Critical covering maps without absolutely continuous invariant probability measure

We consider the dynamics of smooth covering maps of the circle with a single critical point of order greater than \begin{document}$ 1 $\end{document} . By directly specifying the combinatorics of the critical orbit, we show that for an uncountable number of combinatorial equivalence classes of such maps, there is no periodic attractor nor an ergodic absolutely continuous invariant probability measure.


1.
Introduction. In the study of one-dimensional dynamical systems, the question of the existence of an absolutely continuous invariant probability (acip) plays a central role. The importance stems from the fact that an acip provides a complete description of the asymptotic distribution of typical orbits of the system.
The existence of an acip has been much studied for unimodal maps of the interval. Jakobson [8] showed that maps in the quadratic family have an acip for a set of parameters of positive measure. For example, parameters for which the critical point is strictly preperiodic, as studied by Misiurewicz [14]. Keller [10] proved that the existence of an acip is equivalent to almost everywhere positivity of the Lyapunov exponent. Collet and Eckmann [5] showed that exponential growth of the size of the derivative along the critical orbit is a sufficient condition for the existence of an acip. This growth condition was weakened to a summability condition by Nowicki and van Strien [15], and weakened still further to a lower bound condition by Bruin, Shen and van Strien [4]. This result was later extended to multimodal maps [3].
There are also results on non-existence. Arnold [1] showed that for a dense set of irrational rotation numbers, the conjugacy from an analytic diffeomorphism of the circle to a rotation with the same rotation number is not absolutely continuous. In particular, such a diffeomorphism has no acip. Johnson [9] was the first to construct quadratic interval maps without an acip and topologically conjugate to a tent map. This was further developed by Hofbauer and Keller [7] using the tools of kneading theory.
Nevertheless, existence of an acip has been shown to be the more prevalent situation for certain families of maps. Herman [6] proved that for almost every irrational rotation number, the conjugacy to a rotation is smooth, and so an acip exists. Lyubich [11] proved that for almost every map in the quadratic family, there is either an acip or an attracting periodic cycle. This result was extended to real analytic quadratic unimodal maps by Avila, Lyubich and de Melo [2].
In this article, we consider smooth critical covering maps of the circle without an acip. As in the non-existence results for other classes of maps, we specify the dynamics at the topological level. For clarity, we specify suitable combinatorics for the critical orbit directly. The fact that the critical point of a covering map is of inflection type rather than a turning point means that the techniques used are significantly different to the unimodal case because of the lack of dynamical symmetry. In the next section we state the main results after introducing the required notation.

Notation and results.
2.1. Covering maps. Consider the circle S 1 to be defined as the set of complex numbers with modulus one equipped with the topology, orientation and differentiable structure of the real numbers induced by the exponential map τ : R → S 1 given by τ (t) = e 2πit . A distance on S 1 can be defined in the following way: given x = τ (t 1 ) and y = τ (t 2 ), with t 1 , t 2 ∈ [0, 1], the distance between x and y, denoted by |x − y| is the minimum of |t 1 − t 2 | and 1 − |t 1 − t 2 |. Many times in this article, we choose a convenient value of t and represent the circle by the interval [t, t + 1], where we use the identification t ∼ t + 1.
A surjective locally homeomorphic map f : S 1 → S 1 is said to be a covering map of the circle of (topological) degree d, |d| ≥ 1, if the pre-image of each point consists of exactly |d| points and f is order-preserving or order-reversing for d > 0 or d < 0 respectively.
Given a covering map f : S 1 → S 1 of degree d, we can find a lift: a map F : R → R for which τ •F = f •τ . A lift F has the property that F (t+1) = F (t)+d for all t ∈ R. Conversely, any homeomorphism of the real line with this property is a lift of some covering map of the circle. For example, the linear function L d : R → R given by L d (t) = d t is a lift of the uniform covering map of degree d which we denote by l d : We introduce an equivalence relation for covering maps. Given two covering maps f and g of the circle and two marked points a, b ∈ S 1 , we say that the pairs (f, a) and (g, b) are combinatorially equivalent if there exists a homeomorphism h : S 1 → S 1 such that h(f n (a)) = g n (b) for all n ≥ 0.
2.2. Topological dynamics. We begin our study by recalling some of the basic topological dynamics of covering maps. If d ∈ {−1, 1}, then f is a homeomorphism. For d = −1, f is an order-reversing homeomorphism, which necessarily has a fixed point and so the dynamics are well understood. For d = 1, f may have no periodic points, and the dynamical behaviour is an area of current interest.
For d / ∈ {−1, +1}, covering maps are not injective and possess periodic points of all periods and also many compact invariant sets: that is, compact sets Λ ⊂ S 1 such that f (Λ) ⊆ Λ. Given x ∈ S 1 , the set of all accumulation points of the forward orbit (f n (x)) n≥0 is called the ω-limit set of x and is denoted by ω(x). The ω-limit set of a point is a compact invariant set that is important for the analysis of the dynamics. If x is periodic, then ω(x) is a finite set, but there are other possibilities for ω-limit sets, such as Cantor sets or the whole circle S 1 .
