Conversely, every two-tile subdivision rule can berealized by a Thurston map (see Proposition 12.3). It is unique up to Thurston equivalence. In this way, two- tile subdivision rules give simple combinatorial models for Thurston maps. There is no obvious difference between the two-tile subdivision rules realized by rational maps and the ones that are not. This is another motivation for investigating general Thurston maps.
The mapgin Section 1.1 and the maphin Section 1.3 were constructed from Figure 1.1 and Figure 1.2, respectively. This really means that the pictures repre- sent two-tile subdivision rules, and the maps realize these subdivision rules accord- ing to Proposition 12.3. This is our preferred way to define Thurston maps. To be able to discuss examples, we will use this way of constructing Thurston maps in an informal way even before we provide the theoretical foundations in Chapter 12.
Our concept of a two-tile subdivision rule is closely related to the generalsub- division rules that have been studied extensively by Cannon, Floyd, and Parry (see for example [CFP01]).
The main consequence of Theorem 15.1 is that we have a combinatorial de- scription by a two-tile subdivision rule for sufficiently high iteratesF =fnof every
expanding Thurston mapf.
Corollary 15.2(Thurston maps and subdivision rules). Let f:S2→S2 be an expanding Thurston map. Then for each sufficiently large n∈N there exists a two-tile subdivision rule that is realized by F =fn.
In particular, we obtain a cellular Markov partition for F. There are several other approaches to providing combinatorial models for certain classes of maps. For example, a postcritically-finite polynomial can be described by its Hubbard tree (see [DH84]) or a rational map with three critical values by a dessin d’enfant (see [Gro97] and [LZ04]). A very general setting that allows one to address similar questions is the recently developed theory of self-similar group actions. In this context one investigates algebraic objects such as the iterated monodromy group and the biset (or bimodule) defined for a Thurston map (see [Ne05], in particular Chapter 6).
Our approach is more geometric and adapted to Thurston maps. One of its main features is that we have good geometric control for the cells in the decompo- sitionsDn =
Dn(f,
C) ifC is f-invariant. In particular, with respect to any visual metric the curveC is actually a quasicircle (Theorem 15.3) and the boundaries of the tiles inDn are quasicircles with uniform parameters independent of the leveln
(Proposition 15.26). For a rational expanding Thurston map f:Cb →Cb the tiles in Dn are in fact uniform quasidisks with respect to the chordal metric σ on Cb
(Theorem 18.4 (iii)).
1.6. Miscellaneous results
In this section we collect various noteworthy results that may be useful for the orientation of the reader.
The concept ofThurston equivalencefor Thurston maps was already mentioned before. We record its slightly technical definition in Section 2.4. At first sight the concept does not seem to be adapted to the dynamics under iteration. However, in Theorem 11.1 we will prove the important fact that two expanding Thurston maps are Thurston equivalent if and only if they are topologically conjugate.
In Chapter 9 we make a brief excursion to symbolic dynamics. The properties of visual metrics are essential for proving the following statement (see Chapter 9 for the relevant definitions).
Theorem 9.1. Let f:S2 →S2 be an expanding Thurston map. Then f is a factor of the left-shift Σ :Jω→Jω on the space Jω of all sequences in a finite set
J of cardinality#J = deg(f).
The proof of this theorem does not use invariant Jordan curves for f or its iterates; so in a sense it is independent of Theorem 15.1 mentioned above. It can be used to obtain another Markov partition for f, but we have very little control for the geometric shape of the “tiles”.
An immediate consequence of this theorem and its proof is the fact that the periodic points of an expanding Thurston mapf:S2
→S2 form a dense subset of
S2 (Corollary 9.2).
In Chapter 13 we investigate equivalence relations∼on the sphereS2, and the
question when a Thurston map f:S2
→S2 descends to a Thurston map on the
quotient spaceS2/
∼. Here we assume that∼is ofMoore-type (see Definition 13.7), which implies thatS2/∼is a 2-sphere. Under this assumption the relevant condition
is that∼isstrongly invariantforf in the sense thatf maps each equivalence class onto another equivalence class (see Definition 13.1 and Lemma 13.19). We will prove that for a given equivalence relation∼of Moore-type onS2 a Thurston map
f:S2→S2 descends to a Thurston map if and only if∼is strongly invariant for
f (see Theorem 13.2 and Corollary 13.3).
Often it is desirable to promote a given Thurston mapf that is not expanding to an expanding one. More precisely, we want to find an expanding Thurston map
e
f that is Thurston equivalent tof. In general, the existence offeis not guaranteed. However, iff iscombinatorially expanding(see Definition 12.4) such a mapfedoes exist. Roughly speaking, it is constructed by defining an equivalence relation that collapses the sets wheref fails to be expanding to points (see Chapter 14).
We will also investigate some measure-theoretic aspects of expanding Thurston maps. Each such map has a natural measure adapted to its dynamics.
Theorem 17.1. Let f:S2
→S2 be an expanding Thurston map. Then there exists a unique measureνf of maximal entropy forf. The mapf is mixing forνf.
This theorem follows from results due to Ha¨ıssinsky-Pilgrim [HP09, Theo- rem 3.4.1]. We will present a different proof and give an explicit description ofνf
in terms of the cell decompositions Dn(F,
C), where F =fn is a suitable iterate
and C is an invariant curve as in Theorem 15.1. In particular, νf = νF assigns
equal mass to all tiles in the cell decompositionsDn(F,
C) of a given “color” (see Proposition 17.12 and Theorem 17.13).
The measure νf can be used to study the topological and measure-theoretic
dynamics off under iteration. For example, we will see thathtop(f) = log(deg(f)),
where htop(f) is the topological entropy and deg(f) the topological degree of f
(Corollary 17.2).
Ifµis a Borel measure on a metric space (X, d), then we call the metric measure space (X, d, µ)AhlforsQ-regularforQ >0 if