Smooth Combs Inside Hedgehogs

We use techniques of tube-log Riemann surfaces due to R.Perez-Marco to construct a hedgehog containing smooth $C^{\infty}$ combs. The hedgehog is a common hedgehog for a family of commuting non-linearisable holomorphic maps with a common indifferent fixed point. The comb is made up of smooth curves, and is transversally bi-H\"older regular.


Introduction.
We consider the dynamics of a holomorphic diffeomorphism f (z) = e 2πiα z + O(z 2 ), α ∈ R − Q, defined in a neighbourhood of the indifferent irrational fixed point 0. The map f is said to be linearisable if there is a holomorphic change of variables z = h(w) = w + O(w 2 ) such that in a neighbourhood of the origin, where R α (w) = e 2πiα w is the rigid rotation.
The problem of linearisation, or determining when such an f is linearisable, has a long and interesting history, including the work of H. Cremer (see [2], [3]) in the 1920's, C.L.Siegel (see [10]) in 1942, A.D.Bruno in the 1960's, and Yoccoz (see [11]), who resolved the problem of the optimal arithmetic condition on α for linearisability in 1987. In this article however we are primarily concerned with the structure of invariant sets for the dynamics near the fixed point, rather than the issue of linearisability.
When f is linearisable, f behaves like the rotation by angle 2πα around 0 on the maximal domain of linearisation called the Siegel disk of f , which turns out to be a domain of the form h({|w| < R}) where R is less than or equal to the radius of convergence of h around 0. In this case the compacts h({|w| ≤ r}) for r < R are clearly completely invariant under f . However, even in the more general case where f is not necessarily linearisable, R.Perez-Marco found completely invariant continua for f which persist: Theorem (Perez-Marco, [4]). Let f (z) = e 2πiα z + O(z 2 ), α ∈ R − Q be such that f and f −1 are defined and univalent on a neighbourhood of the closure of a Jordan domain Ω ⊂ C containing 0. Then there exists a full compact connected set K contained in Ω such that 0 ∈ K, K ∩ ∂Ω = φ and f (K) = f −1 (K) = K.
The invariant compacts K thus obtained are called Siegel compacta. A classical topological theorem of G.D.Birkhoff (see [1]) does in fact guarantee, for planar homeomorphisms near Lyapunov unstable points, the existence of compacts, but which are either positive or negative invariant, not necessarily totally invariant. In the holomorphic setting one can thus improve to obtain totally invariant compacts. Moreover, though this will not be used in what follows, if the boundary of the Jordan domain Ω is C 1 smooth, then Perez-Marco has shown that such a compact K is in fact unique (see [8]). If K is not contained in the closure of a linearisation domain it is called a hedgehog. A hedgehog is called linearisable or non-linearisable depending on whether it contains a linearisation domain or not. We will be mostly concerned with non-linearizable hedgehogs in this paper. Perez-Marco has shown that these have no interior (see [8]). The structure of such hedgehogs is topologically complex; for example, Perez-Marco shows in [5] that they are not locally connected at any point different from the fixed point. Objects such as combs (homeomorphs of the product of a Cantor set and an interval) had been expected to be found within hedgehogs. Indeed this is stated by Perez-Marco in [7], where he also sketches how the techniques of "tube-log Riemann surfaces" used in [7] may be adapted to construct a hedgehog containing combs. We carry out this suggested construction in this article. The comb obtained is quite regular, being made up of smooth curves containing a dense set of analytic curves. Thus hedgehogs can exhibit some smoothness. We have the following: Main Theorem. There exists a Cantor set C ⊂ R and a family of commuting holomorphic maps (f t (z) = e 2πit z + O(z 2 )) t∈C , all defined and univalent on a common neighbourhood of the closed unit disk D, which have for irrational times t a common hedgehog K ⊂ D, such that K contains a comb E, the homeomorphic image of C 0 × [1,2] for an explicit Cantor set C 0 . The comb is made up of smooth C ∞ curves, and is bi-Hölder regular. More precisely, for the homeomorphism we have: (a) For each fixed θ, the map t → Ψ(θ, t) is smooth.
The maps f t are non-linearisable for t ∈ C ∩ (R − Q).
We note that since the hedgehog is non-linearisable, the comb obtained is non-trivial, in the sense that it is not contained inside the interior of the hedgehog. We first describe the construction, explaining how the hedgehog and comb are formed, and then carry out the proofs of the properties listed above.
For other examples of hedgehogs, we refer the reader to the articles of Perez-Marco [6,7], in the first of which non-linearisable hedgehogs are constructed along with an explicit coding of the dynamics on the hedgehog, while in the second linearisable hedgehogs containing Siegel disks with smooth boundary are constructed.

