WHEN AN INFINITELY-RENORMALIZABLE ENDOMORPHISM OF THE INTERVAL CAN BE SMOOTHED^{(♡)}

Charles Tresser^{(1)} and Amie Wilkinson^{(2)} ^{(∗)}

Abstract

Let K be a closed subset of a smooth manifold M, and let f : K → K be a continuous selfmap of K. We say that f is smoothable if it is conjugate to the restriction of a smooth map by a homeomorphism of the ambient space M. We give a necessary condition for the smoothability of the faithfully infinitely interval-renormalizable homeomorphisms of Cantor sets in the unit interval. This class contains, in particular, all minimal homeomorphisms of Cantor sets in the line which extend to continuous maps of an interval with zero topological entropy.

I. Introduction

We consider the faithfully infinitely interval-renormalizable homeomorphisms of Cantor sets embedded in the unit interval. We begin with some definitions (a detailed topological description of these maps is given in [BORT]). If K is a nonempty topological space and f : K → K is continuous, we say that (K, f) is a dynamical system. A morphism from (K, f) to (K′, f′) is defined by a continuous map ϕ : K → K′ such that f′ ∘ϕ = ϕ∘f. The dynamical system (K, f) is n-renormalizable if there is a morphism from (K, f) to (ZZ/nZZ, m ↦ m + 1), where ZZ denotes the group of integers, and n-interval renormalizable if K is a compact subset of the real line and the fibers of ϕ belong to disjoint closed intervals. Let IRen(f) be the set of n ∈ {1, 2, …} for which (K, f) is n-interval renormalizable. We say that (K, f) is infinitely interval-renormalizable if IRen(f) is an infinite set. (K, f) is faithfully infinitely interval-renormalizable if it is infinitely interval-renormalizable, K is a Cantor set and f is a homeomorphism of K that is conjugate to a generalized adding machine. ^{2} For the sake of brevity, we shall use the term infinitely renormalizable for “faithfully infinitely interval-renormalizable.” If (K, f) is infinitely renormalizable, then the set IRen(f) = {q_{0} < q_{1} < …} has the properties:

(♡) Dedicated to Benoit Mandelbrot on his seventieth birthday: the present paper is one small step toward understanding “universality at the transition to turbulence”, as first conjectured by P. Coullet and one of the authors (see [CT] and [TC]: these researches were deeply influenced by Benoit Mandelbrot’s writings.)

(1) I.B.M., PO Box 218, Yorktown Heights N.Y. 10598, U.S.A.

(2) Department of Mathematics, University of California, Berkeley, CA 94720, U.S.A.

(∗) Supported by NSF Fellowship.

q_{0} = 1 , (a) q_{n+1} = q_{n} · a_{n}, where a_{n} ∈ IN, a_{n} ≥ 2 . (b)

(For the proof, see [BORT]). We call a sequence of natural numbers {q_{n}}_{0}^{∞} a renormalization sequence if it satisfies properties (a) and (b).

We consider the question of whether the regularity of an infinitely-renormalizable map can be altered by changing the geometry, but not the order structure, of the embedded Cantor set. All infinitely-renormalizable homeomorphisms are conjugate by a homeomorphism of the ambient interval to a map with a continuous extension to the interval that has bounded variation on the interval. There are examples of infinitely-renormalizable homeomorphisms that are not conjugate in this sense to any Hölder continuous endomorphism of the interval. In Section III, we state a necessary condition for an infinitely-renormalizable homeomorphism to be C^{1}-smoothable and a necessary condition for such a map to be C^{1+α}-smoothable, for some 0 < α ≤ 1. In Section IV we compare our results to what is known about the smoothability of infinitely-renormalizable embeddings of the two-disk into itself.

Acknowledgements. The authors have benefited from the support of the C.N.L.S., I.B.M., I.H.E.S., and the Mathematics Department of Orsay. Fruitful discussions with Jean Marc Gambaudo, Lisa Goldberg, Charles Pugh, Sebastian van Strien and Dennis Sullivan are also gratefully acknowledged.

