Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps

We consider skew-products of quadratic maps over certain Misiurewicz-Thurston maps and study their statistical properties. We prove that, when the coupling function is a polynomial of odd degree, such a system admits two positive Lyapunov exponents almost everywhere and a unique absolutely continuous invariant probability measure.


Introduction
A quadratic polynomial Q c (x) = c − x 2 (1 < c ≤ 2) induces a unimodal map on the interval [c − c 2 , c]. If the critical point 0 is strictly pre-periodic under iteration of Q c , then we say that Q c is Misiurewicz-Thurston. These maps are the simplest non-uniformly expanding dynamical systems, and their properties are well understood. See for example [8]. When considered as a holomorphic map defined on the Riemann Sphere, the complex dynamics generated by Q c is also exhaustedly studied. In this article, we shall make use of its subhyperbolicity as was studied in, for example, [6, §V.4] or [7, §19] .
In Viana [11], Misiurewicz-Thurston quadratic polynomials were used to construct nonuniformly expanding maps in dimension greater than one. He considered the following skew-product G : (R/Z) × R , (θ, y) → (d · θ, Q c (y) + α sin(2πθ)), where d ≥ 16 is an integer, and Q c (1 < c < 2) is Misiurewicz-Thurston. He proved that G has two positive Lyapunov exponents almost everywhere, provided that α > 0 is small enough. The assumption that d ≥ 16 was weakened to d ≥ 2 in [5]. See also [9] for a similar result in non-integer case of d.
In Schnellmann [10], the following skew product was considered: F : [a − a 2 , a] × R , (x, y) → (g(x), Q b (y) + αϕ(x)), where g = Q m 1 a , Q a (1 < a ≤ 2) and Q b (1 < b < 2) are Misiurewicz-Thurston, and m 1 is a large positive integer. For certain coupling function ϕ, with singularity and depending on a, the author proved that F has two positive Lyapunov exponents almost everywhere, provided that α > 0 is small enough.
In this paper, we shall also consider systems in the form of (1), but the coupling function ϕ will be taken to be a nonconstant polynomial independent of a. The main result is the following: Main Theorem. Let Q a (1 < a ≤ 2) and Q b (1 < b < 2) be Misiurewicz-Thurston and let ϕ be a polynomial of odd degree. Assume also that Q a is topologically exact on [a − a 2 , a]. Then there exists a positive integer m 0 = m 0 (a) such that for each positive integer m 1 ≥ m 0 the following holds: For any α > 0 sufficiently small, the map F defined in (1) has two positive Lyapunov exponents. Moreover, F has a unique invariant probability measure that is absolutely continuous with respect to the Lebesgue measure.
Remark. The assumption that ϕ is of odd degree seems quite artificial. Unfortunately, in our argument, this assumption cannot be fully gotten rid of. See the proof of the claim in Lemma 2.6 and the remark after that lemma for details.
We shall use many ideas appearing in the works cited above. In particular, we choose m 0 large enough, so that the map F is partially hyperbolic, i.e. the expansion along the x-direction is stronger than the y-direction. The exact choice of m 0 will be specified in (5), at the end of §2.1.
To better visualize the partial hyperbolicity, we shall introduce a conjugation map u : [a − a 2 , a] → I a in §2.2, so that the conjugated map h 0 = u • Q a • u −1 : I a becomes uniformly expanding. Our construction of u is based on subhyperbolicity of Q a near its Julia set. A similar construction, using instead the absolutely continuous invariant density, was used in [10]. We shall show that u −1 and inverse branches of h 0 have very nice analytic properties, see Lemma 2.3. Then the map F defined in (1) can be conjugate to the following map F via u × id R : where h = h m 1 0 and φ = ϕ • u −1 . We shall prove the equivalent statement of the Main Theorem for F instead of F . An important advantage of this construction is that in the θ-coordinate, the derivative of an admissible curve, i.e., the image of a horizontal curve under iteration of F, can be approximated by an analytic function αT , for some T chosen from a certain compact space of holomorphic maps.