General idea of the construction.
The construction we will present is based on the techniques of "tube-log Riemann surfaces" invented by Perez-Marco in [9], where they were first used. They were used again in [6] and [7]. It is strongly advised to read this section in conjunction with [6] and [7]. As these techniques are explained in detail in [6] and especially in [7], and the basic elements of the construction are the same as in [7], to avoid repetition the description below is shorter and slightly informal.
We start with the linear flow of translations (F t (z) = z + t) t∈R on the upper half-plane H (note : for convenience, instead of working with dynamics and dynamical objects such as Siegel disks, hedgehogs etc defined on the unit disk D, we will actually work instead with their lifts to the upper half-plane via the universal covering E : H → D − {0}, E(z) = e 2πiz ).

2.1.1) The Riemann surface S 0 .
The idea is 'fold' the dynamics by introducing nonlinearities in the dynamics, such as fixed points; these result in a shrinking of the linearisation domain. This is accomplished by means of the following Riemann surface with distinguished charts S 0 (see Perez-Marco [7]  The surface S 0 is formed from a copy of C with 'slits' [n, n − i∞] at each of the integers n ∈ Z, above and below each of which is pasted isometrically along the slits a family (indexed by Z) of slit cylinders C/Z with slits [0, 0 − i∞]. This construction gives not only a Riemann surface, but also a canonical set of charts for the surface, which allow us to write formulas for functions defined on the surface in terms of these canonical coordinates. We note here that the surface S 0 has 'ramification points of infinite order' at the points corresponding to Z in the 0-plane and to 0 in the cylinders. These can be seen as points added in the completion of the flat metric on S 0 (induced from the flat metrics on its building blocks). However it is important to note that these points do not belong to the surface S 0 itself, nor can the complex structure on S 0 be extended to these points.

2.1.2) Uniformization of the Riemann surface S 0 .
The upper ends of the cylinders on the other hand do correspond to points for the complex structure. Adding in these points gives a simply connected Riemann surface which it is then not too hard to see is biholomorphic to C. So S 0 should be biholomorphic to the complex plane minus a doubly infinite discrete set of points; in fact, there is an explicit formula for the uniformization (see Perez-Marco [7], section 1.2.3), which in terms of the distinguished charts on S 0 is given by The map K satisfies the properties for the appropriate choices of the branches of the logarithms.
The images under K of horizontal lines just above the real line R in the 0-plane are thus 1-periodic (with respect to parametrization by the x-coordinate) curves in C; since the ramification points at Z in the 0-plane correspond to ends at infinity under the uniformization, these curves oscillate with amplitudes which tend to infinity as we take lines tending to the real line. The upper ends of the cylinders correspond under K to a discrete set of points P ⊂ C; amongst these, those that correspond to the upper ends of the 1-cylinders lie below, but close to, the image of the real line. The figure below illustrates the image of the foliation of horizontal lines on S 0 under K. Consider the linear flow of translations (F t (z) = z + t) t∈R on the upper half-plane H. The upper half-plane of the 0-plane of S 0 projects univalently onto the upper half-plane H, so we can lift each F t to a mapF t (z) from the upper half-plane of the 0-plane of S 0 to itself, given in terms of the distinguished chart by the same formulaF t (z) = z + t. We would like to extend the lift analytically to a univalent mapF t of the surface to itself; however, it is not possible to extend to the points Z − t of the 0-plane or the points 0 − t of the cylinders, as these would have to be mapped to the ramification points, and moreover there is a non-trivial monodromy when continuing analytically around these points.
To overcome this problem, we can remove from the surface a small ǫ-neighbourhood (for the flat metric on S 0 ) I(ǫ) ⊂ S 0 of the ramification points. On S 0 − I(ǫ), for |t| < ǫ, the aforementioned problem does not arise for the mapsF t , each of which then extends to a univalent mapF t : S 0 − I(ǫ) → S 0 , which is expressed in terms of the distinguished charts on S 0 by the same formulaF t (z) = z + t (it suffices in fact to remove only a set of horizontal slits Z + (ǫ, ǫ) near the ramification points, but for consistency with the rest of the article we consider removing neighbourhoods).
Let T : S 0 → S 0 be the automorphism given in the charts on S 0 by T (z) = z + 1. We note that the lifts of the integer translations z → z + n extend to all of S 0 as the automorphisms T n : S 0 → S 0 . These automorphisms commute with the mapsF t , hence we can also define for times t ′ = n+t ∈ Z+(−ǫ, ǫ) the lifts F t ′ : S 0 − I(ǫ) → S 0 byF n+t := T n •F t . Hence we obtain a semi-flow (F t ) defined for times t ∈ Z + (−ǫ, ǫ) on a large part S 0 − I(ǫ) of the surface.
The 'upper ends' of the cylinders on the surface are points for the complex structure, which become fixed points (modulo the automorphism T : S 0 → S 0 ) for the dynamics on the surface. Hence lifting the dynamics to the surface has the effect not only of extending its domain of definition, but also of introducing nonlinearities, such as fixed points.
To see this dynamics on the surface as dynamics on the plane, we use the uniformization K : S 0 → C−P : Conjugating the liftsF t by K we obtain dynamics K •F t • K −1 on a large part K(S 0 − I(ǫ)) ⊂ C of the plane. In particular any half-plane { Im z ≥ −y} ⊂ C where y ≥ 0 is contained in K(S 0 − I(ǫ)) for ǫ small enough (see [7], section 1.2.4), so we can always ensure a large domain of definition for the dynamics. The dynamics flows along the foliation shown in Figure 2, and has as fixed points (modulo Z) the points P corresponding under K to the upper ends of the cylinders. This dynamics is what we mean by the 'folded' dynamics.