For |d| ≥ 2, the uniform covering map l d is expanding and points in a residual set have orbits that are dense on the whole circle S 1 . Covering maps that are not expanding may have periodic attractors. Also there may exist intervals I ⊂ S 1 whose orbit consists of pairwise disjoint intervals: that is, f n (I) ∩ f m (I) = ∅ for 0 ≤ n < m. Such an interval could be contained in the basin of a periodic attractor. If this it not the case, then it is called a wandering interval. The existence of wandering intervals is a factor that complicates the topological understanding of the dynamics.
There exists a semi-conjugacy between a covering covering map f of degree d and the uniform covering map l d : that is, a monotone surjective map h : The map h is locally constant on wandering intervals and on connected components of the basin of a periodic attractor, if either of these exist. In the absence of wandering intervals, periodic attractors, and intervals consisting entirely of periodic points, the semi-conjugacy is, in fact, a conjugacy: h is a homeomorphism. In which case, we can conclude that for f , orbits of points in a residual set are dense on the whole circle S 1 . Also, we can conclude that if Per(f ) is the set of all periodic points, then the closure Per(f ) is the whole circle.

2.3.
Measure-theoretic properties. The second step in understanding the dynamics of critical covering maps depends on the ergodic aspects. Let us assume from now on that there are no wandering intervals, periodic attractors nor intervals consisting entirely of periodic points.
The Lebesgue measure on the circle, which we denote by λ, is invariant by the uniform covering map l d : that is, for any Borel set B ⊂ S 1 , λ(l −1 d (B)) = λ(B). Using the homeomorphism h : S 1 → S 1 conjugating a critical covering map f to l, we can define an invariant probability measure µ for f : for every Borel set B ⊂ S 1 , µ(B) = λ(h(B)). This can also be formulated in terms of the regularity of the conjugating homeomorphism.
Let ν be a Borel probability measure on the circle that is invariant for f . Assume that ν is ergodic with respect to f : that is, for any Borel set B ⊂ S 1 , the ν-measure of the symmetric difference f −1 (B) B is either 0 or 1. The basin of ν is the set where C 0 (S 1 , R) denotes the class of continuous functions from S 1 to R. The support of ν is a compact invariant set for f . The basin of ν is a totally invariant set: that is, f (B(ν)) ⊆ B(ν) and f −1 (B(ν)) ⊆ B(ν). By the Ergodic Theorem of Birkhoff, the ν-measure of B(ν) is 1. However, the Lebesgue measure of B(ν) could be smaller, even zero. If the invariant probability measure ν is absolutely continuous with respect to Lebesgue measure λ (or an acip measure, for short), then the support of ν has positive Lebesgue measure. Moreover, under our assumptions the map f is ergodic with respect to λ (see [16]) and this implies that the basin of ν has Lebesgue measure 1. This guarantees that the acip measures are of great relevance for the description of the dynamics of f .

Critical covering maps.
A critical covering map of the circle is a covering map f : S 1 → S 1 of class C r , r ≥ 1, with a unique critical point of inflection type, which we denote by c f or simply by c if no confusion will occur. The local inverse of a critical covering map is also C r , except at the critical value f (c). If f and g are critical covering maps for which (f, c f ) is combinatorially equivalent to (g, c g ), where c f and c g are the critical points of f and g respectively, then we shall say that f is combinatorially equivalent to g.
As a source of examples of critical covering maps, we consider the Arnold family of maps g α : As we show below the Arnold family is an example of a full family of critical covering maps. Consider a family {g α } α∈∆ of critical covering maps, where ∆ is an interval and α → g α (x) is continuous for each x ∈ S 1 . Such a family is said to be a full family if, any critical covering map f is combinatorially equivalent to g α , for some α ∈ ∆.
Let us show some basic properties of each map g α in the Arnold family. A straightforward computation shows that G α has negative Schwarzian derivative, which implies that g α has at most one periodic attractor, see [13]. The point c = 1/2 and its translations by integers are the critical points of G α and all of them project to the unique critical point of g α , which is a fixed point for g −1/2 .
it is easy to see that g −1/2−β and g −1/2+β have an indifferent fixed point which is attracting from one side and repelling from the other. For α ∈ (−1/2−β, −1/2+β) the map g α has an attracting hyperbolic fixed point whose immediate basin of attraction, bounded by two repelling hyperbolic fixed points, contains the critical point. For α ∈ (−1/2 + β, 1/2 − β), the map g α has only one fixed point which is a hyperbolic repeller, say p α . This fixed point and its pre-image define a partition of the circle with two arcs, each of which are mapped onto the circle by g α . So we can conclude that g α is topologically conjugate to the shift on {0, 1} N . Now we observe that the derivative with respect to α is 1 for the critical value G α (1/2) and is negative for the repelling fixed point p α . Then we can conclude that for any sequence in {0, 1} N , there is some value of α ∈ [−1/2 + β, 1/2 − β] such that the itinerary of the critical value of g α realises this sequence. Thus the Arnold family {g α } α∈∆ , with ∆ = [−1/2 − β, 1/2 − β], is an example of a full family of critical covering maps of the circle.