The basic strategy to prove our Main Theorem for the system F is the same as that in previous works [11,5,10]. The main effort is to control recurrence of typical orbits to the critical line y = 0, see Proposition 4.3. This proposition, together with the "Building Expansion" Lemmas in [11], which are summarized in Lemma 2.7, implies that F is nonuniformly expanding in the sense of [3], and hence the results proved therein complete the proof of our Main Theorem. Following Viana's original argument, Proposition 4.3 is reduced to showing that under iteration of F, a horizontal curve becomes non-flat and widely spreads in the y-direction. In our argument, these two properties are obtained in Proposition 3.1 and Lemma 4.1 respectively. Lemma 2.6, serving as an intermediate step, is proved by combining subhyperbolicity of Q a with the super attracting behavior of Q a near infinity.
It remains an interesting open problem whether the statement of our Main Theorem holds for (a, b) chosen from a positive subset of [1,2] × [1, 2).

Preliminaries
2.1. Subhyperbolicity of Q a . Let us review some useful properties of the complex dynamics of the Misiurewicz-Thurston quadratic polynomial Q a (1 < a ≤ 2), which are well discussed in many literatures, see for example [6, §V.3] Denote the post-critical set of Q a by PC a , i.e.
An important feature of Q a as a Misiurewicz-Thurston map is the so-called subhyperbolicity. That is to say, there exists a conformal metric ρ a (z) |d z| around J a (with mild singularities at PC a ), such that with respect to this metric, Q a is uniformly expanding in some neighborhood of J a . We shall give an explicit construction of ρ a (z) |d z| below for further usage. For this purpose, let us start with the following lemma. Because this lemma can be essentially found in [6, §V, Theorem 3.1] or [7, §19], we merely provide an outline of its proof.
Sketch of Proof. Fix q ∈ V a \ PC a as a base point. For each v ∈ PC a , let l v ∈ π 1 (V a \ PC a , q) be represented by a simple closed curve based on q whose winding number around ν is one and whose winding number around ν is 0 for each ν ∈ PC a \ {v}. Then π 1 (V a \ PC a , q) is the free group generated by { l v } v∈PC a . Let G be its normal subgroup generated by {l 2 v } v∈PC a . Then there exists a normal unbranched covering π : D/G → V a \ PC a . For any small disk B(v, r), v ∈ PC a , each connected component of π −1 (B(v, r) \ {v}) is a punctured disk, and the restriction of π to it is a double covering. By filling up the punctured points into D/G, it is easy to verify that π can be extended to a branched covering from some simply connected Riemann surface to V a . Since V a is hyperbolic, we obtain a branched covering π : D → V a . According to the construction of π, assertions (i) and (ii) hold automatically.
Because Q a has a unique critical point 0 with Q a (0) 0, the construction of π exactly guarantees that Q −1 a can be locally lifted to a holomorphic map from D to D with respect to π. Since D is simply connected, the global existence of the lift follows from monodromy theorem. It remains to check the statement about critical points ofτ. Since Q a • π •τ = π and since π (ĉ) = 0 implies that π (ĉ) 0,ĉ is a critical point ofτ if and only ifĉ is a critical point of π andτ(ĉ) is not a critical point of Q a • π. The conclusion follows.
(3) Therefore, ρ a is is a strictly positive analytic function on V a \ PC a , and it has a singularity of order |z − v| − 1 2 at v for each v ∈ PC a . It follows that the function ρ a (Q a (z)) ρ a (z) |(Q a ) (z)| is actually continuous on Q −1 a (V a ). Moreover, by (iii) of Lemma 2.1 and Schwarz lemma, it is strictly larger than 1 on Q −1 a (V a ). In particular, since Q −1 a (V a ) is a compact subset of V a , Here is the right position to specify our choice of m 0 in the statement of the Main Theorem. By definition, m 0 is the minimal positive integer such that By considering the derivative of Q a at its fixed point √ 1+4a− 1 2 , it follows that For some technical reason, we shall also need a constantλ a ∈ (4 1 m 0 , λ a ). To be definite, let Remark. In fact our argument is valid for a smaller m 0 . On the one hand, we can replace λ a with a larger number µ a > 1, which only needs to satisfy that , a], ∀n ≥ 1, for some C a > 0. On the other hand, the upper bound 4 of |Q b | can be replaced by some 1 < R b < 2, which only needs to satisfy that Lemma 3.1] for the existence of R b . Then to guarantee that our argument works, m 0 only needs to satisfy that µ m 0 a > R b , but several details will be more redundant. For simplification, let us work with the assumption λ m 0 a > 4. 2.2. Expanding coordinates. Define where I a := u([a − a 2 , a]). Let h 0 be the conjugated map of Q a : [a − a 2 , a] via u, i.e.
