76 research outputs found
A periodicity criterion and the section problem on the Mapping Class Group
Some years ago, V. Markovic proved that there is no section of the Mapping
Class Group for a closed surface of genus g larger than 5 (in the case of
homeomorphims) and more recently generalized this result with D. Saric to the
case where g is larger than 1. We will state a periodicity criterion and will
use it to simplify some of the arguments given by Markovic and Saric in the
proof of their theorem. The periodicity criterion tells us that a homeomorphism
of a connected surface must be periodic if the set of connected periodic open
sets generates the topology of the surface.Comment: 40 page
A finite dimensional proof of a result of Hutchings about irrational pseudo-rotations
We prove that the Calabi invariant of a pseudo-rotation of the unit
disk, that coincides with a rotation on the unit circle, is equal to its
rotation number. This result has been shown some years ago by Michael Hutchings
(under very slightly stronger hypothesis). While the original proof used
Embedded Contact Homology techniques, the proof of this article uses generating
functions and the dynamics of the induced gradient flow
Prime ends rotation numbers and periodic points
We study the problem of existence of a periodic point in the boundary of an
invariant domain for a surface homeomorphism. In the area-preserving setting, a
complete classification is given in terms of rationality of Carath\'eordory's
prime ends rotation number, similar to Poincar\'e's theory for circle
homeomorphisms. In particular, we prove the converse of a classic result of
Cartwright and Littlewood. This has a number of consequences for generic area
preserving surface diffeomorphisms. For instance, we extend previous results of
J. Mather on the boundary of invariant open sets for -generic area
preserving diffeomorphisms. Most results are proved in a general context, for
homeomorphisms of arbitrary surfaces with a weak nonwandering-type hypothesis.
This allows us to prove a conjecture of R. Walker about co-basin boundaries,
and it also has applications in holomorphic dynamics.Comment: 50 pages, 15 figure
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