Poisson brackets, quasi-states and symplectic integrators (2009)
Entov, Michael, Polterovich, Leonid, Rosen, Daniel
This paper is a fusion of a survey and a research article. We focus on certain rigidity phenomena in function spaces associated to a symplectic manifold. Our starting point is a lower bound obtained...
Entov, Michael, Polterovich, Leonid
Lie quasi-states on a real Lie algebra are functionals which are linear on any abelian subalgebra. We show that on the symplectic Lie algebra of rank at least 3 there is only one continuous...
On continuity of quasi-morphisms for symplectic maps (2009)
Entov, Michael, Polterovich, Leonid, Py, Pierre
We discuss $C^0$-continuous homogeneous quasi-morphisms on the identity component of the group of compactly supported symplectomorphisms of a symplectic manifold. Such quasi-morphisms extend to the...
C0-Rigidity of the Double Poisson Bracket (2009)
Entov, Michael, Polterovich, Leonid
This paper is devoted to function theory on symplectic manifolds. We study a natural class of functionals involving the double Poisson brackets from the viewpoint of their robustness properties with...
C^0-rigidity of the double Poisson bracket (2008)
Entov, Michael, Polterovich, Leonid
The paper is devoted to function theory on symplectic manifolds. We study a natural class of functionals involving the double Poisson brackets from the viewpoint of their robustness properties with...
C^0-rigidity of Poisson brackets (2007)
Entov, Michael, Polterovich, Leonid
Consider a functional associating to a pair of compactly supported smooth functions on a symplectic manifold the maximum of their Poisson bracket. We show that this functional is lower...
Fraczek, Krzysztof, Polterovich, Leonid
Given a bi-Lipschitz measure-preserving homeomorphism of a compact metric measure space of finite dimension, consider the sequence formed by the Lipschitz norms of its iterations. We obtain lower...
Symplectic quasi-states and semi-simplicity of quantum homology (2007)
Entov, Michael, Polterovich, Leonid
We review and streamline our previous results and the results of Y.Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states on symplectic manifolds with semi-simple quantum...
Rigid subsets of symplectic manifolds (2007)
Entov, Michael, Polterovich, Leonid
We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We...
An "anti-Gleason" phenomenon and simultaneous measurements in classical mechanics (2006)
Entov, Michael, Polterovich, Leonid, Zapolsky, Frol
We report on an "anti-Gleason" phenomenon in classical mechanics: in contrast with the quantum case, the algebra of classical observables can carry a non-linear quasi-state, a monotone functional...
Quasi-morphisms and the Poisson bracket (2006)
Entov, Michael, Polterovich, Leonid, Zapolsky, Frol
For a class of symplectic manifolds, we introduce a functional which assigns a real number to any pair of continuous functions on the manifold. This functional has a number of interesting properties....
Nodal inequalities on surfaces (2006)
Polterovich, Leonid, Sodin, Mikhail
Given a Laplace eigenfunction on a surface, we study the distribution of its extrema on the nodal domains. It is classically known that the absolute value of the eigenfunction is asymptotically...
Geometry of contact transformations and domains: orderability versus squeezing (2005)
Eliashberg, Yakov, Kim, Sang Seon, Polterovich, Leonid
Gromov's famous non-squeezing theorem (1985) states that the standard symplectic ball cannot be symplectically squeezed into any cylinder of smaller radius. Does there exist an analogue of this...
Quasi-states and symplectic intersections (2004)
Entov, Michael, Polterovich, Leonid
We establish a link between symplectic topology and a recently emerged branch of functional analysis called the theory of quasi-states and quasi-measures. In the symplectic context quasi-states can...
Sign and area in nodal geometry of Laplace eigenfunctions (2004)
Nazarov, Fedor, Polterovich, Leonid, Sodin, Mikhail
The paper deals with asymptotic nodal geometry for the Laplace-Beltrami operator on closed surfaces. Given an eigenfunction f corresponding to a large eigenvalue, we study local asymmetry of the...
Propagation in Hamiltonian dynamics and relative symplectic homology (2003)
Biran, Paul, Polterovich, Leonid, Salamon, Dietmar
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...
Calabi quasimorphisms for the symplectic ball (2003)
Biran, Paul, Entov, Michael, Polterovich, Leonid
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...
Calabi quasimorphism and quantum homology (2003)
Entov, Michael, Polterovich, Leonid
We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a nontrivial homogeneous quasimorphism to the real numbers with the following property: its value on any...
Paternain, Gabriel P., Polterovich, Leonid, Siburg, Karl Friedrich
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...
Calabi quasimorphism and quantum homology (2002)
Entov, Michael, Polterovich, Leonid
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...
A growth gap for diffeomorphisms of the interval (2002)
Polterovich, Leonid, Sodin, Mikhail
Given an orientation-preserving diffeomorphism of the interval [0;1], consider the uniform norm of the differential of its n-th iteration. We get a function of n called the growth sequence. Its...
Growth of maps, distortion in groups and symplectic geometry (2001)
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...
Diffeomorphisms with an anomalous growth of the differential (2001)
Polterovich, Leonid, Sodin, Mikhail
The paper is withdrawn.
Propagation in Hamiltonian dynamics and relative symplectic homology (2001)
Biran, Paul, Polterovich, Leonid, Salamon, Dietmar
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...
A linear isoperimetric inequality for the punctured Euclidean plane (2001)
Polterovich, Leonid, Sikorav, Jean-Claude
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...
An obstruction to conservation of volume in contact dynamics (2000)
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...
Stable mixing for cat maps and quasi-morphisms of the modular group (2000)
Polterovich, Leonid, Rudnick, Zeev
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...
Kick stability in groups and dynamical systems (2000)
Polterovich, Leonid, Rudnick, Zeev
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...
Partially ordered groups and geometry of contact transformations (1999)
Eliashberg, Yakov, Polterovich, Leonid
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...
On the asymptotic geometry of area-preserving maps (1999)
Polterovich, Leonid, Siburg, Karl Friedrich
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...
Hamiltonian loops from the ergodic point of view (1998)
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...
Hofer's diameter and Lagrangian intersections (1998)
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.
Topological rigidity of Hamiltonian loops and quantum homology (1997)
Lalonde, François, McDuff, Dusa, Polterovich, Leonid
This paper studies the question of when a loop $\phi$ in the group Symp$(M,\omega)$ of symplectomorphisms of a symplectic manifold $(M,\omega)$ is isotopic to a loop that is generated by a...
On the Flux Conjectures (1997)
Lalonde, Francois, McDuff, Dusa, Polterovich, Leonid
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...
Symplectic aspects of the first eigenvalue (1997)
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...