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Cohomology-free diffeomorphisms of low-dimension tori

Published online by Cambridge University Press:  01 August 1998

RICHARD U. LUZ  and
NATHAN M. DOS SANTOS
RICHARD U. LUZ
Affiliation:
Instituto de Matemática-Universidade Federal Fluminense, 24020-005, Niterói, RJ, Brazil Present address: Universidad Catolica del Norte, Facultad de Ciencias, Departamento de Matemáticas, Casilla 1280, Antofagasta, Chile.
NATHAN M. DOS SANTOS
Affiliation:
Instituto de Matemática-Universidade Federal Fluminense, 24020-005, Niterói, RJ, Brazil
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Abstract

We study cohomology-free (c.f.) diffeomorphisms of the torus $T^n$. A diffeomorphism is c.f. if every smooth function $f$ on $T^n$ is cohomologous to a constant $f_0$, i.e. there exists a $C^{\infty}$ function $h$ so that $h-h\circ\varphi=f-f_0$. We show that the only c.f. diffeomorphisms of $T^n$, $1\le n\le3$, are the smooth conjugations of Diophantine translations. For $n=4$, we prove the same result for c.f. orientation-preserving diffeomorphisms.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 18 , Issue 4 , August 1998 , pp. 985 - 1006
Copyright
© 1998 Cambridge University Press

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