This is the point where theorem of K.Pilgrim and T.Lei comes in use. It basically says that obstructions don’t intersect the preimages of arc systems.

The main difficulty in our case was to eliminate the case when the obstruction does intersect the arc system itself. Due to the theorem mentioned above this would imply that it cannot intersect the preimages of the arc systems – that is exactly the point where we aim at a contradiction.

Suppose it happens. Then as follows from the theorem the obstruction must be a Levy cycle. It is also known (BFH) that the complement components of the curves in the Levy cycles can only contain periodic mark points which in our case would be periodic points on extended Hubbard trees lying in the Julia set.

Such points being repelling for corresponding polynomial-like maps, have at least one external ray landing on it, i.e. there exists a preimage of the arc system which intersects the Thurston obstruction.

Contradiction.

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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 together with the shift map, such that the obtained dynamical system is *topologically conjugate* to the dynamical system , 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 , where – topological spaces, is a finite covering map, is a continuous map.

In the case of partial self-coverings, when , 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 is defined as a pullback of the map under the map . .

In the same way

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**Definition. **In some sense, the topological entropy describes the growth of orbits all at the same time. To be precise, for two points one can define the metric

for a given positive integer . Let , the ball of radius . Let be the minimal number of such ball, sufficient to cover the whole of .

Then

.

There are several other definitions for the topological entropy. Sometimes it is not quite useful to use covering sets for computation reasons. For example, one can use the so-called *separating sets, *i.e. the sets such that the pairwise distance between any two points in the metric . The cardinality of the maximal separating set is denoted by and can be used similarly for the definition of the topological entropy, substituting by in the formula above.

For example, for the expanding map , the topological entropy .

*Proof.* In general, for expanding maps the distance between two points growth until it becomes larger than some constant ( for ). Let us choose two points with .Then

.

If we want to have

, then we must have .

Hence if we choose all the points from will have pairwise distances between each other at least . And the set can serve as a separating set with , we obtain that

.

Bowen has asked to find a topological entropy of the complex polynomial of degree . More generally M.Lyubich gave the answer for any rational function in the complex plane. The lower bound was known before due to Misiurewisz-Przytycki theorem.

The answer is as follows:

**Theorem. **Let be a rational function, that is not a constant. Then the topological entropy

.

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**Solution.** Let be a covering map from the unit disk to . First of all, notice that due to Schwarz-Pick theorem such an is either

*A global isometry, i.e. a conformal isomorphisms of**A covering map, but is not one-to-one, i.e. a local isometry**Strictly decreases the Poincare metric*

It is easy to see that in the last case cannot have two fixed points. Therefore let’s assume that is a covering map. Since the covering is universal, one can lift the map to the map .

Also note that . Assume that are the fixed points of and , perhaps composing with an automorphism of we can also assume that . Remember that is not strictly decreasing Poincare metric, the same applies to its lift , hence must be a rotation around the origin (it is not difficult to classify all automorpisms of the unit disk), for some real .

Note that , hence is in the fiber of and therefore the set is a discrete set which implies that is rational and we are done.

P.S. Actually, if we were talking about self-coverings of compact surfaces, the situation would be easier: for compact Riemann surfaces we do have Riemann-Hurwitz formula:

,

where is a ramified covering of degree . If $latex X=Y$ then we would obtain that either or . Inother words, for compact Riemann surfaces of genus the only self-covering is a biholomorphism.

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**Theorem**. The rational functions doesn’t have Levy cycles.

**Proof.** Suppose it does and denote by the corresponding Levy cycle. Choose the geodesics representatives in . Then the map is the covering map, hence a local isometry and preserves the distances locally. Therefore

where is the component of which is in the same homotpy class with . From the other hand and

and we get that each is strictly shorter than , which is impossible.

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In fact, according to results of P.Marco, for every such there is a **fully** invariant set , which is called a Siegel continua, or Hedgehog (in the Cremer case) such that if we consider the conformal representation of the complement to the complement of the unit disk, then it conjugates the map to an anylitc circle diffeomorphism, which actually even has the same roattion number.And the last property is crucial!

The idea of the proof of the existence of such is basically the following: in case of the rational multiplier there are quite a lot of things known about the topology in the neighborhood of the fixed point : Leau-Fatou flower theorem. Which states that one can construct attracting and repelling petals, all have on the boundary, which interchange and form the whole neighborhood of . As a candidate for in the rational case one can take the union of intersections of attracting and repelling petals: they are clearly fully invariant and the other properties can also be established. Then it left to use the density argument and show that the density of rational translates to our case in terms of existence of these continua .

It follows that we have a dictionary between these two theories. Even though a lot of things are the same in this dictionary, there are differences. For example (the only one i know so far), the set of lineariziable quadratic maps with irrationally indifferent fixed point – is the set of Bruno numbers (one way – Bruno, Siegel, the necessity – Yoccoz). However, for circle diffeomorphisms, the linearizability condition is for the rotation number to be Herman!

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