LECTURE Itineraries We start with a simple example of a dynamical system obtained by iterating the quadratic polynomial

Transcription

1 LECTURE. Itineraries We start with a simple example of a dynamical system obtained by iterating the quadratic polynomial f λ : R R x λx( x), where λ [, 4). Starting with the critical point x 0 := /2, we consider the sequence P λ := {x 0, x, x 2,...}, where x i := f λ (x i ) = f i λ (x 0). This sequence is called the post-critical set for f λ. There is a graphical way to construct this sequence, where from the point (x i, x i ), we draw a vertical segment to the graph of f λ ending at (x i, x i ), then a horizontal segment to the diagonal line y = x ending at (x i, x i ), and so on, as depicted below. y x 2 x 0 x 4 x 3 x 5 x x Figure. Post-critical set P λ for λ = We say that f λ is critically periodic if x n = x 0 for some n > 0. Suppose that you and I each have a critically periodic map and are talking over the phone. How can we figure out if we have the same map? The first invariant of a critically periodic map f λ is the period n = P λ. The map f λ permutes the set P λ cyclically : it sends x i to x i+, with indices taken modulo n. More information is obtained from the ordering that P λ inherits as a

2 2 LECTURE subset of the real line. We thus relabel the elements of the post-critical set by the order-preserving map h : {, 2,..., n} P λ. Because f λ is strictly increasing before x 0 and then strictly decreasing, the induced permutation σ := h f λ h on {, 2,..., n} is unimodal, meaning that there is a unique i such that Of course, σ is cyclic as well. σ() <... < σ(i ) < σ(i) > σ(i + ) >... > σ(n). σ i Figure 2. Graph of the permutation σ = ( ) induced by f It turns out that every such permutation σ arises from a unique λ. Theorem. (Milnor-Thurston-Sullivan). Let n and let σ S n be a unimodal cyclic permutation. Then there exists a unique λ [, 4) such that f λ is critically periodic and induces the permutation σ. Below, we describe some of the ideas involved in the proof of this theorem. 2. Uniqueness 2.. Grötzsch s question. Let 0 < a b and let g : R a R b be a diffeomorphism between the rectangles R a = [0, a] [0, ] and R b = [0, b] [0, ] which takes the left, right, top and bottom sides of R a to the corresponding sides of R b. R a g R b a b Theorem 2.. If g is conformal, then a = b. Proof. One can extend g to a conformal map from the plane to itself via repeated Schwarz reflections. Therefore, g is a euclidean similarity. Since we require that it fixes the vertices (0, 0) and (0, ), g is the identity.

3 LECTURE 3 If a < b, then Grötzsch s question asks which admissible maps g : R a R b are the closest to being conformal. This requires the notion of dilatation, a measurement of the failure of a map to be conformal. For an orientation-preserving diffeomorphism g : U V between open subsets of the plane, the derivative D z g : R 2 R 2 at any point z U is a linear map. It thus sends the unit circle onto an ellipse, say with major axis of length M and minor axis of length m. The dilatation of g at z is defined as the eccentricity of this ellipse, that is, Dil z (g) := M/m. D z g m M If in complex coordinates the derivative is given by D z g(w) = Aw + Bw, then we have alternatively A + B Dil z (g) = A B. The dilatation of g is defined as Dil(g) := sup Dil z (g). z U Grötzsch s question then becomes : What is inf g Dil(g) over the set of admissible maps g : R a R b? When is it realized? Theorem 2.2 (Grötzsch). We have with equality only for the stretching map Dil(g) b/a (x, y) g (bx/a, y) Teichmüller s question. Let X, Y be finite subsets of the Riemann sphere Ĉ := C { } with the same cardinality and let h : Ĉ Ĉ be an orientationpreserving homeomorphism with h(x) = Y. Call a homeomorphism g : Ĉ Ĉ admissible if g X = h X, g is homotopic to h rel X, and g is a diffeomorphism off of a finite set. Teichmüller s question is : What is the infimum of Dil(g) over admissible maps g and when is it realized? Theorem 2.3 (Teichmüller). There exists a unique admissible map g : Ĉ Ĉ, called the Teichmüller map, having minimal dilatation. Let K := Dil(g). If K =, then g is a Möbius transformation. If K >, then for every point p Ĉ at which g is non-singular, there is a conformal chart ϕ about p and a conformal chart ψ about g(p) such that ψ g ϕ (x, y) = (Kx, y).

