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Semigroups containing proximal linear maps. (English) Zbl 0845.22004
Let $$V$$ be a finite dimensional vector space. A linear automorphism $$g$$ of $$V$$ is called a proximal element if $$g$$ has a unique eigenvalue $$\lambda = \lambda (g)$$ of maximal absolute value and the corresponding weight space $$V_n = \{v \in V; (g - \lambda E)^n v = 0$$ for some $$n\}$$ is one-dimensional. Goldsheid and Margulis noticed in 1981 that if a subgroup $$G$$ of $$GL(V)$$ contains a proximal element then so does every Zariski dense subsemigroup $$H$$ of $$G$$, provided $$V$$, considered as a $$G$$-module, is strongly irreducible. The authors show further that if $$H$$ is a strongly irreducible subsemigroup of $$GL(V)$$ and if the algebraic closure $$G$$ of $$H$$ contains a proximal element, then there is an $$\varepsilon > 0$$ and for every $$r \geq 1$$ there is a finite subset $$M$$ of $$H$$ such that for every $$g \in GL(V)$$ there is an $$r \in M$$ such that $$\gamma g$$ is $$(r, \varepsilon)$$-proximal. Thus, under the above hypotheses, the subsemigroup $$H$$ provides a rich supply of proximal elements. Extensions and refinements of the above result are obtained in the following directions: a quantitative version of proximality; reducible representations; several eigenvalues of maximal modules. The authors also point out that proximal elements play a crucial role in the dynamics of linear maps because of their simple structure. Generalizations of linear proximal maps to Lyapunov filtrations and to several irreducible summands are obtained.

##### MSC:
 22A15 Structure of topological semigroups
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##### References:
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