Lemma 2.2. For the notations introduced above, the following statements hold. Proof. Since u = ρ a , by (3), u > 0 on J 0 , and u is real analytic on J 0 with singular points of order 1 2 at both end points. Due to (i) of Lemma 2.1, π 0 onω 0 and π has critical points of local degree 2 at both end points ofω 0 . Therefore, it is easy to see that assertion (i) holds.
Given a bounded interval I ⊂ R and ξ > 0, let B ξ (I) = {z ∈ C : dist(z, I) < ξ}. The following two lemmas are the main results of this subsection, which motivate us to replace g with h = u • g • u −1 as our base dynamics for taking advantage of nice analytical properties of h −1 . Lemma 2.3. There exist ξ > 0 and D ξ,i > 0, ∀i ≥ 1, such that the following statements hold.
Lemma 2.4. There exists C d > 0, such that for any ω n ∈ Q n (n ≥ 1) and any measurable set E ⊂ ω n , we have Proof. Let N = N(a) ≥ 1 be the minimal integer such that for each ω ∈ Q N , ∂ω contains at most one point in h −1 0 (u(PC a )). For each n ≥ 0, denote Q n+N byQ n . It suffices to prove the lemma for {Q n } n≥0 instead of {Q n } n≥0 , because of the reasons below: • there exists an integer M = M(a), such that for each n ≥ 0, every element in Q n is a union of no more than M elements inQ n with a finite set; • once the left inequality in (9) has been proved for {Q n } n≥0 , it follows that there exists p > 0, such that for every n ≥ 0 and every pair (ω,ω) ∈ Q n ×Q n with ω ⊂ ω, we have |ω| ≥ p|ω|.
Let J be the element in Q 0 containing ω 0 . By definition, σ n can be extended to a univalent holomorphic Since (u −1 ) 0 on J and sinceQ 0 is a finite set, there exists δ 0 > 0, independent of ω 0 , such that u −1 can be extended to a univalent holomorphic function on B 2δ 0 (ω 0 ), which satisfies that I 2δ 0 ⊂ J and u −1 (B 2δ 0 (ω 0 )) ⊂ C u −1 (J) . Then σ n • u −1 is univalent on B 2δ 0 (ω 0 ), so by Koebe distortion theorem, the distortion of σ n • u −1 on B δ 0 (ω 0 ) is only dependent on δ 0 . It follows that there exists C > 0, determined by δ 0 only, such than for either component of σ n • u −1 (I δ 0 \ ω 0 ), its length is no less than C · |u −1 (ω n )|, where u −1 (ω n ) = σ n • u −1 (ω 0 ). Also note that σ n • u −1 (I δ 0 ) does not intersect PC a , the singular set of u. Because all the singularity points of u are of square root type, we can conclude that there exists > 0, only dependent on δ 0 , such that u can be extended to a univalent function on B (u −1 (ω n )). Since σ n • u −1 has bounded distortion on B δ 0 (ω 0 ), finally we can find 0 < δ ≤ δ 0 , determined by only, such that τ n = u • σ n • u −1 is univalent on B δ (ω 0 ). Then the conclusion follows from Koebe distortion theorem.
is univalent, and hence the conclusion also follows from Koebe distortion theorem. 3. There exists 1 ≤ i ≤ n, such that 0 ∈ ∂ω i . Then ∂ω j ∩ u(PC a ) consists of a unique point Q i− j a (0), j = 0, 1, . . . , i, and according to the definition ofQ 0 , On the other hand, similar to the discussion in Case 3, is in the situation of Case 1. Then by Koebe distortion theorem, there exists a constant C > 1, only dependent on δ , such that for any measurable sets E i−1 ⊂ ω i−1 and E n ⊂ ω n , Combining the estimates above, (9) follows in this case.