In order to ensure that a critical covering map has no wandering intervals, see [13] and also [16], it is enough to assume that it is of class C 2 (outside the critical point where it is C 1 ) and the critical point has finite order: that is, there exist a C 1 map ψ defined near c satisfying lim x→c ψ(x) = 0 and real constants ϑ > 0 and β > 1 such that for every x in a neighbourhood of c. The constant β, when it exists for some ψ as above, is called the order of the critical point.
From now on, we shall concentrate on full families of C 1 critical covering maps {g α } α∈∆ with the following properties for all α ∈ ∆: (H1) the critical point c has order β > 1; (H2) g α has topological degree 2; (H3) Restricted to S 1 \{c}, the map g α is C 3 and has negative Schwarzian derivative. Remark 1. Assumptions (H2) and (H3) are to simplify the exposition: the same methods can be used for any degree d ≥ 2, and the arguments could be adapted so as not to use the negative Schwarzian assumption, but this would increase the technicality of the proofs.
2.5. Statement of results. Our main result, Theorem 2.1 below, concerns the absence of acip measures, which we will prove in the Section 5. According to this theorem, the measure µ defined above may not describe the statistical behaviour of a significant set of orbits. This property depends on a strong recurrence of the critical point and to understand it, we will describe in Section 3 an uncountable set of combinatorics of the critical orbit associated to it.
Theorem 2.1. Within any full family of critical covering maps satisfying the above hypotheses (H1)-(H3), there are uncountably many combinatorially non-equivalent maps with no absolutely continuous invariant measure and no periodic attractor.
In the case of any full family, we have the following corollary.
Corollary 1. Within any full family of critical covering maps {g α } α∈∆ satisfying the above hypotheses (H1)-(H3), there are uncountably many values of α ∈ ∆ for which g α has no absolutely continuous invariant measure and no periodic attractor.
We specify the combinatorics used in Theorem 2.1 above in the following section.

Combinatorics.
3.1. Kneading sequences. For our results we need to find critical covering maps with strongly recurrent critical points. Then, when iterating the map, the strong contraction near to the critical point can overcome the expansion that occurs on the rest of the circle. We specify the strongly recurrent behaviour at the combinatorial level.
There are different choices of partition that can be made for defining the kneading sequence of a critical covering map. For example, the partition defined by the unique fixed point p and its pre-image q = p is a Markov partition with two intervals. However, the itineraries given by this partition are linked with the recurrence of the critical point in a complicated way.
Given a critical covering map f consider instead the circle represented by the interval [c, c + 1], where c and c + 1 are identified. Then choose the partition . We shall use this partition to define a coding of the orbits of f . Let A denote the alphabet consisting of the three symbols 0, 1 and 2.
We say that x ∈ S 1 is coded by the symbol i ∈ A if x ∈ I i . Similarly, given x ∈ S 1 and n ≥ 1, the piece of orbit (x, f (x), . . . , f n−1 (x)) is coded by the finite Extending to forward orbits, we have the coding map κ : Of special importance is the coding κ(f (c)) of the forward orbit of the critical value f (c), which is called the kneading sequence of f and denoted by κ f . Clearly, combinatorially equivalent critical covering maps have the same kneading sequence and vice-versa. However, as the partition I 0 , I 1 , I 2 is not Markov, not every element of A N is the kneading sequence of some critical covering map. We say a sequence ω ∈ A N is admissible if there exists some critical covering map f with κ f = ω.
We now define the kneading sequences required for Theorem 2.1. Given any two sequences ( n ) ∞ n=1 and (m n ) ∞ n=1 of natural numbers with ( n ) ∞ n=1 strictly increasing and m n ≥ 2, we inductively define a sequence ω ∈ A N . We first define the words ω 1 = 0 and ν 1 = 1 1 0. Then, assuming that the words ω n and ν n are already defined, we define the words ω n+1 and ν n+1 by setting ω n+1 = ω mn n ν n and ν n+1 = ω mn n · · · ω m1 where, for a finite word u and integer k ≥ 1, u k denotes the word obtained by concatenating k copies of the word u. Similarly, we will write u ∞ for the infinite word obtained by concatenating countably many copies of a finite word u.
Notice that the symbol 2 is forbidden from occurring in the kneading sequences. As a result, the critical orbit avoids the interval I 2 and so the ω-limit set ω(c) is a Cantor set (see Theorem 4.1).
We also define r n = |ω n |. We denote by [ω n ] ⊂ A N the cylinder of all infinite words beginning with the word ω n . By construction, for each n the word ω n+1 begins with the word ω n , and so we have a nested sequence [ consists of a unique infinite word, which we denote by ω ∈ A N .
As there are uncountably many distinct choices for the sequences ( n ) ∞ n=1 and (m n ) ∞ n=1 , and each pair of sequences gives rise to a different sequence in A N , we obtain an uncountable collection of infinite words ω that we denote by K.
Remark 2. We will show below that the sets H(ω n ) are non-empty and that they nest down to the non-empty set H(ω) = ∞ n=1 H(ω n ). Every critical covering map in this set has the same kneading sequence ω in K. We also can see that H(ω n ) contains H(ω ∞ n ) and, as a consequence, any critical covering map in H(ω) can be approached by critical covering maps in H(ω ∞ n ). The critical point of each map in H(ω ∞ n ) is asymptotic to a periodic attractor of period r n = |ω n |. 3.2. Admissibility of combinatorics. We will now show that H(ω) is non-empty. We start by showing that H(ω n ) is non-empty: indeed, we will construct by induction, a sequence of critical covering maps f n ∈ H(ω n ), each with critical point c = 0. The way we construct f n , the sequence will converge to a map in H(ω), showing that this set is also non-empty. Alternatively, each f n is topologically conjugate to a critical covering map g αn in a given full family {g α } α∈∆ , then g αn converges to a map in H(ω).