II. Preliminaries

Let I = [0, 1] be the unit interval. Let {q_{n}}_{0}^{∞} be a renormalization sequence and set a_{n} = \frac{q_{n+1}}{q_{n}}. We say that a Cantor set K ⊂ I is \{q_{n}\}-presented if

K = ⋂_{n=0}^{∞}( ⋃_{i=0}^{q_{n}−1} K_{n,i}), where K_{n,i} = [l_{n,i}, r_{n,i}],

for an indexed family of real numbers r_{n,i}, l_{n,i} ∈ I satisfying:

l_{0,0} = 0, r_{0,0} = 1, r_{n,i} < l_{n,i+1}, l_{n,i} = l_{n+1,i·a_{n}}, r_{n,i} = r_{n+1,(i+1)·a_{n}−1},

2 A generalized adding machine is a quasi-periodic motion obtained by adding 1 on the compact abelian group ẐZ_{Q} = lim_{←} ZZ/q_{i}ZZ, where q_{0} < q_{1} < … is a sequence of natural numbers ordered by divisibility. and

lim_{n→∞} max_{i∈{0,...,q_{n−1}}} |r_{n,i} − l_{n,i}| = 0.

These conditions imply that K_{n+1,j} ⊂ K_{n,i} for j ∈ {ia_{n}, …, (i + 1)a_{n} − 1}. In the presentation above, we refer to the set K_{n,i} as an interval at level n. An open interval of the form G_{n,i} = (r_{n,i}, l_{n,i+1}) is called a gap in K at level n. Let K̃_{n,i} = K ∩K_{n,i} be the piece of the Cantor set K corresponding to the interval K_{i,j}.

The following condition is equivalent to the definition of infinitely-renormalizable stated in the Introduction (see [BORT]). Given a Cantor set K ⊂ I, a homeomorphism f : K → K is infinitely renormalizable if and only if there exists a renormalization sequence {q_{n}} and a {q_{n}}-presentation of K, such that for every n and for every 0 ≤ i ≤ q_{n} − 1,

f(K̃_{n,i}) = K̃_{n,σ_{n}(i)},

where σ_{n} is a cyclic permutation of the set {0, …, q_{n} − 1}.

To each infinitely-renormalizable homeomorphism we associate a renormalization sequence {q_{n}} (or, equivalently, the sequence {a_{n}}), and a sequence of permutations σ = (σ_{1}, σ_{2}, …), where |σ_{n}| = q_{n}. The sequence σ is called the order type of f. We remark that this combinatorial information completely describes the infinitely-renormalizable homeomorphism as an ordered dynamical system, the ordering arising from the embedding of K in the interval. When a_{n} = 2 for all n, we say that f is a period-doubling homeomorphism.

Example 1.1. The unimodal period-doubling attractor (see, e.g, [CT] and [TC].

Suppose that {f_{a}} is a suitably smooth 1-parameter family of quadratic-like maps on the interval whose critical value increases as the parameter increases. An example is the family {x ↦ ax(1 − x)}. Then the family will undergo a process of bifurcation of its periodic orbits (called period-doubling) as the parameter value increases. At a distinguished parameter value ā, the map will have an attracting Cantor set with a {2^{n}}- presentation on which fā is a period-doubling homeomorphism (see [S]).

Example 1.1 illustrates how an infinitely-renormalizable homeomorphism can arise as the restriction of a continuous map on the interval to an invariant Cantor set. We take the synthetic perspective, starting with a map on an embedded Cantor set. With this information alone, it is possible to reconstruct the behavior of such a map outside the Cantor set. Given a map f : K → K, we say that g : I → IR is an extension of f to I if g|_{K} = f. Any extension of an infinitely-renormalizable map to a continuous endomorphism of the interval will have distinguished periodic points whose existence is forced by the dynamics on the Cantor Set. Their period and ordering is determined by the order type σ.

Lemma 1.2: Let f : K → K, K ⊂ I, be an infinitely-renormalizable homeomorphism with order type σ. If g is an extension of f to a continuous endomorphism of I, then:

(a) for every n ≥ 0, there exists an x_{0} ∈ K_{n,0} such that g^{q_{n}}(x_{0}) = x_{0} and for k < q_{n}, g^{k}(x_{0}) ∈ K_{n,σ_{n}^{k}(0)}. In particular, x_{0} is periodic with period q_{n}.

(b) g acts minimally on K. In particular, g has no periodic points in K.

The proof relies on elementary applications of the intermediate value theorem, and is found in [BORT].

With infinitely-renormalizable maps, there is no obstruction to extending continuously from the Cantor set to the entire interval. Given any infinitely-renormalizable homeomorphism f with order type σ, we can construct a presented Cantor set K and a map g_{σ,K} : I → IR such that g_{σ,K}|_{K} is infinitely-renormalizable with order type σ.