2.3. Backward shrinking of Q a . For further usage, let us give some detailed estimates on the accumulation rate of points in C \ J a to J a under iterated action of Q −1 a . Lemma 2.5. For each m ∈ N, there exists K m > 0, such that the following statement holds. Let z 0 ∈ C \ J a and z i := Q mi a (z 0 ), ∀i ≥ 0. Assume that z n V a for some n ∈ N and let σ be the branch of Q −mn a with σ(z n ) = z 0 . Then there are two cases: Proof. Denote F m : Then F m is a forward invariant compact subset of the Fatou set of Q a . Note that on the Fatou set of Q a , σ can be conjugated to some branch of z → z 2 −mn via the Böttcher coordinate about ∞. In case (i), z i ∈ F m , ∀0 ≤ i ≤ n, and hence the conclusion follows from the compactness of F m . For is bounded from above, so the conclusion in case (ii) follows.

2.4.
A family of functions. From now on, let us fix an arbitrary integer m 1 ≥ m 0 and choose g = Q m 1 a as our base dynamics. As mentioned in the introduction, we shall study F defined in (2) rather than F . Let us summarize some frequently used notations and their basic properties here. • Without loss of generality, assume that |φ| ≤ 1 on I a .
]. Q b maps I b to its interior. • Assume that α is sufficiently small, so that F maps I a × I b into itself. • P n := Q m 1 n , ∀n ≥ 0. i.e. {P n } n≥0 is a nested sequence of Markov partitions of h.
• Denote F n (θ, y) by (h n (θ), f n (θ, y)). Due to (i) of Lemma 2.3 and the definition of F, for every ω ∈ P 0 and every n ≥ 1, f n is real analytic on ω × R. Given a non-constant polynomial ϕ, let us introduce a family of functions as below, which is inspired by considering the α-linear approximation of the derivative of high Fiteration of a horizontal curve. The importance of this family will become clear later. See Lemma 3.2 and Lemma 4.1.
All the useful properties of T are listed in the lemma below. The following notations are used in its statement.
• ω c denotes the unique element in P 0 containing 0.
• ω ± c ⊂ ω c denote the only two elements in Lemma 2.6. Let ϕ be a non-constant polynomial. For each ω ∈ P 0 , the family T ω defined for ϕ, considered as a space of holomorphic functions defined on B ξ (ω), is compact with respect to the compact-open topology. Moreover, (i) There exist A n > 0, n = 0, 1, 2, . . . , such that for each T ∈ T ω , (ii) 0 T ω . More specifically, there exist l 0 ∈ N and B > 0, such that for each T ∈ T ω , (iii) If, additionally, ϕ is of odd degree, then for each T ∈ T ω c and each D ∈ [−4, 4], the two functions are not identical to each other.
Proof. By (8) and (12), functions in the family T ω are uniformly bounded on B ξ (ω). Then according to Montel's theorem, T ω is a pre-compact subset of holomorphic functions on B ξ (ω) with respect to the compact-open topology. On the other hand, in (12), for each n ≥ 1, there are only finitely many choices of τ n , so by the definition of T ω , it is also a closed subset. Therefore, T ω is compact. Assertion (i) follows from the compactness of T ω immediately.
To prove (ii) and (iii), let us change back to the x coordinate from the θ coordinate to make use of the complex dynamics of Q a on the whole Riemann sphere. (ii) and (iii) will be deduced from: Claim. Given J ∈ u −1 (P 0 ), let { σ n } n≥0 be a sequence of real analytic maps defined on J with σ 0 = id J and g • σ n = σ n−1 , ∀n ≥ 1, and let { c n } n≥1 be a sequence of numbers with |c n | ≤ 4 n , ∀n ≥ 1. Then is not identically zero. Moreover, if ϕ is of odd degree and J = u −1 (ω c ) 0, then S is not an odd function on J ∩ (−J).