2.1.4) Normalization of the uniformization.
The images under K of horizontal lines in the upper half-plane of the 0-sheet of S 0 are graphs over the x-axis which are 1-periodic with respect to parametrization by the x-coordinate. For the image of a horizontal line at a height δ > 0 in the 0-sheet, its peaks and troughs occur at the points K(n + 1/2 + iδ) and K(n + iδ) respectively (where n runs over all integers). The amplitude of oscillation (the difference in height between peaks and troughs) is a strictly decreasing function of δ which tends to +∞ as δ tends to 0, and tends to 0 as δ tends to +∞.
We define a family of normalized uniformizations as follows: First we place the ramification points of S 0 slightly below the real axis instead of on the real axis; more precisely, for δ > 0 we denote by S δ the Riemann surface obtained by the same construction as S 0 , pasting a plane and cylinders isometrically along slits, but with ramification points placed at Z − iδ in the 0-plane and 0 − iδ in the cylinders. The uniformization of this surface is given by the map z → K(z + iδ) instead of z → K(z). The image under this uniformization of the real line R in the 0-sheet of S δ is a 1-periodic graph with an amplitude of oscillation that is large when δ is small. For small δ, the troughs of this curve have large negative imaginary parts and lie well below the real axis; in applications however, we would like the image of the real line to lie above the x-axis, with the troughs touching the x-axis, and so we define the uniformization K δ : where by the "height of the troughs" we mean "height of the troughs" = Im K(n + iδ) = − 1 2π log(− log(1 − e −2πδ )) (where n here is any integer). The curve K δ (R) then lies in the upper half-plane of C with troughs lying on the real axis at the points K δ (n + 0i) = n + 0i, n ∈ Z; the image of the upper half-plane of the 0-sheet of S δ under K δ , which we denote by K δ (H) is the region in the upper half-plane of C bounded by K δ (R).
In applications it will also be useful to have the image of the real line oscillate at higher frequencies, more precisely with period 1/a instead of period 1 for integers a ≥ 1; to do this we consider the uniformizations 1 a K δ obtained by dividing K δ by integers a ≥ 1. The image of the real line, the curve 1 a K δ (R), is then a 1/a-periodic curve. Finally, in order to control the amplitude of oscillation, we note that given a ≥ 1 integer and h > 2 real, there exists a unique δ = δ(a, h) (as is shown in Perez-Marco [7] section 1.2.5), with 0 < δ < 1/2, such that the amplitude of oscillation of K δ (R) is a · h, and hence that of 1 a K δ (R) is h. Moreover δ(a, h) → 0 for fixed h when a → ∞. The constant δ satisfies (see Perez-Marco [7] section 1.2.5) The peaks and troughs of the curve 1 a K δ (R) occur at the points 1 a K δ (n + 1/2 + 0i) = n + 1/2 + ih and 1 a K δ (n + 0i) = n + 0i respectively, and we have Max x∈R Im 1 a K δ (x) = Im 1 a K δ (n + 1/2) = h, Min x∈R Im 1 a K δ (x) = Im 1 a K δ (n) = 0, for any integer n ∈ Z. Figure 3.
We thus have a family of normalized uniformizations 1 a K δ depending on two parameters a ≥ 1 an integer and h > 2 a real, where here δ is understood to be δ = δ(a, h). The maps 1 a K δ have the following properties : There exists a constant C independent of a and h > 2 such that for all z with Im z > 1.
(5) Let z 0 ∈ C denote the limit of 1 a K δ (z) when z → ∞ in S δ through the upper end of one of the 1-cylinders. There exists a constant C 1 independent of a and h > 2 such that (note that the images under 1 a K δ of the upper ends of the 1-cylinders differ from each other by 1/atranslations, so Im z 0 is independent of the choice of 1-cylinder). We note that, for the dynamics obtained by conjugation by 1 a K δ , these points are no longer necessarily fixed points (modulo Z), but rather periodic points (modulo Z) of period a.
We note that the properties (1)-(4) refer to 1 a K δ (z) for z in the 0-plane of S δ , while (5) refers to the 1-cylinders. These properties appear in Perez-Marco [7], section 1.2.5, but can also be checked by elementary computations . We note that condition (5) allows us to control the heights in the upper half-plane of the non-linearities of the dynamics obtained by conjugation by 1 a K δ ; indeed (5) says that the heights of the periodic orbits is approximately h, hence larger choices of h result in somehow smaller linearisation domains.

