Abstract.
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;¶• The word length of its n-th iteration, where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms.¶We find lower bounds for 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. As an application, we prove non-existence of certain non-linear symplectic representations for finitely generated groups.
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Oblatum 6-XII-2001 & 19-VI-2002¶Published online: 5 September 2002
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ID="*"Supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.
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Polterovich, L. Growth of maps, distortion in groups and symplectic geometry. Invent. math. 150, 655–686 (2002). https://doi.org/10.1007/s00222-002-0251-x
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DOI: https://doi.org/10.1007/s00222-002-0251-x