On dynamical Teichmüller spaces.

Sullivan introduced the study of a dynamical Teichmüller space, which we denote by $T_{1}(R)$, associated to a rational function $R$. The space of orbits of $T_{1}(R)$, under the action of an associated modular group $Mod_{1}(R)$, coincides with the space $QC(R)$ of quasiconformal deformations of $R$. We modify Sullivan’s definition to get another Teichmüller space $T_{2}(R)$, with its corresponding modular group $Mod_{2}(R)$. In this situation, the $J$-stability component is the space of orbits of $T_{2}(R)$, under the action of $Mod_{2}(R)$. When $R$ is hyperbolic, the $J$-stability component is the hyperbolic component of $R$.

There are natural inclusions of the space $T_{1}(R)$ into $T_{2}(R)$, and from the group $Mod_{1}(R)$ into $Mod_{2}(R)$. We find that, properties of these inclusions are related to the dynamics of $R$. With this at hand, we can establish relations between the dynamics of $R$ and topological properties of $T_{2}(R)$.

When the Julia set of $R$ is totally disconnected. The space $T_{2}(R)$ has a laminated structure. In this way, we also realize $T_{2}(R)$ as the space of deformations of the natural extension of $R$. The structure of the paper is as follows.

In Section 2, we recall basic definitions and facts of the classical Teichmüller space $T_{1}(R)$.

In Section 3, we introduce $T_{2}(R)$ and show that, as in the case of $T_{1}(R)$, is a complete metric space. In Theorem 4, we establish characterizations for the path connectivity of $T_{2}(R)$. Using this, we prove Theorem 5 stating that, when $R$ is a polynomial, the connectivity of Julia set $J(R)$ is equivalent to the path connectivity of $T_{2}(R)$.

In Section 4, we restrict to the case where $J(R)$ is homeomorphic to a Cantor set. In this case, $T_{2}(R)$ is a trivial product of $T_{1}(R)$ times a totally disconnected space. We finish the section giving a characterization of the property that $J(R)$ is homeomorphic to a Cantor set in terms of properties of $T_{2}(R)$.

Finally, in Section 5, we construct a realization of $T_{2}(R)$ as the space of deformations of the natural extension of $R$.

Given a rational map $R$, let us define the space $T_{1}(R)=T(S_{R})\times B_{R}$, where $S_{R}$ is a Riemann surface associated to the Fatou set $F(R)$, $T(S_{R})$ denotes the classical Teichmüller’s space of $S_{R}$, and $B_{R}$ is the space of invariant Beltrami differentials, defined on the Julia set $J(R)$, which are compatible with the dynamics of $R$. That is, $B_{R}$ is the space of measurable $(1,-1)$ forms $\mu$ with $L^{\infty}$ norm bounded by $1$, satisfying the conditions that $\mu$ is $0$ outside the Julia set $J(R)$ and $R^{*}(\mu)=\mu$. For a more detailed account of the definitions see [4] and [5].

An equivalent way to define $T_{1}(R)$ is as the set of isotopy classes of pairs $\langle[R_{1}],[\phi]\rangle$ where $\phi$ is a quasiconformal conjugation of $R$ to the rational map $R_{1}$. The first modular group $Mod_{1}(R)$, is the group of all isotopy classes of quasiconformal homeomorphisms of $\mathbb{C}$ commuting with $R$. The group $Mod_{1}(R)$ acts on $T_{1}(R)$ with the action given by

A theorem due to Sullivan and McMullen states that $Mod_{1}(R)$ acts on $T_{1}(R)$ as a group of isometries, for more details see [4] and [5]. The formula $T_{1}(R)/Mod_{1}(R)=QC(R)$, where $QC(R)$ is the space of quasiconformal deformations of $R$, will play an important role in what follows.

We will define a Teichmüller space that generalizes the formula

for the $J$-stability component of $R$.