2.1.5) Construction of the limit dynamics.
The main idea is, roughly, to iterate the process described in section 2.1.3 above. We take a surface S δ1 , lift the linear flow of translations to S δ1 and conjugate by a uniformization K 0 = 1 a0 K δ1 to obtain folded dynamics as described in section 2.1.3. Having fixed S δ1 and K 0 , we take a surface S δ2 , lift the linear flow of translations to S δ2 and conjugate by a uniformization K 1 = 1 a1 K δ2 ; we then lift this folded dynamics to S δ1 and conjugate by the uniformization K 0 , to obtain a twice-folded dynamics, given by conjugating by the map K 0 K 1 . We continue this way, choosing at each stage a surface S δn and a uniformization K n−1 , to obtain at stage n a dynamics given by conjugation by K 0 . . . K n−1 . At each stage the dynamics obtained is linearisable, but the linearisation domain decreases; the point is that, by choosing the appropriate uniformizations and controlling the geometry of the domains obtained at each stage, we should be able to obtain, in the limit, dynamics having an invariant domain with the desired geometry.
In the construction that follows, we define a sequence K n = 1 a n K δn+1 , δ n+1 = δ(a n , h n ), n ≥ 0 of uniformizations depending on two sequences of parameters (a n ) n≥0 and (h n ) n≥0 . The whole construction is completely determined by the sequences of parameters a n and h n , which we will indicate how to choose in the course of the construction. For the moment we assume that they are given, and construct the desired limit dynamics as follows: Let n ≥ 1.
We let F n,n = (F n,n,t (z) = z+t) t∈R be the linear flow of translations. As explained in section 2.1.3 above, we can lift the elements of this flow to univalent mapsF n,n,t defined on the part of the surface S δn obtained by deleting an ǫ-neighbourhood I(ǫ) of the ramification points and for times t ∈ Z+ (−ǫ, ǫ). We take ǫ = 3a n −1 , and conjugate by the uniformization K n−1 of S δn to obtain a semi-flow F n−1,n = (F n−1,n,t ) t∈An−1,n defined by (we could in this step have taken a bigger set of times for A n−1,n such as a n−1 −1 (Z + (−3a n −1 , 3a n −1 )), but in the steps that follow it will be necessary to restrict to smaller sets of times). We note that the time has been reparametrized in order to have the right behaviour at +i∞, when Im z → +∞, These maps are defined on K n−1 (S δn − I(3a n −1 )) ⊂ C, which contains a large half-plane H −y = { Im z ≥ −y}, where y → +∞ as a n → +∞.
We would now like to lift the semi-flow F n−1,n to the surface S δn−1 and conjugate by the uniformization K n−2 to obtain a semi-flow F n−2,n . The difference with the previous case is that we are now starting with a nonlinear flow instead of the linear flow of translations. However, by giving up some space in the lower region of the half-plane of definition H −y , we can work with maps close to linear ones; more precisely we have (see [7] Lemma 1.1.6 and Proposition 2. Since y → +∞ as a n → +∞, we observe that once a n−1 is fixed, by choosing a n large enough we can ensure that this inequality holds in as large a half-plane H −ỹ as desired, where −ỹ = −y + 1 2π log a n−1 + C. Now consider the region J (−ỹ) = {z ∈ S δn−1 : Im z ≤ −ỹ} ⊂ S δn−1 (note that there is a well-defined function Im z on S δn−1 ). Since the estimate above holds on the complement S δn−1 − J (−ỹ) of this region, it should be clear that if we now remove as well an ǫ-neighbourhood I(ǫ) of the ramification points, this time with ǫ = 3a n−1 −1 , then we can define, for , the lifts of elements of F n−1,n as univalent mapsF n−1,n,t : We can now proceed as before and define the semi-flow F n−2,n = (F n−2,n,t ) t∈An−2,n by (the times have again been reparametrized appropriately so that F n−2,n,t (z) = z+t+o(1) when Im z → +∞).
