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Conjugacy class of homeomorphisms and distortion elements in groups of homeomorphisms. (arXiv:1405.1000v2 [math.DS] UPDATED)
来源于:arXiv
Let S be a compact connected surface and let f be an element of the group
Homeo\_0(S) of homeomorphisms of S isotopic to the identity. Denote by
\tilde{f} a lift of f to the universal cover of S. Fix a fundamental domain D
of this universal cover. The homeomorphism f is said to be non-spreading if the
sequence (d\_{n}/n) converges to 0, where d\_{n} is the diameter of
\tilde{f}^{n}(D). Let us suppose now that the surface S is orientable with a
nonempty boundary. We prove that, if S is different from the annulus and from
the disc, a homeomorphism is non-spreading if and only if it has conjugates in
Homeo\_{0}(S) arbitrarily close to the identity. In the case where the surface
S is the annulus, we prove that a homeomorphism is non-spreading if and only if
it has conjugates in Homeo\_{0}(S) arbitrarily close to a rotation (this was
already known in most cases by a theorem by B{\'e}guin, Crovisier, Le Roux and
Patou). We deduce that, for such surfaces S, an element of Homeo\_{0}(S) is
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