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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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