Following this same procedure of lifting and conjugating the dynamics, we can define successively the semi-flows F n−3,n , F n−4,n , . . . , F 0,n , with F j,n = (F j,n,t ) t∈Aj,n defined for times t ∈ A j,n where A j,n is defined inductively by for j = n − 1, . . . , 0 and A n,n = R. We note here that the sets A 0,n form a decreasing sequence, ie A 0,n+1 ⊂ A 0,n , n ≥ 1; indeed, Thus we obtain a sequence of semi-flows F 0,n = (F 0,n,t ) t∈A0,n , where each one is given by 'folding' the linear flow n times. Moreover, the a n 's can be chosen inductively to grow fast enough so that all the semiflows F 0,n are defined in the upper half-plane H, or indeed in any fixed half-plane H −M = { Im z > −M }, as large as desired; indeed we can state as a proposition the following (which follows directly from Perez-Marco [7] there exists a sequence of conditions (C n ) n≥1 on the integers a n of the form (C n ) a n ≥ C n (a 0 , . . . , a n−1 , h 0 , . . . , h n−1 , M ) such that when they are fulfilled then all the semi-flows We observe thus a feature of the construction which is crucial when passing to the limit, namely that it allows us to successively paste nonlinearities to the dynamics without decreasing its domain of definition too much, allowing us to retain a half-plane H −M .
Assume that the a n 's have been chosen so that all the semi-flows F 0,n are defined on H −M . From the sequence of semi-flows F 0,n we can extract a convergent subsequence (it always exists, see Perez-Marco [7], section 1.1.2) converging normally for times t ∈ A = ∩ n≥0 A 0,n to get a limit semi-flow F = (F t ) t∈A defined on the half-plane H −M for times t ∈ A. We note that elements of the limit semi-flow do indeed commute, since we can pass to the limit in the equation (using uniform convergence in compact neighbourhoods of the points z, F t (z), F s (z)).
The Cantor set of times C in the main theorem is given by If all the a n 's are large enough, say for example a n ≥ 5 for all n, then the set C is indeed a Cantor set.
The semi-flow (f t ) t∈C of the main theorem is defined on a common neighbourhood E(H −M ) of the closed unit disk D via the equation As stated in section 2.1.4, each uniformization K n maps the upper half-plane of the 0-plane of S δn+1 into the upper half-plane H. So we can consider its restriction to the upper-half plane of the 0-plane as a map K n : H → H of H into itself. If we consider the composition of these maps K 0 • K 1 • . . . • K n , then it is clear that the set K 0 K 1 . . . K n (H) is totally invariant under the semi-flow F 0,n+1 , and hence the set is then totally invariant under the limit semi-flow F . Moreover, H, being the decreasing limit of the domains K 0 K 1 . . . K n (H), is full, and H ∩ ∂H = φ (since 0 ∈ H). Thus H is in fact a common hedgehog for irrational elements of the limit semi-flow F . The hedgehog K of the semi-flow (f t ) t∈C in the main theorem is given by It is important to note in this construction that the only requirement imposed on the a n 's so that the limit dynamics is defined on the upper half-plane is that they grow fast enough, as expressed by Proposition 2.1.1 above. Subject to this restriction, the construction still offers enough flexibility to construct hedgehogs with quite different geometries by varying the choices of the parameters a n and h n .
In the following sections we show how to choose the uniformizations K n appropriately (by proper choices of the parameters a n , h n ), and control the geometries of the domains K 0 K 1 . . . K n (H), so that the hedgehog H contains a comb with the desired geometry.