Before this, note that, for a critical covering map f , the open interval I 1 = (z + 1 , z − 1 + 1) contains a unique fixed point p of f and where L n , R n , n ≥ 1, are the left and right fundamental domains of p: that is, they are the maximal open intervals satisfying Remember that we identify the intervals (z − 1 + 1, c + 1) and (z − 1 , c).
Now we start the construction by induction of the sequence of critical covering maps (f n ) ∞ n=1 in H(ω n ). The easy first step of induction below is just the construction by hand shown in Figure 3 3.3. Induction argument. We use an induction argument to show that all sequences in K are admissible. First step of induction. There exists a critical covering map f 1 for which the following properties hold: For y ∈ E k 1 , the piece of orbit (f 1 (y), . . . , σ 1,k (y)) is coded by the word 1 k 0. In particular the entry time of for all k > 1 . Induction hypothesis. For n ≥ 1, there exist critical covering maps f 1 , . . . , f n such that, for 1 ≤ i ≤ n, the following properties hold: Figure 2. The first entry map to the interval V + 2n−1 , showing the critical branch φ n and some of the branches σ k, k , with k ≥ n.
For y ∈ E k i , the piece of orbit (f i (y), . . . , σ i,k (y)) is coded by the word ω mi−1 i−1 · · · ω m1 1 1 k 0. In particular the entry time of y to V + 2i−1 is t i,k := m i−1 |ω i−1 | + · · · + m 1 |ω 1 | + k; In Figure 3.3 we show the intervals E n n , E n+1 n , V + 2n inside V + 2n−1 and the branches σ n, n , σ n, n+1 and φ n of first entry map to V + 2s−1 defined on them. Induction step. We assume that the induction hypothesis holds true for n ≥ 1 and prove it for n + 1. Since c ∈ D n ⊂ V 2n and φ mn n (D n ) = V + 2n−1 \ V + 2n , we define V 2n+1 ⊂ V 2n to be the closed neighbourhood D n . For k > n−1 , there is a closed interval I k n ⊂ E k n which is mapped diffeomorphically onto V + 2n+1 by σ n,k . If necessary, we modify f n on the interior of D n = V 2n+1 to obtain a critical covering map f n+1 for which φ mn n (c) ∈ I n n . Then we define the interval V 2n+2 ⊂ V 2n+1 around the critical point such that φ mn n (V 2n+2 ) = I n n . The first entry map φ n+1 from V 2n+2 onto V + 2n+1 is defined by φ n+1 = σ n, n • φ mn n and properties (i)-(ii) are satisfied for 1 ≤ i ≤ n + 1. For k > n , I k n ⊂ E k n ⊂ φ mn n (V + 2n+1 ) and we can define the interval E k n+1 ⊂ V + 2n+1 \ V + 2n−2 such that φ mn n (E k n+1 ) = I k n . The first entry map σ n+1,k from E k n+1 onto V + 2n+1 is defined by σ n+1,k = σ n,k •φ mn n . In this way we have property (iii) satisfied for 1 ≤ i ≤ n + 1. To ensure that property (iv) is satisfied, we can adjust f n+1 further on the interior of V 2n+1 a little more. Since V 2n+1 = D n , the map f n+1 we have constructed also satisfies property (v).
The induction above is the main step in proving Proposition 1 that guarantees that H(ω) is non-empty and all sequences in K are admissible. Proposition 1. Given a sequence ω ∈ K, in any full family of critical covering maps there exists a map f with kneading sequence ω.
Proof. Each critical covering map f n in the induction argument above has kneading sequence κ fn contained in the cylinder [ω n ] of all words starting with the word ω n , n ≥ 1. We have that f n+1 = f n , on S 1 \D n . Since we have the freedom to choose the sizes of the closed neighbourhoods D n of c so that ∞ n=1 D n = {c}, this guarantees that the sequence (f n ) ∞ n=1 converges to a continuous coveringf such that the kneading sequence of the marked point c is ω. In any full family there is a critical covering f with a marked critical point which is combinatorially equivalent tof with the point c marked. This implies that the kneading sequence of f is ω and we can conclude that ω is admissible.
Alternatively, each f n is topologically conjugate to a critical covering map g αn in a full family {g α } α∈∆ . Then g αn converges to a map in H(ω), also showing that ω is admissible.

Combinatorics in full families. The sets H(ω n ) and H(ω) =
∞ n=1 H(ω n ) are non-empty. In any full family {g α } α∈∆ of critical covering maps with g α satisfying hypotheses (H1)-(H3), there is a non-degenerate interval of parameters α such that g α ∈ H(ω n ). These intervals accumulate on a parameter α ω such that g αω ∈ H(ω). One way to reach this interval of parameters is starting with maps g αn in H(ω ∞ n ), that is, parameters such that the critical point is asymptotic to a periodic attractor associated to the word ω i .