Proof of Claim. Note that S can be analytically continued to a holomorphic function defined on C J := (C \ R) ∪ J. The basic idea is to show that around ∞, the ϕ term in the series expression of S is dominating. To this end, take an arbitrary z ∈ C J \ V a , and let n z be the minimal integer such that σ n z (z) ∈ Q −1 a (V a ). The existence of n z is guaranteed by the fact that the Fatou set of Q a equals to the attracting basin of infinity, and the backward Q a -invariance of V a implies that σ n (z) Q −1 a (V a ) if and only if n < n z . Then we can apply Lemma 2.5 to z for m = m 1 . Firstly, when n ≤ n z , by (10), Secondly, when n > n z , by (11), Here C > 0 is independent of z or n. Since |c n | ≤ 4 n−1 and 2 m 1 ≥ λ g > 4, combining the two inequalities above, we have: where C > 0 is independent of z. On the other hand, as |z| → ∞, |ϕ (z)| |z| deg ϕ−1 . Therefore, Now we can complete the proof. Since ϕ is non-constant, (18) implies that S cannot be identically zero, i.e. the first statement of the claim holds. When ϕ is of odd degree, |ϕ (z) + ϕ (−z)| is bounded away from zero as |z| → ∞. When 0 ∈ J, C J∩(−J) is a domain containing 0 and symmetric about 0. These facts together with (18) imply that for z ∈ C J∩(−J) , |S (z) + S (−z)| is also bounded away from zero as |z| → ∞. The second statement of the claim follows. Now let us prove (ii) and (iii) with the help of the claim. To start with, let us clarify the relation between the family T and functions in the form of (17). Given ω ∈ P 0 and has the form of (17) and it is automatically well defined on J ⊃ J 1 with J ∈ u −1 (P 0 ).
To prove assertion (ii), for each T ∈ T ω , let . The claim says that S is not identically 0, which implies that 0 T ω . The rest of assertion (ii) follows from this fact and the compactness of T ω easily by reduction to absurdity.
To prove assertion (iii), by reduction to absurdity, for T ± appearing in (16), suppose that . By definition, S has the form of (17), and . By the assumption T + ≡ T − on h(ω ± c ), we have: , where γ − is defined in (13) and γ − = − id on u −1 (ω + c ) . It implies that S is an odd function on J ∩ (−J), which contradicts to the claim and completes the proof.
This choice of η is only used once for proving the first Claim in Proposition 4.2.

Admissible curves
In this section, following previous works, we shall introduce the concept of admissible curves, which are images of horizontal curves under iteration of F. Then we shall study analytical properties of admissible curves and show that they are nearly horizontal but non-flat.
To begin with, let us give some frequently used notations. Given a curve X : I → R defined on some interval I, denote its graph by X, i.e. X = {(θ, X(θ)) : θ ∈ I}. Note that X and X are determined by each other, and by abusing terminology, both of them will be called curves. For h :Î →Ĵ, diffeomorphic, I ⊂Î, J = h(I) and X :Î → R, F( X| I ) denotes the graph of the curve defined on J by h(θ) → f 1 (θ, X(θ)), θ ∈ I. Now we can specify the precise meaning of admissible curves in our situation.

Definition 3.1 (Admissible Curve
). An analytic function X defined on some ω ∈ P 0 is called an admissible cuvrve, if there exist y ∈ I b , n ≥ 1 and ω n ∈ P n , such that for the horizontal curve Y ≡ y, X = F n ( Y| ω n ).
Remark. By definition, if X is an admissible curve, then F n ( X) splits into a union of admissible curves for each n ∈ N.
The main result of this section is: There exist l 0 ∈ N and A > B > 0, such that when α is sufficiently small, any admissible curve X : ω → R satisfies that To prove Proposition 3.1, the basic idea is to approximate the derivative of admissible curves by functions in the family T , which is guaranteed by: Lemma 3.2. For each l ∈ N, there exist C i > 0, i = 0, 1, . . . , l, such that the following statement holds when α is small enough. Let ω ∈ P 0 and let X : ω → I b be an admissible curve. Then there exists T ∈ T ω , such that Moreover, for each ω 1 ∈ P 1 with ω 1 ⊂ ω and each θ 0 ∈ h(ω 1 ), if we denote X 1 = F( X| ω 1 ), τ = (h| ω 1 ) −1 and then T 1 ∈ T h(ω 1 ) and (20) still holds when (X, T, ω) is replaced by (X 1 , T 1 , h(ω 1 )).