4 4 LECTURE -prong not a singularity 3-prong 4-prong Figure 3. Pronged singularities Note how similar the answers to Teichmüller s question and Grötzsch s question are. One can indeed think of a Teichmüller map as several Grötzsch s stretch maps glued together. The charts mentioned in the above theorem are unique up to translation and rotation by angle π. This implies that one can form a foliation on the complement in Ĉ of the singular set of g, the leaves of which are the curves along which g is stretching maximally, by pulling-back horizontal lines under ϕ-charts. The singularities of this foliation are pronged singularities, which locally look like some number of rectangles glued around a point along horizontal edges. -pronged singularities can only occur on the set X Idea for uniqueness. Here is how one can apply Teichmüller s theorem to proving the uniqueness part of the Milnor-Thurston-Sullivan theorem. Suppose that f λ and f η are critically periodic and induce the same permutation σ. Let h : R R be an increasing homeomorphism such that h(p λ ) = P η. We can take h to be piecewise linear for example. Then extend h to R 2 by setting h(x, y) = (h(x), y). Let X := P λ { } and let Y := P η { }. Let g : Ĉ Ĉ be the map such that g X = h X and g is homotopic to h rel X with minimal dilatation. Since the map z g(z) is also admissible and has the same dilatation as g, it is equal to g by Teichmüller s uniqueness theorem. In particular, g(r) = R. Note that g conjugates the actions of f λ and f η on their respective critical sets, since the two maps induce the same permutation σ by hypothesis. It follows that we can lift g to a homeomorphism g : Ĉ Ĉ such that the diagram g (C, P λ ) (C, P η ) f λ f η g (C, P λ ) (C, P η ) commutes. If g has dilatation greater than, then it has a finite positive number of singularities. A combinatorial argument shows that the stretching foliation for g must have more singularities counting multiplicities than the stretching foliation for g. This is because f λ has degree two, so a typical singularity has two preimages, although a -prong singularity at the critical value of f λ would unfold to a regular point.

5 LECTURE 5 Since f λ and f η are locally conformal on C \ P λ and C \ P η respectively, the dilatation of g is the same as the dilatation of g. Moreover, g is admissible for the same problem as g, since it maps (C, R, P λ ) to (C, R, P η ) and is increasing on R. By Teichmüller s uniqueness theorem we must have g = g, a contradiction. Therefore, g has dilatation and is thus conformal, hence a euclidean similarity. The equality g f λ = f η g implies that g maps the fixed points of f λ to the fixed points of f η, and in particular g(0) = 0. Since g f λ attains its maximum at /2, f η attains its maximum at g(/2) = /2. Therefore, g is the identity and λ = η. 3. Existence Given a unimodal cyclic permutation σ S n, we have to find a set of n points X σ R and a quadratic polynomial f inducing the permutation σ on X σ. It is then a simple matter to conjugate f by an affine map to a polynomial of the form f λ (x) = λx( x). To do this, we start with any set X 0 R of cardinality n and apply the following procedure. Given the set X i, let h : {, 2,..., n} X i be the order-preserving bijection, m := max(x i ) = h(n) and τ := h σ h the permutation induced by σ on X i. For x X i, let g(x) := ± m x, with positive sign if τ (x) τ (m) and negative sign otherwise. Then define X i := g(x i ) and repeat. By construction, the polynomial f(y) = m y 2 maps the j-th element of X i to the σ(j)-th element of X i. As we iterate this process, the sets X i will converge, up to translation and rescaling, to a set X σ solving the initial problem. y y 2 y 3 y 4 y 5 y 6 y x x 2 x 3 x 4 x 5 x 6 x Figure 4. Pull-back map for σ = ( ) applied to the set {, 2, 3, 4, 5, 6}

6 6 LECTURE Let us make this more precise. Define F n := {X R : X = n}/, where X Y if Y = ax + b for some a > 0 and b R. The process described above yields a map F σ : F n F n, called the pull-back map. For [X], [Y ] F n, let h : C C be an orientationpreserving homeomorphism such that the restriction h : X Y is order-preserving. The Teichmüller distance between [X] and [Y ] defined as d([x], [Y ]) := log inf {Dil(g) g is admissible with respect to h}. Theorem 3.. There exists an [X σ ] F n such that () F σ ([X σ ]) = [X σ ] and Fσ k ([X]) [X σ ] as k for all [X] F n. Moreover, (2) d(f σ ([X]), F σ ([Y ])) < d([x], [Y ]) for all [X], [Y ] F n with [X] [Y ]. In particular, the fixed-point [X σ ] is unique. We discuss briefly how the contraction property (2) can be used to to prove (). If F σ was a uniform contraction, then the existence of a fixed-point would follow from the fact that the Teichmüller metric is complete. In order to prove (2), we show that F σ is smooth and its derivative has norm less than on every compact subset of F n with respect to the Teichmüller metric. It matters then to understand the behavior of F σ near infinity. Note that a sequence [X k ] in F n goes to infinity (i.e. escapes every compact set) if the points of X k are clumping together. If for some [X] F n the sequence F k σ ([X]) was going to infinity, this would mean that a certain clumping phenomenon is invariant under the dynamics, but we will see that this is impossible.

7 LECTURE 7 y y 2 y 3 y 4 y 5 y 6 y x x 2 x 3 x 4 x 5 x 6 x Figure 5. Second iteration of the pull-back map