Let us recall that two rational maps $R_{1}$ and $R_{2}$ are $J$-equivalent, if there is a homeomorphism $h:J(R_{1})\rightarrow J(R_{2})$, which is quasiconformal in Pesin’s sense and conjugates $R_{1}$ to $R_{2}$.

Given a family of maps $\{R_{w}\}$ depending holomorphically on a parameter $w\in W$, a map $R_{w_{0}}$ in $\{R_{w}\}$ is called $J$-stable if, there is a neighborhood $V$ of $w_{0}$ such that, $R_{w}$ is $J$-equivalent to $R_{w_{0}}$ for all $w\in V$, and the conjugating homeomorphisms depend holomorphically on $w$.

We denote by $QC_{J}(R)$, the $J$-stability component of a rational map $R$. This is the path connected component of the $J$-equivalence class of $R$ containing $R$. In [3], Mañe, Sad and Sullivan proved that for every holomorphic family of rational maps, the union of the $J$-stability components is open and dense. When $R$ is hyperbolic, an application of the $\lambda$-Lemma, for holomorphic motions around $J(R)$, shows that $QC_{J}(R)$ coincides with $Hyp(R)$, the hyperbolic component of $R$.

Let $R$ be a rational map, we define the space $X_{n}(R)$ as the set of pairs $(h,U)$, where $U$ is an open neighborhood of the Julia set $J(R)$, and $h:U\rightarrow\mathbb{C}$ is a quasiconformal embedding such that

is the restriction of rational map, with $\deg R\leq\deg R_{h}\leq n$, wherever the conjugacy is well defined.

We say that $(h_{1},U_{1})\sim(h_{2},U_{2})$ in $X_{n}(R)$ if, and only if, there exists open sets $V_{1}$ and $V_{2}$, satisfying $V_{i}\subset U_{1}\cap U_{2}$, $J(R)\subset V_{i}$, for $i=1,2$, and a Möbius transformation $\gamma:\mathbb{C}\rightarrow\mathbb{C}$ such that the following diagram commutes

and so that, $F$ is a map homotopic to the identity, with a homotopy that commutes with $R$.

With this equivalence relation on $X_{n}(R)$, we can take representatives $(h,U)$ such that, $U$ has nice dynamical properties. For instance, if $R$ is hyperbolic we can always choose $U$ satisfying $R^{-1}(U)\subset U$.

Following classical Teichmüller theory, the map $\gamma$ would be a holomorphic map. However next proposition, which is actually a folklore fact, justifies our definition.

Let $T_{2,n}(R)=X_{n}(R)/\sim$, this definition generalizes the notion of the Teichmüller space for a rational function. The space $X_{n}(R)$ is extremely big, note that we can change the neighborhood $U$, arbitrarily in the pair $(h,U)$, and still get the same point in $T_{2,n}(R)$. For instance, the restriction of $h$ on a smaller neighborhood. Consider the space $T_{2,n}(z^{2})$ with $n\geq 3$, this space contains all maps of the form $z^{2}+\lambda z^{3}$ for $\lambda$ small enough. In this paper, we will restrict to the case where $n=\deg(R)$ and, in this situation, we will omit the subindex $n$.

Two quasiconformal maps $f:U\rightarrow V$ and $g:U^{\prime}\rightarrow V^{\prime}$, defined on neighborhoods of $J(R)$, are equivalent $f\sim g$, if there exist $W\subset U\cap U^{\prime}$ on which $f$ and $g$ are homotopic, with a homotopy that commutes with $R.$ We can define a modular group

Let $R$ be a hyperbolic rational map, one can check that

Note that the group $Mod_{2}(R)$ does not depend on $n$. For $n>\deg R$, the quotient $T_{2,n}(R)/Mod_{2}(R)$ forms a much bigger space containing $Hyp(R)$, it also contains other components, coming from higher degrees, arranged on the boundary of $Hyp(R)$. This construction allow us to consider, as basic points of the Teichmüller space, points that “belong” to the boundary of other $T_{2}(R^{\prime})$. For instance, $z^{2}$ “belongs” to the boundary of the space $T_{2}(\lambda z^{3}+z^{2})$ for $\lambda$ close to zero, but not zero. In fact, the same is true for $T_{1}(\lambda z^{3}+z^{2})$. Nevertheless, the complete picture is yet to be understood.

