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Cohomology-free diffeomorphisms of low-dimension tori. (English) Zbl 0922.58038

Let \(M\) be a \(C^{\infty }\) manifold and \(\varphi \) a smooth diffeomorphism of \(M\). Then \(\varphi \) is cohomology-free if every smooth function on \(M\) is cohomologous to a constant, i.e. for every function \(f\) on \(M\) there exists a smooth function \(h\) on \(M\) and constant \(f_{0}\) so that \(h-h\circ \varphi =f-f_{0}\). A. Katok [Constructions in ergodic theory, Preprint (1983)] proposed the problem: for a given manifold \(M\) what are cohomology-free diffeomorphisms? The aim of this article is to give an answer to Katok’s question when the manifold is a low-dimensional torus.
Theorem. The cohomology-free diffeomorphisms of the torus \(T^{n}\) , \(1\leq n\leq 3\), are \(C^{\infty }\) conjugate to diophantine translations.
For \(n=4\) the same result is proved for orientation preserving diffeomorphisms. A translation \(T_{\alpha }\) of \(T^{n}\) is diophantine if \(\alpha \in \mathbb R^{n}\) satisfies the condition \[ \| k\cdot \alpha \| \geq \frac{C}{| k| ^{n+\beta }},\quad C,\beta >0 \] for all \(k=(k_{1},\dots,k_{n})\in \mathbb Z^{n}-\{ 0\} \), where \(\| x\| =\inf \left\{ | x-l| ,l\in \mathbb Z^{n}\right\} \), \(| x| =\sup_{j}| x_{j}| \) and \( k\cdot \alpha =k_{1}\alpha _{1}+\dots +k_{n}\alpha_{n}\). The following result was known earlier: the only cohomology-free affine diffeomorphisms of the torus \(T^{n}\) are the diophantine translations.

MSC:

37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37E99 Low-dimensional dynamical systems
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