The lemma below guarantees that H(ω ∞ n+1 ) and H(ω mn n ν ∞ n ) are also non-empty. Therefore, the infinite words ω ∞ n+1 and ω mn n ν ∞ n are admissible. Below, for a given f n ∈ H(ω n ), as in the induction above, let us use the notation E n instead of E n, n and σ n instead of σ n, n . Lemma 3.1. Given f n ∈ H(ω n ), the following properties hold true: 1. There are x n , y n ∈ E n such that σ n (x n ) = c and σ n (y n ) = y n . For y = x n or y = y n , the piece of orbit (f n (y), . . . , σ n (y)) is coded by the word ν n ; 2. f n can be modified on D n to obtain either a critical covering map G − n or G + n , both in H(ω n ), for which φ mn n (c) is either x n or y n , respectively. The corresponding kneading sequences of G − n and G + n are ω ∞ n and ω mn−1 n−1 ν ∞ n−1 , respectively.
Proof. As E n ⊂ V + 2n−1 and the map σ n : E n → V + 2n−1 is a surjective diffeomorphism, there are x n , y n ∈ E n such that σ n (x n ) = c and σ n (y n ) = y n . By part (iii) of the induction hypothesis, for y ∈ E n , the piece of orbit (f n (y), . . . , σ n (y)) is coded by the word ω mn−1 n−1 · · · ω m1 1 1 n 0 = ν n . This finishes the proof of the first part of the lemma. To prove the second part of the lemma we observe that changes of f n in D n do not affect the branches σ i : E i → V + 2i−1 and the points x i , y i ∈ E i , for 1 ≤ i ≤ n. There exists a closed neighbourhood H n of c that is properly contained in D n with the property that φ mn n maps H n diffeomorphically onto [x n , y n ]. Moreover, the closed intervals H n , φ n (H n ), . . . , φ mn−1 n (H n ) are pairwise disjoint and all are properly contained in V 2n . So by subtracting from f n a smooth bump function supported on D n we can reduce each of the points φ n (c), φ 2 n (c), . . . , φ mn n (c) so as to get a map G − n for which φ mn n (c) = x n . We then have that c is a periodic point since φ n+1 (c) = σ n • φ mn n (c) = c, and so the kneading sequence for G − n is (ω mn n ν n ) ∞ = ω ∞ n+1 . Alternatively, by adding to f n a smooth bump function supported on D n we can increase each of the points φ n (c), φ 2 n (c), . . . , φ mn n (c) in order to get a map G + n for which φ mn n (c) = y n . We then have that φ mn n (c) is the periodic point y n and so the kneading sequence of G + n in this case is ω mn n v ∞ n . We have the following immediate corollary.
Corollary 2. Given f n ∈ H(ω n ), there exists G n ∈ H(ω ∞ n ) with the property that G n = f n on S 1 \ D n with critical point asymptotic to a parabolic periodic attractor.
Proof. This is an immediate consequence of Lemma 3.1. Indeed, any continuous one parameter family in H(ω n ), starting in G − n and ending in G + n , has a critical covering map as stated. Now, looking in a full family, we have the following lemma.
Lemma 3.2. Let {g α } α∈∆ be a full family of critical covering maps that satisfies hypotheses (H1)-(H3). Then, for each n ≥ 1, there are parameters a(n) and b(n) which satisfy the following: 1. The critical point is a periodic superattractor for g a(n) , with period r n = |ω n |; 2. The critical point is asymptotic to the orbit of a parabolic periodic attractor for g b(n) , with period r n ; 3. For α in the interval defined by a(n) and b(n), the map g α has a periodic attractor of period r n and its kneading sequence is ω ∞ n . Proof. According to Lemma 3.1 there is a critical covering G − n−1 whose kneading sequence is ω ∞ n . This can be realized in any full family and then we get the parameter a(n) of the statement. Instead of this, we can get a parameterb(n) corresponding to the kneading sequence ω mn n ν ∞ n . Then, by an intermediate value argument, there is a parameter b(n) in between a n andb n as stated.
3.5. Extension of first entry branches. The lemma below assures that the branches of first entry maps σ n and φ n associated to a critical covering map f n in H(ω n ) have big diffeomorphic extensions. Their ranges cover at least the interval (p n − 1, f n (c) + 1), where p n is the unique fixed point of f n in (c, c + 1). These extensions depend only on the combinatorial properties given by the word ω n . In fact they depend only on the form of the word and in particular they do not depend on the sequences ( n ) ∞ n=1 and (m n ) ∞ n=1 . As we will see in Lemma 5.1, as a consequence of Koebe Principles, these extensions imply a uniform control of distortion for these branches. Proof. We proceed by induction on n and to start let us assume that n = 1. For Part 1 of the lemma we chooseV 1 = V 2 and to define V 1 we remember that r 1 = 1, f has only one critical point in (c − 1, c + 1) and the image of this interval by f n contains (p n − 1, f n (c) + 1). Then we can choose V 1 as wanted.