In this construction, the Cantor set K is uniformly-branched with the ratio of the size of an interval at level n + 1 to that of an interval at level n determined by the sequence {a_{n}}. We start with

K_{0,0} = [0, 1].

At level n, we divide each of the q_{n−1} intervals at level n − 1 into 2a_{n−1} − 1 intervals of equal length. From each interval at level n−1 we remove every second open interval in the division, leaving a total of a_{n−1} · q_{n−1} = q_{n} closed subintervals. Each of these subintervals is now an interval at level n. We label these intervals K_{0,0}, …, K_{0,q_{n}−1}, from left to right in I.

The intersection of these intervals is a Cantor set K. The presentation of K described above and the order type σ of f induces the action of our map g_{σ,K} on K. To define g_{σ,K} at a point x in a gap of K, interpolate linearly between the values of g_{σ,K} at the two nearest points in K to the left and to the right of x. This procedure yields a map from the interval to itself.

Proposition 1.3:

(a) g_{σ,K} is continuous for all σ, K.

(b) If {a_{n}} is bounded above, then g_{σ,K} is Lipschitz.

(c) a_{n} = 2 for all n if and only if g_{σ,K} has topological entropy zero.

Proof: The proofs of (a) and (b) are not difficult, so we prove only (c).To show that h_{g_{σ,K}} = 0, we use the fact that an endomorphism of the interval has positive topological entropy if and only if it has a periodic point of period k · 2^{n}, where k is an odd integer greater than one and n is a nonnegative integer. (See [BF], [M]). It is straightforward to check that if f is period-doubling, then g_{σ,K} has no periodic points of this form. Lemma 1.2 implies that period-doubling is a necessary condition.♢ Remark: It is possible to modify the construction of g_{σ,K} to get a map g_{σ,K′} with bounded variation on the interval; it is enough to enlarge the gaps G_{n,i} and shrink the K_{n,i} according to the magnitude of a_{n+1}. Details are left to the reader.

III. An Obstruction to Smoothability

If there exists a homeomorphism h : I → I and a C^{k} map g : I → IR such that g ∘ h = h ∘ f, then we say that f is C^{k} - smoothable. Alternately, we say that g is a C^{k}- smoothing of f. Observe that we can transport the presentation of K to a presentation of h(K) so that g|_{h(K)} is infinitely-renormalizable.

Proposition 1.3 implies that any infinitely renormalizable map f of bounded type (that is, one in which {a_{n}} is bounded) is Lipschitz-smoothable. By contrast, if we allow f to have unbounded type, we have:

Proposition 2.1:

(a) There exists an infinitely-renormalizable map f : K → K (of unbounded type) such that any C^{0}-smoothing of f has infinite topological entropy. Hence f cannot be Lipschitzsmoothed.

(b) There exists an infinitely-renormalizable map f : K → K (of unbounded type) that cannot be C^{α}-smoothed for any 0 < α ≤ 1.

The proof of Propostion 2.1 is left to the reader: a simple way is to use examples from [MN] and [GT].

The order type of an infinitely-renormalizable map f : K → K forces any continuous extension of f to have turning points, and so forces a smoothing of f to have critical points. Hence the order type might present a combinatorial obstruction to smoothability. In this section, we formulate this notion more precisely.

Let σ be the order type of f and let {q_{n} = |σ_{n}|}_{0}^{∞} be the associated renormalization sequence. Let E(f) denote the set of all continuous extensions of f to the interval. We define the critical sequence {k_{n}} of f by

k_{n} = inf_{g∈E(f)} #{0 ≤ i ≤ q_{n} − 1 | g is not monotone on K_{n,i}}.

Theorem 2.2: Let f : K → K have critical sequence {k_{n}}. If f is C^{1}-smoothable, then

lim_{n→∞} \frac{k_{n}}{q_{n}} = 0.

Remark: Recently, de Melo and van Strien [MS] have shown that when the sequence {k_{n}} is bounded above, the map f is smoothable by a polynomial map. Our result also applies to those maps which are, by necessity, infinitely modal (i.e., have infinitely many turning points) in any continuous extension. Some of these maps can be quite smooth; using methods from [T], it is easy to construct examples of C^{∞} maps with infinitely many turning points, in particular a one-dimensional analogue to the embeding of the disk constructed in [BF].

