Archive for August, 2009

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.
Contradiction.

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Hubbard Trees

August 7, 2009

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) \}

Topological Entropy

August 3, 2009

There are several qualitative invariants of dynamical systems, such as density of periodic points, topological mixing, hyperbolicity. The answer whether a concrete dynamical system has this property or not would be “Yes” or “No”. There do exists quantitative invariants, for example the growth of period points. But perhaps the most important quantitative invariant is a topological entropy.

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

\displaystyle d_n^f(x)=max_{0 \leq i \leq n-1} d(f^i(x),f^i(y))

for a given positive integer n. Let B(x,r,n)=\{y \in X: d_n^f(x,y)<r \}, the ball of radius r. Let S_d(f,r,n) be the minimal number of such ball, sufficient to cover the whole of X.

Then

\displaystyle h_{top}(f)=\lim_{r \to 0} \lim_{n \to \infty} \frac{ \log S_d(f,r,n)}{n}  .

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 d_n^f > r  . The cardinality of the maximal separating set is denoted by N_d(f,r,n) and can be used similarly for the definition of the topological entropy, substituting S_d(f,r,n) by N_d(f,r,n) in the formula above.

For example, for the expanding map E_m(x)=mx \quad mod (1), the topological entropy h_{top}(E_m)=\log|m|.

Proof. In general, for expanding maps the distance between two points growth until it becomes larger than some constant (1/2m for E_m). Let us choose two points x,y with d(x,y)<\frac{1}{2m^n}.Then

d_n^{E_m}(x,y)=d(E_m^{n-1}(x),E_m^{n-1}(x)) = m^{n-1}d(x,y).

If we want to have

d_n^{E_m}(x,y)> m^{-k}, then we must have d(x,y)>m^{-k-n}.

Hence if we choose S=\{i\cdot m^{-k-n}: 0\leq i \leq m^{k+n}-1 \} all the points from S will have pairwise distances between each other at least m^{-k}. And the set S can serve as a separating set with |S|=m^{k+n}, we obtain that

h_{top}(E_m)=\log |m|.

Bowen has asked to find a topological entropy of the complex polynomial of degree d. More generally M.Lyubich gave the answer for any rational function in the complex plane. The lower bound h_{top}(f) \geq \log (deg f) was known before due to Misiurewisz-Przytycki theorem.

The answer is as follows:

Theorem. Let f: S^2 \to S^2 be a rational function, that is not a constant. Then the topological entropy

h_{top}(f)=\log (deg f).