Formation of the comb.
The purpose of this section is to describe informally the idea behind the construction of the comb within the hedgehog; precise statements and proofs are given in the following section, 2.3.

t) + it
This definition is made conditionally since the existence of these parametrizations depends on the choice of the parameters (a n ), (h n ); that these parameters can indeed be chosen appropriately will be justified in the following section.
We define the set (which will be our comb)

Γǫ
(this definition is also made conditionally since the existence of the curves Γǫ for infinite sequencesǫ depends on the choice of the parameters (a n ), (h n )); again, justification will be provided in the following section). The set E which will be the comb of the main theorem is given by E = E(C) (where E(z) = e 2πiz ). That these sets are indeed combs and contained in the hedgehog H will be shown in section 3.
We note that points of C can be represented naturally by pairs (ǫ, t), thus exhibiting for C a product structure of the form Cantor set × interval, withǫ belonging to the abstract Cantor set {−1, 0, 1} N with the product topology, and t to the interval [1,2].
To study the regularity of the comb C however, we will parametrize it by the metric comb C 0 × [1, 2], where the Cantor set C 0 ⊆ R is given by embedding {−1, 0, 1} N into R, via the map (the map is a homeomorphism onto its image C 0 if the a n 's are large enough, for example if a n ≥ 4 for all n). For sequencesǫ = (ǫ 0 , . . . , ǫ n , 0, 0, 0, . . .) with finitely many non-zero terms we will denote θ(ǫ) by θ(ǫ 0 , . . . , ǫ n ).
The homeomorphism Φ : C 0 × [1, 2] → C is then given by (the homeomorphism Ψ in the main theorem is given by composing with the exponential, Ψ = E • Φ). We remark again that this definition is conditional, depending on the choice of parameters a n , h n ; in the following section 2.3, we show how to choose the parameters so that this map Φ is well-defined, a homeomorphism, and satisfies the Hölder and Lipschitz conditions stated in the main theorem, which will be proved in section 3. We note also that, ensuring that the comb is strictly contained in a period strip for the exponential, such as { −1/2 < Re z < 1/2 } (this will be achieved by proper choice of the parameters), it is not hard to check that these properties for Φ imply the same for Ψ.