A strictly weaker obstruction to smoothness, but one with a “finite depth” description, is the measure of the growth of turning points in f determined by its order type σ. For each n, consider those continuous maps from the interval [0, q_{n} − 1] to itself that agree with σ_{n} on the integer points {0, …, q_{n} − 1}. Let c_{n} be the minimal number of turning points of such an extension of σ_{n}. We have the following Corollary:

Corollary 2.3: If f is C^{1}-smoothable, then

lim_{n→∞} \frac{c_{n}}{q_{n}} = 0.

We need the following proposition, whose proof we defer until after the proof of the Theorem.

Recall that the orbit of a periodic point x of a smooth map g of period k is attracting if |Dg^{k}(x)| < 1, repelling if |Dg^{k}(x)| > 1 and indifferent if |Dg^{k}(x)| = 1.

Proposition 2.4: Suppose that g : I → IR is C^{1} and there exists a {q_{n}}-presented Cantor Set K such that g|_{K} is infinitely renormalizable. Then for every n ≥ 0, g has a periodic point z with period p_{n} · q_{n}, where 1 ≤ p_{n} ≤ a_{n}, so that the orbit of z is either repelling or indifferent. Further, every gap G_{n,i} of the presented Cantor set K contains precisely p_{n} points of the orbit of z, for i = 0, …, q_{n} − 1.

Proof of Theorem 2.2: Suppose that g is a C^{1}-smoothing of f. Since it will not alter the dynamics, we may assume without loss of generality that g is an extension of f : K → K to the interval. Let k′_{n} be the number of intervals K_{n,i} at level n that contain a critical point of g. Note that k_{n} ≤ k′_{n} for all n. We show that \frac{k′_{n}}{q_{n}} converges to 0 as n tends to infinity.

Because I is compact and g is C^{1}, we can pick δ > 0 such that, for all x, y ∈ I, if |x − y| < δ then |g′(x) − g′(y)| < ϵ. For the same reason, there exists M > 0 such that | g′(x) |≤ M for all x ∈ I. Pick N so that when n ≥ N, we have: | G_{n,j} |<| K_{n,j} |< δ.

The sequence \{\frac{k′_{n}}{q_{n}} \} is nonincreasing and bounded below, so the limit

lim_{n→∞} \frac{k′_{n}}{q_{n}}

exists. We need only show that it converges to 0. Assume the contrary to the conclusion; that is, assume that there exists a k > 0 such that the number of critical intervals at stage n is bounded below by k · q_{n}, for every n ≥ 0.

By Proposition 2.4, we can pick n > N such that g has a periodic point x_{n} ∈ G_{n,0} of period p_{n} · q_{n}, where p_{n} ≤ a_{n}, whose orbit is either repelling or indifferent. Further, p_{n} points of the orbit of x_{n} lie in G_{n,i}, for i = 0, …, q_{n} − 1. By our assumption, at least k · p_{n} · q_{n} points on the orbit of x_{n} are within δ of a critical point of g. That is, there exist distinct j_{1}, …, j_{k·p_{n}·q_{n}} ∈ {1, …, q_{n}} such that

|g′(g^{j_{i}}(x_{n}))| < ϵ

for i = 1, …, k · p_{n} · q_{n}. Since x_{n} is repelling or indifferent,

|Dg^{p_{n}·q_{n}}(x_{n})| = | ∏_{i=1}^{p_{n}·q_{n}} g′(g^{i−1}(x_{n}))| ≥ 1.

Hence,

M^{p_{n}·q_{n}−k·p_{n}·q_{n}} · ϵ^{k·p_{n}·q_{n}} > |Dg^{p_{n}·q_{n}}(x_{n})| ≥ 1.

This implies that

M ≥ (\frac{1}{ϵ})^{\frac{k}{1−k}} .

Since ϵ was arbitrary, this gives a contradiction.♢

We use the following lemma to prove Proposition 2.4. Assume that g|_{K} = f is infinitely renormalizable, and K is presented with intervals K_{n,i} and gaps G_{n,j} at level n.

Lemma 2.5: Given g as above. For every n there is a map g̃ : I → R satisfying the following two properties:

(a) g̃(K_{k,i}) = K_{k,σ_{k}(i)}, for k = 0, …, n + 2 and i = 0, …, q_{k} − 1

(b) There exists a γ > 0, such that for every k = 0, …, n + 2,

| g̃^{q_{k}}(x) − x |< γ ⟹ g^{q_{k}}(x) = g̃^{q_{k}}(x).

