We study the minimally displaced set of irreducible automorphisms of a free group. Our main result is the co-compactness of the minimally displaced set of an irreducible automorphism with exponential growth \(\phi \), under the action of the centraliser \(C(\phi )\). As a corollary, we get that the same holds for the action of \( <\phi>\) on \(Min(\phi )\). Finally, we prove that the minimally displaced set of an irreducible automorphism of growth rate one consists of a single point.

中文翻译：

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$$ F_N $$ FN的不可约自同构的极小位移集是互紧的

我们研究了一个自由群体的不可约自同构的最小位移集。我们的主要结果是在扶正器\（C（\ phi）\）的作用下，具有指数增长\（\ phi \）的不可约自同构的最小位移集的紧致性。作为推论，我们得出\（<\ phi> \）对\（Min（\ phi）\）的作用也是如此。最后，我们证明了一个不可约的增长率自同构的最小位移集由一个点组成。