For the proof of Part 2 of the lemma we defineÊ k 1 = E k 1 and to define E k 1 we remember that E k . We consider three cases: . In this case t 1,1 = 2 and the diffeomorphic image of E 1 ) is the minimal interval which contains f n (φ m1 1 (c)) and R 1 . In this case t 1,k = 1 + 2 and the diffeomorphic image of E 1+1 The interval (p n − 1, c) has no points from the forward orbit of the critical point and, for k > 0 , t 1,k = k + 1. In any one of the three cases above, we can shrink E k 1 so that f k+1 n (E k 1 ) = (p n − 1, f n (c) + 1) as required. To prove Part 3 of the lemma we remember that the intervalÊ k 1 is a connected component of the first entry map to V + 1 and, for y ∈Ê k 1 , the piece of orbit {f n (y), . . . , σ 1,k (Ê k 1 )} corresponds to the word 1 k 0. This implies that the intervalsÊ k 1 ,Ê k+1 . The fact that E k 1 is contained in the convex hull of φ m1−1 1 (c) and V + 1 \ V + 2 also follows from the definition in one of the cases (i)-(iii) above.
To prove Part 4 of the lemma we observe that, for k > 1 , E k 1 is defined by the case (ii) or case (iii) above and we get that E k 1 ⊂ φ m1 1 (V + 3 ) as required. To prove Part 5 of the lemma we remember thatÊ k 1 = E k 1 is the domain of a branch of the first entry map to V + 1 . Then the collection of intervals {Ê k 1 , . . . , f t 1,k n (Ê k 1 )} has multiplicity of intersection equal to two. The collection of intervals {V 1 , . . . , f tr 1 n (V 1 )} also has multiplicity of intersection at most two because r 1 = 1. This finishes the first step of induction. Now we assume that the lemma is true for n ≥ 1 and prove it for n + 1. To define the pairV n+1 ⊂ V n+1 required to prove Part 1 we remember that φ mn and E n n is contained in the convex hull of φ mn−1 n (c) and V + 2n−1 \V + 2n . All these properties imply that we can take the pairV n+1 ⊂ V n+1 as the intervals around the critical point such that f mnrn n (V n+1 ) =Ê n n and f mnrn n (V n+1 ) = E n n . With this choice we have that f rn+1 n = f t n, n n •f mnrn n maps the triple V 2n+2 ⊂V n+1 ⊂ V n+1 homeomorphically onto the triple V + 2n+1 ⊂ V + 1 ⊂ (p n − 1, f n (c) + 1). We also have that f jrn n (V n+1 ) ⊆ V + 2n−1 , for 1 ≤ j ≤ m n , therefore f mnrn n has only one critical point in V n+1 , the point c.
Since f t n, n n is a diffeomorphism from E n n = f mnrn n (V n+1 ) onto (p n − 1, f n (c) + 1), we are done.
For Part 2 of the lemma we assume that k > n , in which case E k n ⊆Ê k n ⊂ E k n are all contained in φ mn n (V + 2n+1 ) and there is a closed interval I k n ⊂ E k n that is mapped diffeomorphically onto V + 2n+1 by σ n,k . Then we can take E k n+1 ⊂Ê k n+1 ⊂ E k n+1 to be the intervals mapped diffeomorphically onto I k n ⊂Ê k n ⊂ E k n by φ mn n . Since f t n+1,k n (y) = σ n,k • φ mn n (y), for y ∈ E k n+1 , Part 2 of the lemma follows. For Part 3 we remember that, for k > n , we can conclude that the intervalŝ E k n+1 ,Ê k+1 n+1 ,Ê k+2 n+1 , . . . are pairwise disjoint because they are mapped by φ mn n diffeomorphically onto the pairwise disjoint intervalsÊ k n ,Ê k+1 n ,Ê k+2 n , . . .. The interval E k n is on the right ofÊ n n and this implies thatÊ k n+1 ⊂ V + 2n+1 \ V + 2n+2 = φ mn n (V 2n+3 ) as required. We still need to prove that E k n+1 is contained in the convex hull of φ nn+1−1 n+1 (c) ∈ V + 2n+2 and we claim that this point is not in E k n+1 . Indeed, the image of E k n+1 by f tn+1,k n is (p n − 1, f n (c) + 1). Then, if the claim were not true, the interval (p n − 1, c) would contain points from the forward orbit of the critical point which is not true.
To prove Part 4 of the lemma we consider k > n+1 and by the same reasoning as above, the point φ mn+1 n+1 (c) is in E n+1 n+1 , but not in E k n+1 . Otherwise the interval (p n − 1, c) would contain points from the forward orbit of the critical point, which is not true.