Proof of Lemma 2.5: For all i, j, let l_{i,j} and r_{i,j} be the smallest and largest elements, respectively, of K_{i,j}. Note that, for all i, j, we have l_{i,j}, r_{i,j} ∈ K. Define g̃ as follows. Let

g_{0}(x) = \{ \begin{matrix}1 & if g(x) ≥ 1 \\ 0 & if g(x) ≤ 0 \\ g(x) & otherwise. \end{matrix}

Now define, inductively, for k = 1, …, n + 2:

g_{k}(x) = \{ \begin{matrix}l_{k,σ_{k}(j)} & if x ∈ K_{k,j} and g_{k−1}(x) ≤ l_{k,σ_{k}(j)} \\ r_{k,σ_{k}(j)} & if x ∈ K_{k,j} and g_{k−1}(x) ≥ r_{k,σ_{k}(j)} \\ g_{k−1}(x) & otherwise. \end{matrix}

Let g̃ = g_{n+2}. By construction, g̃ satisfies property (a).

To show that g̃ satisfies (b), we first observe that g̃|_{K} = g|_{K}; this means that, g̃ is an infinitely-renormalizable map with the same order type and presented Cantor set as g. Suppose that g(x) ≠ g̃(x), for some x. By definition of g̃, this implies that there exists k, 0 ≤ k ≤ n + 2, such that x ∈ K_{k,j} but g(x) ∉ K_{k,σ_{k}(j)}. We also know that g(K̃_{k,j}) = K̃_{k,σ_{k}(j)}, so if g(x) ∉ K_{k,σ_{k}(j)}, then x ∉ K. Hence, g̃ and g coincide on K, and, in particular, g̃ acts minimally on K. Note further that by our construction, if g̃(x) ≠ g(x), then g̃(x) ∈ K. This fact, combined with the minimality of g̃ on K, implies that if x is a periodic point for g̃, then x is also a periodic point for g (of the same period).

Now consider the following subset of I:

C = ⋃_{i=1,k=0,j=0}^{q_{n+2},n+2,q_{k}−1} g̃^{−i}(l_{k,j}) ∪ ⋃_{i=1,k=0,j=0}^{q_{n+2},n+2,q_{k}−1} g̃^{−i}(r_{k,j}).

The closed set C contains the set of points in the interval where g^{q_{k}} and g̃^{q_{k}} fail to coincide, for k = 0, …, n + 2. By the above considerations, C contains no periodic points for g̃. So, for every 0 ≤ k ≤ n + 2, the set C is disjoint from the set

D_{k} = {x ∈ I | g̃^{q_{k}}(x) = x}.

Because C is a closed subset of I, the D_{k} are uniformly bounded away from C; thus, there exists a γ > 0 so that C is disjoint from a γ-neighborhood of D_{k}, for every 0 ≤ k ≤ n + 2. This proves (b).♢

Proof of Proposition 2.4: By Lemma 1.2, for all k ≥ 0 there exists x_{k} ∈ G_{k,0} of period q_{k}. Assume that for all but finitely many n, the orbit of x_{n} ∈ G_{n,0} is attracting of period q_{n} and the orbit of x_{n+1} ∈ G_{n+1,0} is attracting of period q_{n+1}. Fix such an n, and fix x_{n+1} and x_{n} (note that x_{n+1} < x_{n}). For this n, we may find a map g̃ satisfying the conditions of Lemma 2.5.

For k = 0, …, n + 2, we have that g̃(K_{k,i}) = K_{k,σ_{k}(i)}, for i = 0, …, q_{k} − 1. Consider the map h = g̃^{q_{n+1}} |_{K_{n,0}}. Assume without loss of generality that x_{n} and x_{n+1} are also periodic points for g̃ of period q_{n} and q_{n+1}, respectively. We have that h(x_{n}) = x_{n} and h(x_{n+1}) = x_{n+1}. Since x_{n} and x_{n+1} both have attracting orbits, we also have that | h′(x_{n}) |< 1 and | h′(x_{n+1}) |< 1. This implies that there exists δ such that for any y ∈ K_{n,0}, if 0 < x_{n} − y < δ, then h(y) > y, and if 0 < y − x_{n+1} < δ, then h(y) < y. Hence there exists z ∈ K_{n,0} such that h(z) = z and, for some ϵ with γ > ϵ > 0, (where γ is as in Lemma 2.5, part (b)) we have that h(w) ≤ w for 0 < z − w < ϵ, and h(w) ≥ w for 0 < w − z < ϵ.

