transformations. We study skew products which come from loops in the group of Hamiltonian diffeomorphisms of a symplectic manifold. Our main question is which homotopy classes of loops can be represented by strictly ergodic skew products. We prove an existence result, and find an obstruction which arises from Hofer's geometry on the group of Hamiltonian diffeomorphisms.

math.DG↗
arXiv1998

Hofer's diameter and Lagrangian intersections

We prove that the group of Hamiltonian diffeomorphisms of the 2-sphere has infinite diameter with respect to Hofer's metric. Our approach is based on the theory of Lagrangian intersections.

math.DG↗
arXiv1997

Topological rigidity of Hamiltonian loops and quantum homology

This paper studies the question of when a loop $ϕ$ in the group Symp$(M,ω)$ of symplectomorphisms of a symplectic manifold $(M,ω)$ is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to be Hamiltonian.) Our main result is that Hamiltonian loops are rigid in the following sense: if $ϕ$ is Hamiltonian with respect to $ω$, and if $ϕ'$ is a small perturbation of $ϕ$ that preserves another symplectic form $ω'$, then $ϕ'$ is Hamiltonian with respect to $ω'$. This allows us to get some new information on the structure of the flux group, i.e. the image of $π_1(Symp(M,ω))$ under the flux homomorphism. We give a complete proof of our result for some manifolds, and sketch the proof in general. The argument uses methods developed by Seidel for studying properties of Hamiltonian loops via the quantum homology of $M$.

dg-ga↗
arXiv1997

On the Flux Conjectures

The ``Flux conjecture'' for symplectic manifolds states that the group of Hamiltonian diffeomorphisms is C^1-closed in the group of all symplectic diffeomorphisms. We prove the conjecture for spherically rational manifolds and for those whose minimal Chern number on 2-spheres either vanishes or is large enough. We also confirm a natural version of the Flux conjecture for symplectic torus actions. In some cases we can go further and prove that the group of Hamiltonian diffeomorphisms is C^0-closed in the identity component of the group of all symplectic diffeomorphisms.

dg-ga↗
arXiv1997

Symplectic aspects of the first eigenvalue

There are two themes in the present paper. The first one is spelled out in the title, and is inspired by an attempt to find an analogue of Hersch-Yang-Yau estimate for $lambda_1$ of surfaces in symplectic category. In particular we prove that every split symplectic manifold $T^4 times M$ admits a compatible Riemannian metric whose first eigenvalue is arbitrary large. On the other hand for Kahler metrics compatible with a given integral symplectic form an upper bound for $lambda_1$ does exist. The second theme is the study of Hamiltonian symplectic fibrations over the 2-sphere. We construct a numerical invariant called the size of a fibration which arises as the solution of certain variational problems closely related to Hofer's geometry, K-area and coupling. In some examples it can be computed with the use of Gromov-Witten invariants. The link between these two themes is given by an observation that the first eigenvalue of a Riemannian metric compatible with a symplectic fibration admits a universal upper bound in terms of the size.

dg-ga↗