If we replaceÊ k n+1 byV n+1 and use the same reasoning above we conclude that the collection {V n+1 , . . . , f rn+1 n (V n+1 )} have multiplicity of intersection equal to two. Indeed, by definition, r n+1 = m n r n + t n, n ,V n+1 ⊂ V 2n+1 and f rn+1 n . This implies that the collection {V n+1 , . . . , f mnrn n (V n+1 ) =Ê n n } has pairwise disjoint intervals, all of them contained in V 2n+1 ∪ · · · ∪ f mnrn n (V 2n+1 ). The intervals of the collection {f n (Ê n n ), . . . , φ m1 1 • · · · • φ mn−1 n−1 (Ê n n ) =Ê n 1 } are also pairwise disjoint and they have empty intersection with the intervals from the previous collection. If we put all these intervals together we get that the intervals of the piece of orbit fromV n+1 to φ m1 1 •· · ·•φ mn n (V n+1 ) =Ê n 1 are pairwise disjoint. Now, adding the pairwise disjoint intervals f n (Ê n 1 ), . . . , f t 1, n n (Ê n 1 ) we get the collection {V n+1 , . . . , f rn+1 n (V n+1 )} and conclude it has multiplicity of intersection equal to two as required. 4. The limit set of the critical point. In this section we study the ω-limit set ω(c) of the critical point of a critical covering map f ∈ H(ω). The fact that the sequence ( n ) ∞ n=1 is strictly increasing means that ω(c) contains the repelling fixed point p. So ω(c) is not a minimal set, since it has a proper non-trivial invariant subset {p}. We now prove that ω(c) is a Cantor set of zero Lebesgue measure.
Theorem 4.1. If f is a critical covering map which satisfies hypotheses (H1)-(H3) and has kneading sequence in K, then the ω-limit set of its critical point is a Cantor set of zero Lebesgue measure.
For a better understanding of the set ω(c) and the proof of Theorem 4.1, observe that the symbol 2 does not occur anywhere in sequences in K. This means that the positive orbit of c does not visit the arc of the circle corresponding to [p, c + 1]. Therefore, it will follow that ω(c) is a Cantor set. A more refined study of the relative sizes of the pre-images of this arc will permit us to prove that the Lebesgue measure of this Cantor set is zero and conclude the proof. In this stage we will need to use some Koebe Principles, that may be found, for example, in [13]. They are our main tool for controlling the distortion of iterates of a function, and we state them in the following lemma. 2. min{|L|, |R|} ≥ α|J|.
Remark 3. With the notation of Lemma 4.2 we define the distortion of h in J, that is: We remark that, if α in Lemma 4.2 grows to infinity, then the distortion of h in J decreases to 1.

4.1.
Measure of the limit set of the critical point. To prove that the ω-limit set of the critical point has Lebesgue measure zero, part of Theorem 4.1, we use a well-known procedure of inducing to get rid of the critical point. We consider the function T : and W 0 = V + 2 , such that the coding of the forward orbit of f (x), for x ∈ W 0 , start with the symbol 0. For n ≥ 1 and y ∈ W n ⊂ W , the coding of the forward orbit of f (y) begins with the word 1 n 0. We define T at x ∈ W n by putting T (x) = f n+1 (x). The restriction of T to W n is a branch of the first entry map to W . In particular, the restriction of T to W 0 coincides with φ 1 , which coincides with f restricted to W 0 . Lemma 4.3. Consider the function T : ∞ n=0 W n → W associated to a critical covering map f ∈ H(ω) that satisfies hypotheses (H1)-(H3) for some kneading sequence ω ∈ K. Consider also the set Ω consisting of the points x ∈ W for which T n (x) is well-defined, for all n ≥ 1. Then Ω is a Cantor set of zero Lebesgue measure.
Proof. We denote by T n the branch of T that corresponds to its restriction to the interval W n and we begin by listing some properties of these branches: (i) by definition, for n ≥ 0, we have W n+1 ⊂ [W n , W n+2 ] ⊂ W ; (ii) the left endpoint of W 0 is the critical point c, T 0 is the restriction to W 0 of f and coincides with the branch φ 1 of the first entry map from V + 2 to W = V + 1 . By Lemma 3.3, there is an interval W 0 = V 1 around W 0 which is mapped f homeomorphically (with only one critical point, the point c) onto the interval (p − 1, f (c) + 1); (iii) for n ≥ 1, T n is the restriction to W n of f n+1 and coincides with the branch σ 1,n of the first entry map from E k 1 to W = V + 1 . By Lemma 3.3, there is an interval W n = E k 1 ⊂ W which is mapped by f n+1 diffeomorphically onto the interval (p − 1, f (c) + 1); (iv) for z ∈ [p + , z + 1 ], the coding of the forward orbit of f (z) includes the symbol 2, and so the forward orbit of the critical point doesn't visit this interval and the map T isn't defined in it. We are going to substitute the function T , which has a critical point at c, for another function T that has no critical points. This function is induced and obtained through an inductive process that generates a sequence (T n ) ∞ n=1 of functions T n that possess a critical branch (with critical point c) defined on intervals that nest down to c as n ≥ 1 increases. The sequence (T n ) ∞ n=1 has a limit T that has an infinite number of branches that map their respective domains diffeomorphically onto W . In the initial step of this induction we put T 1 = T . This function T 1 has infinitely many branches that are diffeomorphisms of their respective domains onto W and a single branch with a critical point at c and defined onŴ 1 = W 0 . Then we compose this critical branch