Thus:

\frac{h(z_{1}) − h(z_{2})}{z_{1} − z_{2}} ≥ \frac{z_{1} − z_{2}}{z_{1} − z_{2}} = 1,

for all z_{1}, z_{2} such that z − ϵ < z_{1} < z < z_{2} < z + ϵ. In this neighborhood, h is differentiable (by Lemma 2.5, part (b)), and h′(z) ≥ 1.

Since z is a fixed point of h, z is a periodic point of g̃ (and hence, of g) of period at most q_{n+1}. Because g̃ is aperiodic on K, the point z must lie in some gap G_{k,j} of K, for some k. Since z lies in K_{n,0}, we know that k is at least n. If k is greater than n + 1, then z must also lie in an interval K_{n+2,j′}. By the construction of g̃, however, the q_{n+2} intervals at level n+2 are cyclically permuted by g̃, so if z is in K_{n+2,j′}, then the orbit of z contains at least q_{n+2} distinct points. This is impossible, so k is either n or n + 1. Therefore, z is a periodic point for g of period p_{n} · q_{n}, where 1 ≤ p_{n} ≤ a_{n}. In either case, the orbit of z is either repelling or indifferent.♢

The following two facts are useful:

(1) If an interval of length r contains a zero of an α-Hölder function, then the values of the function on that interval are bounded by c · r^{α}, where c is a uniform constant.

(2) The product of the lengths of k′_{n} disjoint intervals contained in the unit interval is at most (\frac{1}{k′_{n}} )^{k′_{n}}.

Using (1) and (2), we can modify the proof of Theorem 2.2 to show Theorem 2.6.

Theorem 2.6: Let f be infinitely renormalizable with critical sequence {k_{n}}. If f is C^{1+α}-smoothable for some 0 < α ≤ 1,then there exists a c > 0 such that n · \frac{k_{n}}{q_{n}} ≤ c, for all n. That is, the sequence \{\frac{k_{n}}{q_{n}} \} is O(\frac{1}{n}).

This result allows to construct a wealth of examples of period-doubling maps that are C^{1}-smoothable but not C^{1+α}-smoothable for any α > 0.

Question: Are there further obstructions to higher smoothability in terms of the sequence \{\frac{k_{n}}{q_{n}} \}? For k > 1 are there infinitely-renormalizable maps that are C^{k}-smoothable, but not C^{k+1}-smoothable?

IV. Cascades of Periodic Orbits and Homeomorphisms of the 2-disk

Let f be an orientation-preserving embedding of the 2-disk ID^{2}. A cascade of periodic orbits for f is an infinite sequence of periodic orbits {O_{n}} of f with periods {q_{n}} such that, for each n ≥ 1, we have:

- q_{n} = a_{n} · q_{n−1} with q_{0} = 1 and a_{n} > 1,

- there exists a collection of disjoint, simple closed curves C_{n}^{0} , …, C_{n}^{q_{n−1}−1} bounding the disjoint disks D_{n}^{0} , …, D_{n}^{q_{n−1}−1}, with the following properties:

- each D_{n}^{i} contains exactly one point of O_{n−1}, and a_{n} points of O_{n},

- f(C_{n}^{i}) is isotopic to C_{n}^{i+1 mod q_{n−1}} in the punctured disk ID^{2} \ ⋃_{i≤n} O_{i},

- the union of the D_{n}^{i}’s is contained in the union of the D_{n−1}^{i}’s,

- the diameters of the D_{n}^{i}’s converge uniformly to 0 with n.

Let {f_{t}}_{t∈[ 0, 1]} be an arc of embeddings joining the identity map to f = f_{1}, and {f_{t}}_{t∈IR} be the extended arc of embeddings joining the identity map to all iterates of f, with f_{t} = f^{[t]} ∘ f_{{t}}. To each cascade of periodic orbits {O_{n}}, we associate a signature \{(l̃_{n}, q_{n})\}, where l̃_{n} = \frac{l_{n}}{q_{n}}, and l_{n} is defined as follows: In one of the D_{n}^{i}’s, pick the point x_{n−1} of O_{n−1}, and a point x_{n} of O_{n}; then l_{n} is the algebraic number of loops that the vector \frac{f_{t}(x_{n})−f_{t}(x_{n−1})}{∥f_{t}(x_{n})−f_{t}(x_{n−1})∥} performs on the unit circle when t goes from 0 to q_{n}. The number l_{n} is independent of the choice of D_{n}^{i}, and of the choice of the point x_{n} in D_{n}^{i}.