Proof. By definition, X is the F n image of a constant curve Y ≡ y for some n ≥ 1 and y ∈ I b . When n = 1, X = αφ • τ 0 + Q b (y) for some inverse branch τ 0 of h −1 defined on ω. Since T := (φ • τ 0 ) ∈ T ω and X = αT , (20) holds automatically in this case. To prove the lemma in full generality, by induction on n, it suffices to prove the following statement: for each l ∈ N, there exist C i > 0, i = 0, 1, . . . , l, such that if X − αT satisfies (20), then for T 1 defined in (21), T 1 ∈ T h(ω 1 ) and For the θ 0 appearing in the statement of the lemma, denote y 0 = X(τθ 0 ) and D = Q b (y 0 ) = −2y 0 . Since |D| ≤ 4, by the definition of T , obviously T 1 ∈ T h(ω 1 ) . Because it follows that To complete the proof, we only need to show that X 1 − αT 1 satisfies (22). To this end, let us show the existence of C i inductively on i. Firstly, when i = 0, by (23), Here and below · denotes the maximum modulus norm of functions. Note that |D| ≤ 4, τ ≤ λ −1 g , T ≤ A 0 (by (14)) and X ≤ X − αT + α T . Therefore, It follows that, if we choose C 0 large, say C 0 = 3A 2 0 λ g −4 , then when α is small, X 1 − αT 1 satisfies (20) for i = 0. Now assume that for some 1 ≤ j ≤ l, C 0 , C 1 , . . . , C j−1 have been chosen, so that X 1 −αT 1 satisfies (22) for 0 ≤ i ≤ j − 1. Let us determine the choice of C j . Differentiating (23) j times, we obtain that Here P j is a linear combination of (X − αT ) (i) • τ, 0 ≤ i ≤ j − 1, and Q j is a homogeneous quadratic polynomial of (X − y 0 ) ( j) • τ, 0 ≤ i ≤ j. For both P j and Q j , their coefficients are polynomials of τ (i) , 1 ≤ i ≤ j + 1. By (8), τ (i) ≤ D ξ,i . By (14), T (i) ≤ A i . Moreover, (X − αT ) (i) ≤ C i α 2 , i = 0, . . . , j − 1. Therefore, on the one hand, there exists M j > 0, such that On the other hand, Substituting the inequalities above into (24), we can conlude that if (X − αT ) ( j) ≤ C j α 2 , then As a result, if we choose C j large, say C j = 2M j + A 0 A j , then when α is small, (20) holds for X 1 − αT 1 with i = j, which completes the induction, and hence the lemma follows.
Now we can deduce Proposition 3.1 from Lemma 2.6 and Lemma 3.2.
Proof of Proposition 3.1. Let l 0 , B be as in Lemma 2.6, let A = 2 l 0 i=0 A i , where A i 's are given in (14), and let C 0 , C 1 , . . . , C l 0 be determined by Lemma 3.2 for l = l 0 . Assume that α is so small that (l 0 + 1)C i α < min{A/2, B}, i = 0, 1, . . . , l 0 and that Lemma 3.2 holds for l = l 0 .
By Lemma 3.2, there exists T ∈ T ω , such that X − αT satisfies (20) for l = l 0 . Due to Lemma 2.6, T satisfies (14) and (15). Then the choice of constants in the previous paragraph guarantees that (19) holds for X.

Critical return
The crucial ingredient in proving that F is non-uniformly expanding is to control the approximation of a typical orbit to the critical set I a × {0}. For this purpose, we shall show that F satisfies the so called slow recurrence conditions in Proposition 4.3. Following [11,5], the key step to deduce these conditions is to prove Proposition 4.2, which is an analogue of [11,Lemma 2.6]   There exist M * ∈ N and 0 > 0, such that the following statement holds when α is sufficiently small. For each admissible curve X defined on ω 0 ∈ P 0 , there exist ω ± ∈ P M * with ω ± ⊂ ω 0 and h M * (ω + ) = h M * (ω − ), such that for Z ± = F M * ( X| ω ± ), sup θ∈dom(Z ± ) Proof. Recall the notations ω c and ω ± c introduced just ahead of Lemma 2.6. Since h is topologically exact on I a , there exists M 0 ∈ N, such that h M 0 (ω 0 ) = I a for each ω 0 ∈ P 0 . In particular, given ω 0 ∈ P 0 , there exists ω ∈ P M 0 , such that ω ⊂ ω 0 and h M 0 (ω) = ω c . Denote F M 0 ( X| ω ) by Y and denote D = Q b (Y(0)). Since Y is admissible, according to Lemma 3.2, there exists T ∈ T ω c , such that Y − αT satisfies (20) for l = 0.