Inductive choice of the parameters.
We let G n = u n + iv n := K 0 . . . K n , n ≥ 0 We fix an increasing sequence α n ↑ 1 ( we will ensure at each stage n in the formation of the comb that Φ satisfies an α n -Hölder estimate, so that in the limit Φ will be α-Hölder for all α). Fix a constant M = 1 (any M > 0 will do); the parameters will be chosen so that the limit semi-flow F is defined on the half-plane H −M .
The following lemma will be used in the construction to ensure that Φ is injective and Φ −1 Lipschitz: Lemma 2.3.1. Given fixed constants v, h > 0 such that 0 < v < h, for a ≥ 1 and x 0 ∈ Z, let z 0 = x 0 + iy 0 , with y 0 depending on a, be such that Im 1 a K δ(a,h) (z 0 ) = v (note that y 0 doesn't depend on x 0 ). Then for uniformly in x 0 , y.
We can now begin the construction. At stage n = 0 : Let h 0 = 10. We choose a 0 ≥ 5 to satisfy the following: Applying Lemma 2.3.1 above to K 0 = 1 a0 K δ(a0,h0) , with v = h 0 − 1/2 < h = h 0 and a = a 0 , we require that a 0 is chosen large enough so that The parameters a 0 and h 0 having been chosen, the map K 0 is now fixed. We need to choose additional parameters so that the induction can continue: Using property (4) of the maps 1 a K δ , we choose h 1 > 2 such that Im K 0 (x + i(h 1 − 1)) > h 0 + 1 for all x ∈ R. We choose h ′ 1 > 0 such that 2 < Im K 0 (0 + ih ′ 1 ). Also, we have 0 = Im K 0 (0 + i0) < 1; so by continuity, we can choose τ 1 > 0 such that By property (3) of the maps 1 a K δ , ∂v0 ∂y > 0 on the vertical segment [0 + i0, 0 + ih ′ 1 ]; thus by continuity and compactness of the segment, we also take τ 1 small enough to have The reason for imposing the conditions above is to guarantee that for |x| ≤ τ 1 the curves K 0 (ξ x ) pass through the strip S and, in S, are graphs over the imaginary axis, so that in the next stage, we can choose a 1 such that 1/a 1 < τ 1 to ensure the same is true for the curves γ (−1) = K 0 (ξ −1/a1 ), γ (1) = K 0 (ξ 1/a1 ).
(This condition will allow us to ensure that all elements of the semi-flow F 0,n have a common periodic orbit at a height at least h 0 + n, so that elements of the limit flow will have periodic orbits above any given height and hence be nonlinearisable for irrational rotation numbers).
In condition (e) below we will require a further control on the C 0 distance between these parametrizations, but for now it follows from the above condition on a n that for anyǫ = (ǫ 0 , ǫ 1 , . . .) ∈ {−1, 0, 1} N , the sequence {σ (ǫ0,...,ǫm) } m≥0 will be Cauchy in every C r norm on [1,2], and hence convergent in every C r norm to a parametrization σǫ of a C ∞ curve Γǫ. Thus we are guaranteed that the map Φ : (θ(ǫ), t) → σǫ(t) is, firstly, well-defined, and secondly, smooth in t for fixed θ(ǫ) ∈ C 0 , as stated in the Main Theorem.
(f ) For the hedgehog to be non-linearisable : Let P n be the image under K n of the upper ends of the 1-cylinders in S δn+1 , or more precisely, P n := z n + 1 a n Z where z n is the limit of K n (z) as Im z → +∞ in one of the 1-cylinders of S δn+1 (note P n is independent of the choice of 1-cylinder, since their images differ by 1/a n translations), and define the sets Proof : Fix m ≥ n + 1. Consider an F ∈ F n,m . The map F is the conjugate under K n of a liftF t of a map in F n+1,m for a time . Suppose Im z → +∞ in the 1-cylinder of S δn+1 whose ramification point lies at a point q ∈ Z in the 0-sheet; theñ F t (z) escapes to infinity through the 1-cylinder whose ramification point lies at the point q + l in the 0-sheet, and ImF t (z) → +∞. Since P n consists of the images under K n of the upper ends of the 1-cylinders, it follows that the orbits under F of points of O n,n+1 = P n are periodic (modulo Z) and contained in O n,n+1 .
For 0 ≤ i ≤ n − 1, since the elements of F i,m are given by conjugating those of F n,m by the map K i . . . K n−1 , and O i,n+1 is given by transporting O n,n+1 by the same map, the desired conclusion follows. ♦.
Using property (5) of the maps 1 a K δ , we choose a n large enough so that The induction hypothesis 1. then guarantees that O 0,n+1 , the image of P n under K 0 . . . K n−1 , which is a set of periodic orbits (modulo Z) for all the semi-flows F 0,m , m ≥ n + 1, will be at a height at least h 0 + n in the upper half-plane, so the semi-flows (F 0,k ) k≥0 will have non-linearities at increasing heights.
We now choose and fix a n ≥ 5 large enough to satisfy all the conditions (a)-(f ) above. Thus the map K n is determined.
To complete the induction step, it remains to choose the constants h n+1 , h ′ n+1 , τ n+1 and the domain D n+1 so that the induction hypotheses are satisfied for the next step.
We observe that with these definitions all the conditions of the induction hypothesis 4. are satisfied.
Thus the induction hypotheses 1. to 4. required for stage n + 1 are satisfied and the induction can continue.
This ends the inductive choice of the parameters.

Proof of the main theorem.
In the course of the induction above we noted that we get a well defined map Φ : C 0 × [1, 2] → C, which is C ∞ smooth in t ∈ [1, 2] for fixed θ ∈ C 0 . It remains to prove that Φ is a homeomorphism, Φ is α-Hölder regular for all α, 0 < α < 1, Φ −1 is Lipschitz, and the elements f t of the limit semi-flow are non-linearisable for irrational t.

Acknowledgements.
The main result of this article, which is based on part of my Ph.D. thesis, was first stated by my advisor R.Perez-Marco in [7]. I am very grateful to him for proposing this problem to me and also for explaining how to use his techniques of tube-log Riemann surfaces to solve it. I thank also the referee for many helpful suggestions which have improved this article considerably.