For each signature \{(l̃_{n}, q_{n})\} with l̃_{n} = \frac{l_{n}}{q_{n}}, the sequence \{q_{n}\} is strictly increasing and the integers a_{n} = \frac{q_{n}}{q_{n−1}} and l_{n} are coprime for each n.

Remark. When changing the isotopy {f_{t}}_{t∈[ 0, 1]}, all the l_{n} are changed by adding the same integer.

The following result is proven in [GST].

Theorem 4.1: If {O_{n}} is a cascade of periodic orbits for a C^{1} diffeomorphism f of the 2-disk, then lim_{n→∞}(l̃_{n}) exists.

Let σ be the order type of a period-doubling homeomorphism. By Lemma 1.2, the map g_{σ,K} has a cascade of periodic orbits {O_{n}}, where O_{n} = {x_{0}, x_{1}, …, x_{2^{n}−1}}, with x_{0} < x_{1}, … < x_{2^{n}−1}. For each n we associate weights to the points of O_{n} as follows: we consider the minimal (piecewise linear) interpolation f_{n} of the finite cascade {O_{i}}_{0≤i≤n}, and set w(x_{2j}) = w(x_{2j+1}) = 0 if f_{n} is increasing on [x_{2j}, x_{2j+1}], and w(x_{2j}) = w(x_{2j+1}) = 1 otherwise (then f_{n} is decreasing on [x_{2j}, x_{2j+1}]). Then we have

l_{n} = ∑_{x_{j}∈O_{n}} w(x_{j})

and

l̃_{n} = \frac{∑_{x_{j}∈O_{n}} w(x_{j})}{2^{n}} .

In the two-dimensional case when the embedding is smooth, there is also the notion of an infinitessimal winding number: it is the winding number of a vector in the tangent space around an orbit (see [R] and [GST]). Given a cascade {O_{n}}, this yields a sequence of infinitessimal twisting numbers ω_{n}, and average infinitessimal twisting numbers ω̃_{n} = \frac{ω_{n}}{q_{n}}. The average infinitessimal twisting number can also be shown to be a well-defined number ω on the Cantor set in the closure of {O_{n}}, with the following property

Theorem 4.2([GST]). If {O_{n}} is a cascade of periodic orbits for a C^{1} diffeomorphism f of the 2-disk, then lim_{n→∞}(l̃_{n}) = ω.

These infinetessimal quantities can be computed in one dimension by introducing a new system of weights.

Let {O_{n}} be the cascade associated to the period-doubling map g_{σ,K}. The map g_{σ,K} is the minimal interpolation (in terms of turning points) of {O_{n}}. For x_{i} ∈ O_{n}, set s(x_{i}) = 0 if g_{σ,K} is increasing at x_{i} and s(x_{i}) = 1 if g_{σ,K} is decreasing at x_{i} (again, one of these two possibilities holds true). Then

w_{n} = ∑_{x_{j}∈O_{n}} s(x_{j})

and

w̃_{n} = \frac{∑_{x_{j}∈O_{n}} s(x_{j})}{2^{n}} .

On the Cantor set K, the s-weight s(x) is not defined if g_{σ,K} is not monotone at x, but otherwise is defined in the same way as for periodic orbits. In the smooth case, we know by Theorem 1 that the critical points of any smoothing have zero measure for the (unique) invariant probability measure ν on K, and so the set of points in K where s is not defined have zero measure. By the ergodic theorem, the average infinitessimal winding number along orbits

∑_{i=0}^{∞} s(g_{σ,K}^{i}(x))

is thus defined almost everywhere on K, and is equal to the average value of s:

∫_{K} s(x)dν(x).

Theorem 2.2 and Corollary 2.3 thus imply the following

Corollary 4.3: If {O_{n}} is a cascade of periodic orbits with a_{n} ≡ 2 for a C^{1} endomorphism f of the interval, then lim_{n→∞}(l̃_{n}) = ω.

We leave to the reader to check that the restriction to the case a_{n} ≡ 2 is indeed superfluous in Corollaries 4.2 and 4.3. Since Theorem 2.2 implies Corollary 2.3, but is strictly stronger, Theorem 2.3 is stronger than the one-dimensional version of Theorems 4.1 and 4.2.

References

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