Hubbard Trees

Hubbard Tree of z^2+i

It is known that Hubbard trees serve as a model for Julia sets of post-critically finite polynomials. What does it mean precisely? In fact, there is an inverse limit space of sequences (x_1,x_2,\ldots,x_n,\ldots) together with the shift map, such that the obtained dynamical system is topologically conjugate to the dynamical system p:J \to J, where $J$ is the Julia set of the polynomial $p$.

One of the explanations of the fact comes from the theory of IMG (iterated monodromy groups). Perhaps, the fact itself could be proven much easier way without involving the notion of IMG, but this is just one point of view on it.
IMG itself could be defined of partial self-coverings. A bit more general situuation arises when one defines IMG on topological automaton.

Definition. Topological automaton is a quadruple (M,M_1,f,\iota), where M,M_1 – topological spaces, f:M_1 \to M is a finite covering map, \iota:M_1 \to M is a continuous map.

In the case of partial self-coverings, when M_1 \subset M, \iota is just an embedding. But in this general setting we have in fact two maps between two topological spaces. One can iterate this topological automaton and associate an inverse limit system to it.
For example M_2 is defined as a pullback of the map f:M_1 \to M_0=M under the map \iota. M_2 = \{ (x,y) \in M^2_1: f(y)=\iota(x) \}.
In the same way M_n = \{ (x_1,x_2,\ldots,x_n) \in M^n_1 | f(x_{i+1})=\iota(x_i) \}


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