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Rotation measures for homeomorphisms of the torushomotopic to a Dehn twist

Published online by Cambridge University Press:  01 June 1997

H. ERIK DOEFF
H. ERIK DOEFF
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL 60208, USA (e-mail: [email protected])
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Abstract

We extend the theory of rotation vectors to homeomorphisms of the two-dimensional torus that are homotopic to a Dehn twist. We define a one-dimensional rotation number and recreate the theory of the homotopic case to the identity case. We prove that if such a map is area preserving and has mean rotation number zero, then it must have a fixed point. We prove that the rotation set is a compact interval, and that if the rotation interval contains two distinct numbers, then for any rational number in the rotation set there exists a periodic point with that rotation number. Finally, we prove that any interval with rational endpoints can be realized as the rotation set of a map homotopic to a Dehn twist.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 17 , Issue 3 , June 1997 , pp. 575 - 591
Copyright
1997 Cambridge University Press

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