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Let $T$ be a regular rooted tree. Make $T$ into a metric space by making each edge isometric to the unit interval. What is known about what semigroups can act as continuous functions on $T$ such that each element of the semigroup fixes the root of $T$ and each element maps vertices to vertices? Are there any algebraic restrictions on such semigroups, for example? Can every semigroup arise in this way? Thanks!

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It depends how you allow the semigroup to act and if you are assuming the action is faithful. If the semigroup acts faithfully by level preserving endomorphisms of the tree, then it must be residually finite. If you remove the restriction on regular tree, you can obtain all residually finite semigroups. Good references are the papers Automata, Dynamical Systems, and Groups by Nekrashevych, Grigorchuk and Sushchanskii and Monoids acting on trees by Rhodes (in the first issue of IJAC) and Further results on monoids acting on trees by Rhodes and Silva

I guess as long as your endomorphisms preserve the graph structure of the tree (and hence will be contractions) then you will still have residual finiteness since you cannot increase distance from the root and so if you fix a level, then the set of levels less than or equal to that level is invariant. It is natural to look at level preserving maps because they correspond to sequential machines (or transducers) and they induce contractions of the boundary of the tree.

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