Archive for the ‘Uncategorized’ Category

Classification of Newton maps: realization

August 8, 2009

On the way to realize combinatorial models as real Newton maps the plan was to use Thurston’s theorem. In order to do so, one needs to prove that the branched covering, specified by the combinatorial data, doesn’t have Thurston obstructions.
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.

Hubbard Trees

August 7, 2009

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 $M_n = \{ (x_1,x_2,\ldots,x_n) \in M^n_1 | f(x_{i+1})=\iota(x_i) \}$

Rational Maps don’t have Levy cycles

July 13, 2009

Here is a sketch of the following:

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

Proof. Suppose it does and denote by $\Gamma$ the corresponding Levy cycle. Choose $\gamma_1,\gamma_2,\ldots, \gamma_n$ the geodesics representatives in $\Gamma$. Then the map $f: \bar{C}-f^{-1}(P_f) \to \bar{C}-P_f$ is the covering map, hence a local isometry and preserves the distances locally. Therefore

$l_{\bar{C}-f^{-1}(P_f)} (\gamma'_i) = l_{\bar{C}-P_f} (\gamma_{i+1})$ where $\gamma'_i$ is the component of $f^{-1}(\gamma_{i+1})$ which is in the same homotpy class with $\gamma_i$. From the other hand $\bar{C}-f^{-1}(P_f) \subset \bar{C}-P_f$ and

$l_{\bar{C}-P_f} (\gamma_{i+1}) = l_{\bar{C}-f^{-1}(P_f)} (\gamma'_i) > l_{\bar{C}-P_f} (\gamma_i)$ and we get that each $\gamma_i$ is strictly shorter than $\gamma_{i+1}$, which is impossible.

Linearization and circle diffeomorphisms

July 11, 2009

Today, i have learn that there is quite interesing interplay between the theory of linearization of maps $f(z)=\lambda z + O(z^2)$ and analytic circle diffeomorphism theory.

In fact, according to results of P.Marco, for every such $f(z)$ there is a fully invariant set $K$, which is called a Siegel continua, or Hedgehog (in the Cremer case) such that if we consider the conformal representation of the complement $\bar{C} - K$ to the complement of the unit disk, then it conjugates the map $f(z)$ 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 $K$ is basically the following: in case of the rational multiplier $\lambda$ there are quite a lot of things known about the topology in the neighborhood of the fixed point $0$:  Leau-Fatou flower theorem. Which states that one can construct attracting and repelling petals, all have $0$ on the boundary, which interchange and form the whole neighborhood of $0$.  As a candidate for $K$ 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 $K$.

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!

Hello world!

July 11, 2009

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