Now, let us define a third modular group $Mod_{3}(R)$, as the group of maps $\phi:J(R)\rightarrow J(R)$ which are quasiconformal in Pesin’s sense and commute with $R.$

One would be inclined to introduce a third Teichmüller space $T_{3}(R)$. A sensible definition for this space, is to consider the set of quasiconformal maps, in the sense of Pesin, defined just in the $J(R)$ and commuting with $R$. However, it is not clear how to relate this Teichmüller space with the usual quasiconformal theory. In other words, in general, is not clear if the natural map from $Mod_{2}(R)$ to $Mod_{3}(R)$ is surjective. We can carry on this discussion when the map $R$ is hyperbolic and, more generally, when the Julia set is described as limits of telescopes with bounded geometry. In these cases, every quasiconformal map, defined on the Julia set and inducing an isomorphism on telescopes, can be extended to a quasiconformal map defined on a neighborhood of $J(R)$. For definition on telescopes see [6].

We now restrict the discussion to the case where the Julia set $J(R)$ is homeomorphic to a Cantor set. Under these conditions, we show that the Teichmüller space $T_{2}(R)$ has a product structure. We shortly remind the proof of the following known fact.

Let $S$ be a multiply connected Riemann surface with boundary such that, the connected components of $\partial S$ are Jordan curves. The pure mapping class group $Map(S)$ is defined by the set of topological automorphisms of $S$, acting identically on the boundary, modulo a homotopic relation. This homotopic relation is defined as follows, $f\sim g$ are equivalent if, and only if, there exist an isotopy $H$, from $f$ to $g$, such that $H|_{\partial S}=f|_{\partial S}=g|_{\partial S}$. A classical theorem states that, the group $Map(S)$ is generated by Dehn twists along simple closed curves.

Let $P$ be a unimodal polynomial of degree $d$, such that the Julia set $J(P)$ is homeomorphic to a Cantor set. This is equivalent to say that the critical orbit of $P$ escapes to infinity. Let $\gamma$ be a Jordan curve, whose interior contains the critical value and the Julia set. The preimage $P^{-1}(\gamma)$ consists of $d$ disjoint Jordan curves and, together with $\gamma$, defines a $d+1$-connected Riemann surface $S_{1}$ with boundary. Define recursively $S_{n}$ by $S_{n}=P^{-1}(S_{n-1})$. Then $Map(S_{n+1})$ is the $d$-fold product of $Map(S_{n})$. We have the following:

Note that if we consider, instead of $Map(S_{n})$, the group of automorphisms of $S_{n}$, not necessarily acting identically on $\partial S_{n}$. Then, on the corresponding product, it appears the action of a braiding group.

It is not clear that every element in $Mod_{2}(R)$, acting identically on $J(R)$, should be homotopic to some element in $Map(S_{n})$. Moreover, $Mod_{2}(R)$ consists of elements that have a simplicial extension, this relates the modular group $Mod_{2}(R)$ with Thompson’s group of automorphisms of the Cantor set.

In general, $T_{2}(R)$ is not path connected. Since $T_{2}(R)$ contains $H(T_{1}(R))$ and the orbit of $H(T_{1}(R))$ under the action of $Mod_{2}(R)$. Locally, the orbit space is homeomorphic to $Mod_{2}(R)/\alpha(Mod_{1}(R))$. Thus we have

Conceivably, $T_{2}(R)$ is also locally a product when $R$ is a rational map with disconnected Julia set. However, in this case, a non-trivial monodromy can appear along $H(T_{1}(R)).$

Now we will see that the connectivity of $J(R)$ is related to the injectivity of $\alpha:Mod_{1}(R)\rightarrow Mod_{2}(R)$.