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Math. Z. 234, 225239 (2000) c Springer-Verlag 2000

Summary: Math. Z. 234, 225239 (2000)
c Springer-Verlag 2000
On the Ruelle rotation for torus diffeomorphisms
Konstantin Athanassopoulos
Department of Mathematics, University of Crete, GR-71409 Iraklion, Greece
(e-mail: athanako@math.uch.gr)
Received March 22, 1999; in final form May 17, 1999
1. Introduction
In this note we are concerned with the Ruelle rotation number, which is
defined for a C1 diffeomorphism h of the 2-torus T2 that is isotopic to
the identity. Intuitively, the Ruelle rotation number is the mean value, with
respect to an h-invariant Borel probability measure , of the asymptotic rate
at which the derivative of h rotates the tangent planes. In order to measure it,
we need to have fixed in advance a trivialization of the tangent bundle TT2
of T2. The Ruelle rotation number depends on the trivialization, as well as
on the isotopy from id to h. As was mentioned by Ruelle in [7], there is an
independence on the isotopy if we take the number mod Z, i.e. consider the
corresponding point on the unit circle S1.
In Sect. 4 we clarify the dependence on the trivialization of TT2. It
turns out that the two points of S1 taken starting with two different triv-


Source: Athanassopoulos, Konstantin - Department of Mathematics, University of Crete


Collections: Mathematics