Let Z ± = F( Y| ω ± c ). Then both of Z ± are admissible curves defined on h(ω ± c ). Therefore, for T ± ∈ T h(ω ± c ) defined in (16) with T and D given in the previous paragraph, by applying Lemma 3.2 again, one can see that (Z ± ) − αT ± also satisfy (20) for l = 0. According to (iii) of Lemma 2.6 and the compactness of T h(ω ± c ) , there exists 1 > 0, independent of T ± , such that By (20), (Z ± ) − αT ± ≤ C 0 α 2 . Therefore, when α is sufficiently small, it follows that . Then it is easy to see that (26) holds for M * = M 0 + 1 and suitable choice of 0 .
The proposition below is the substitution of [11,Lemma 2.6] or [5,Proposition 5.2] with the same idea of proof. We shall follow the proof in [5]. The main change of ingredients here is to replace [5,Corollary 5.4] and [5, Lemma 5.5] with Corollary 3.3 and Lemma 4.1 in this paper respectively. Denote Note that for N α introduced in Lemma 2.7, we have M α ≤ 2 5 N α . Proposition 4.2. There exists β 0 > 0, such that when α is sufficiently small, for each admissible curve Y defined on ω 0 ∈ P 0 and each r ≥ r 0 (α), we have Proof. When r ≥ 1 2 + 2η log 1 α , √ αe −r ≤ α 1+2η . Since F M α ( Y) splits into a union of admissible curves, the conclusion follows from Corollary 3.3 and Lemma 2.4 immediately by choosing β 0 appropriately. Otherwise, it suffices to consider the case r = r 0 (α) and let us follow the argument in [5]. Without loss of genarality, we can assume that there exists Let us summarize some basic estimates of distortion in [5] with slight modification. Let y i = Q i b (y 0 ), i ≥ 0. Recall the constant A in (19) and let L = max{1, A|I a |}. Denote Following [5], let us define some notations and constants. Firstly, let σ = √ σ and introduce Then obviously λ j λ j+1 = |Q b ( f j (z 0 ))| σ < 4, 0 ≤ j < M α − 1. Secondly, recalling the constant M * appearing in Lemma 4.1, let κ > 4 M * be a constant satisfying that Thirdly, let 0 = t 1 < t 2 < · · · < t q ≤ M α be all the times such that By definition, Following Claim 1 in the proof of [5, Proposition 5.2], we have: Claim. When α is sufficiently small, k 0 (α) ≥ ηr 0 (α)/ log(2κ).
Proof of Claim. Given ω i ∈ Ω i , Y i := F t i ( Y| ω i ) is an admissible curve. According to Lemma 4.1, there exist ω ± ∈ P M * with ω ± ⊂ h t i (ω i ) and Denote t i+1 − t i − M * by n temporarily. By definition, given θ ∈ dom(Y ± i+1 ), there exists θ 0 ∈ J, such that h n (θ 0 ) = θ. Due to (29), Then (32) and the two inequalities above imply that For t i+1 ≤ j ≤ M α , define By (33) and noting that |(Y ± i+1 ) | ≤ Aα, we have: (34) As in the proof of [5,Proposition 5.2], denote D j = min y∈B j |Q b (y)|. Since By (29) and (30) , Noting that λ j < λ t i+1 when j > t i+1 , combing the two inequalities above, it is easy to obtain that Substituting (34) into the inequality above and making use of λ t i+1 ≥ 2κe −r 0 (α) / √ α and the choice of κ, we have: That is to say, ω ± i+1 cannot both intersect E, i.e. the claim holds. Now let us return to the proof of the proposition. Let N ≥ 2M α /k 0 (α) be a large integer independent of α. Then By Lemma 2.4, there exists p > 0, only depending on N, such that for each pair ω ⊂ ω with ω ∈ P n , ω ∈ P n , n < n ≤ n + N, we have |ω | ≥ p |ω|. By the second claim, it follows that when t i+1 ≤ t i + N, Then by (31), |E| ≤ e −β 0 r 0 (α) for some constant β 0 > 0, which completes the proof.