with the diffeomorphic branches. In this way we obtain new branches that are diffeomorphisms onto W and a new critical branch that is defined on an intervalŴ 2 ⊂Ŵ 1 . Continuing in this way we obtain a sequence (T n ) ∞ n=1 and a sequence of nested closed intervals (Ŵ n ) ∞ n=1 with the following properties: (i) ∞ n=1Ŵ n = {c}; (ii) T n has diffeomorphic branches mapping onto W , except for a single critical branch with critical point c and defined on the intervalŴ n ; (iii) the branches of T n have extensions mapping diffeomorphically onto (p − 1, f (c) + 1), except for the critical branch, which has an extension mapping homeomorphically onto (p − 1, f (c) + 1) with critical point c; (iv) outside of the intervalŴ n we have that T n+1 = T n ; (v) the sequence (T n ) ∞ n=1 has a unique limit, which has no critical points and is the function T described above.  Each branch of T is a diffeomorphism that coincides with some iterate f t that maps a neighbourhood of the domain of a branch of T diffeomorphically onto (p − 1, f (c) + 1). Then, by the Koebe Principle in Lemma 4.2, the branches of T and their iterates are diffeomorphisms onto W and have uniformly bounded distortion. Because of this, together with the fact that the complement of the domain of T in W contains the interval [p + , z + 1 ], we can conclude that the set Ω of points x ∈ W such that T n (x) is defined for all n ≥ 1, is a Cantor set and has zero Lebesgue measure and the lemma is complete. Proof of Theorem 4.1. It is clear that ω(c) is a set that is closed and totally disconnected. As c is recurrent, we also have that ω(c) is perfect. Thus ω(c) is a Cantor set. To conclude that the Lebesgue measure of ω(c) is zero, we observe that the forward orbit of the critical point has empty intersection with the interval [p + , z + 1 ]. Here p + ∈ [c, z + 1 ] is the point for which f (p + ) = f (p) = p. Then f (c) ∈ Ω and W ∩ ω(c) ⊂ Ω and, by Lemma 4.3, the Lebesgue measure of W ∩ ω(c) is zero. So we conclude that the Lebesgue measure of ω(c) is zero, proving the theorem. 5. Absence of acip measures. As we have already observed, a critical covering map f has many invariant measures and this is the topic of this section. We know that under our hypotheses (H1)-(H3), with the absence of periodic attractors and non-degenerate intervals consisting of periodic points, f is topologically conjugate to the uniform covering map l = l 2 by a homeomorphism h : S 1 → S 1 : that is, h • f = l • f . As l preserves Lebesgue measure on the circle, denoted by λ, we can define the measure h * λ by setting h * λ(B) = λ(h(B)) for any Borel subset B ⊂ S 1 . This measure h * λ is always invariant for f and we could expect that it would be Figure 4. The graph of T An important step in our proof of the non-existence of an acip measure is the following lemma which, in particular, implies that the Lebesgue measure of rn i=1 f i (V 2n ) tends to zero as n → ∞, where r n = |ω n |. Lemma 5.2. Given K 1 < ∞ as in Lemma 5.1, if n ≥ 1, ω n is a finite word as before and f n ∈ H(ω n ), then In particular the Lebesgue measure of rn i=0 f i n (V 2n ) tends to zero when n grows to infinity.
Proof. One reason for this lemma to be true is the fact that, for 1 ≤ i ≤ r n , we can pull back the space V + 1 \V + 2n−1 on the side of V + 2n−1 to a space on the side of f i n (V 2n ) maintaining the proportion up to a uniform distortion constant. To be more precise observe that Lemma 3.3 implies that there are intervals V 2n ⊂V n ⊂ V n such that, for 1 ≤ i ≤ r n , the map f rn−i n is a diffeomorphism from f i n (V 2n ) ⊂ f i n (V n ) ⊂ f i n (V n ) onto V + 2n−1 ⊂ V + 1 ⊂ (p n − 1, f n (c) + 1), where p n is unique fixed point of f n . By Lemma 4.2 the distortion of f rn−i n on f i n (V n ) is uniformly bounded by K 1 and Taking the sum with 1 ≤ i ≤ r n we get that By Lemma 3.3, the multiplicity of intersection of the collection {V n , . . . , f rn n (V n )} is at most two. Therefore To follow the same ideas as Johnson in [9], we observe that as set out in Lemma 5.2, the Lebesgue measure of the union rn i=1 f i n (V 2n ) tends to zero as n tends to infinity. On the other hand, since the interval V 2n contains the critical point, using the theorem of Mañé (see [12] and [13]) it is easy to conclude that, for every n fixed, the Lebesgue measure of N i=0 f −i n (V 2n ) tends to 1, when N tends to infinity. Therefore an arbitrarily small set is visited by an arbitrarily large set. Moreover, we would like to have the property that the set of points of N i=0 f −i n (V 2n ) that leave rn i=0 f i n (V 2n ) in fewer than N iterates could be made arbitrarily small by increasing m n . For this to hold, we will substitute the interval V 2n by the interval U n ⊂ V 2n . First we consider the point γ n ∈ V 2n such that the derivative Dφ n (γ n ) is equal to 1. Then we define U n to be the connected component of V 2n \ {γ n } which contains c. For the map G n from the Corollary 2 or for the map g b(n) from Lemma 3.2, there is a parabolic periodic attractor and m n = ∞. In this case the point γ n is the parabolic periodic attractor.
The next lemma guarantees that the size of the interval U n has a positive lower bound that is uniform for every f n ∈ H(ω n ) with ω n fixed. In particular, U n does not degenerate to a point when we keep 1 , . . . , n−1 , m 1 , . . . , m n−1 fixed and