4.2.
Slow recurrence conditions. To make the argument slightly simpler, let us adopt the following weakened forms of Corollary 3.3 and Proposition 4.2. There exists 0 < β ≤ min{ 1 2l 0 , β 0 }, such that when α is small enough, for each admissible curve X : ω → R, we have: • If ≤ α 2 , then Based on (35) and (36), following the "Large deviations" argument in [11], we can deduce the following version of slow recurrence conditions. Proposition 4.3. There exists c > 0, such that when α is sufficiently small, the following statement holds. For each ε > 0, there existsδ =δ(ε) ∈ (0, 1 2 ) , independent of α, such that when n ∈ N is sufficiently large, (θ, y) ∈ I a × I b : where δ =δ α 1−2η . In particular, for Lebesgue a.e. (θ, y) ∈ I a × I b , Remark. It is well known that, starting from the estimate shown in (37), kinds of statistical properties beyond the Main Theorem can be obtained. See, for example [1], for a survey on this topic.
Fix ε > 0. For K ∈ N, let By Fubini's theorem, Proposition 4.3 can be reduced to the following lemma.

Lemma 4.4.
When α is small enough, the following statement holds. Given ε > 0, there exist 0 <δ < 1 2 , independent of α or y 0 , and K 0 ∈ N, independent of y 0 , such that when K > K 0 , Proof. Let Y denote the constant curve Y ≡ y 0 . We shall make use of the admissibility of pieces of F n ( Y) for appropriate n repeatedly.
Given K ∈ N, let For each k ≥ 1 and each ω ∈ P k , X ω := F k ( Y| ω ) is an admissible curve, and Therefore, when K is large, say K > 4 log α log λ g 2 , by (35) together with (9), It follows that, when K is large, Given K ≥ 4M 2 α /ε 2 , to estimate the size of E 1 K , let us denote L := 2[ √ K], and for every 0 ≤ p ≤ L − 1, define Claim. Given ε > 0, there exists 0 <δ < 1 2 , such that when K is large(independent of y 0 ), Proof of Claim. Since each nonzero r k is no less than ∆, where the first binomial coefficient counts the position of nonzero r k 's, and the second counts the distribution of their values. By Stirling's formula, R −1 log R [R/∆] → 0 uniformly in R as ∆ → ∞. Therefore, by the definition of ∆, whenδ is sufficiently small, for 1 ≤ i ≤ [R/∆], we have: Since #M p ≈ Noting that #M p < R/ε, in this case the conclusion also follows from Stirling's formula, provided thatδ is small, and ∆/ε is large accordingly.
By combining the last two claims, we have: It follows that, for K large, which, together with (39), completes the proof of the lemma.

Proof of the main theorem
To prove the Main Theorem, it suffices to prove the same statements for F instead of F . Recall that, when α is small enough, F maps I a × I b into itself. Apparently the interesting dynamics of F is concentrated on the invariant set Λ = ∞ n=0 F n (I a × I b ), because for (θ, y) outside Λ, f n (θ, y) → ∞ as n → ∞.
Let us follow some terminology in [3]. To start with, note that the base dynamics h : I a → I a is a C 2 local diffeomorphism outside a finite set C θ of singularities. Let S θ = C θ × I b and S y = I a × {0}. Then F is a C 2 local diffeomorphism outside S = S θ ∪ S y , the so called singular set in [3], and the conditions (S1)-(S3) in [3] hold for β = 1. The following subsections are devoted to showing that F is nonuniformly expanding in the sense of [3].

5.2.
Existence of a.c.i.p. We shall apply the results in [3, Theorem C] to obtain the existence of an a.c.i.p. for F. By definition, .
We have checked that all the conditions of [3, Theorem C] are satisfied provided that α is small enough. Thus F has an absolutely continuous invariant measure.

5.3.
Uniqueness of a.c.i.p. As shown in Lemma 6.1 of [4], Λ = F n (I a × I b ) when n ≥ 2. By a similar argument as in [4,Proposition 6.2], it is easy to prove that F is topologically exact on Λ. Moreover, by [3,Lemma 5.6], up to a set of zero Lebesgue measure, the basin of each a.c.i.p. of F n contains some disk. Therefore the a.c.i.p. of F is unique.