Nonuniformly Expanding 1D Maps

This paper attempts to make accessible a body of ideas surrounding the following result: Typical families of (possibly multi-model) 1-dimensional maps passing through “Misiurewicz points” have invariant densities for positive measure sets of parameters. The simplest paradigms of chaotic behavior in dynamical systems are found in uniformly expanding and uniformly hyperbolic (or Anosov) maps. Allowing expanding and contracting behaviors to mix leads to a multitude of new possibilities. In spite of much progress, the analysis of most nonuniformly hyperbolic systems has remained hopelessly difficult. One-dimensional maps are an exception. The situation in 1 dimension is made tractable by the fact that the worst enemy of expansion is the critical set, i.e., the set on which f ′ vanishes, and for typical 1D maps, this set is finite. It has been shown that by controlling the orbits starting from this finite set, the dynamics on the rest of the phase space can be tamed. Anonuniform theory for 1D maps was developed in a series of papers in the late 1970s and 1980s ([J, M, CE, BC1, R, BC2, NS], and others). These ideas are later exploited in the study of attractors with a single direction of instability, beginning with the Hénon maps ([BC2, BY] etc.) and culminating recently in a general theory of rank-one attractors that can live in phase spaces of arbitrary dimensions [WY2]. In the course of these developments, some of the original 1D arguments have been extended and improved. This paper is written in response to numerous requests from the dynamical systems community to make more accessible a certain body of ideas in 1 dimension, both for its independent interest and as an introduction to the study of rank-one maps in higher dimensions. The content of this paper can be summarized as follows: Let I be a closed interval or a circle, and let C2(I, I ) denote the set of C2 maps from I to itself. We seek to identify a reasonably large class of maps G ⊂ C2(I, I ) with controlled nonuniform expansion, The research of both authors is partially supported by grant from the NSF Q. Wang, L.-S. Young and to give a description of its dynamical properties. This is carried out in the following 3 steps: (1) First we identify a set M ⊂ C2(I, I ) defined by strong expanding conditions. (2) Our class of “good maps" G ⊂ C2(I, I ) is obtained by relaxing these conditions. Maps in G are shown to have absolutely continuous invariant measures. (3) We show that G is “large" in the sense that for every typical 1-parameter family {fa} passing through M, the set {a : fa ∈ G} has positive Lebesgue measure. We first cite the main references directly related to (1)–(3): The class M is a slight generalization of the maps studied in [M]. In the special case of the quadratic family fa(x) = 1−ax2, the existence of absolutely continuous invariant measures for a positive measure set of parameters is the well known theorem of Jakobson [J]; for other proofs of Jakobson’s theorem, see [BC1, BC2] and [R]. A key idea used in [BC1] and [BC2], namely the exponential growth of derivatives along critical orbits, is first introduced in [CE]. An analysis along the lines of (1)–(3) above for unimodal maps was carried out in [TTY]. A version of Jakobson’s theorem for multimodal maps is given in [T]. While the results of this paper as stated have not been published before, we do not claim that the ideas of the proofs are new. In outline, our proofs follow those in [BC1] and Sect. 2 of [BC2]. The generalization from fa(x) = 1 − ax2 to more general maps is along the lines of [TTY]. We have also borrowed heavily from [WY1] and especially [WY2], both in terms of setting and the way in which the arguments are carried out. More detailed references are given at the end of each section. Organization of paper. The class M in (1) above is discussed in Sect. 1; the class G is introduced in Sect. 2. The result on invariant measures (Theorem 1) is stated and proved in Sect. 3. The result on positive measure sets of parameters (Theorem 2) is stated in Sect. 4.1 and proved in Sects. 4–7. Part I. Dynamical Properties 1. The Class M 1.1. Definition and expanding property. For f ∈ C2(I, I ), let C = C(f ) = {f ′ = 0} denote the critical set of f , and let Cδ denote the δ-neighborhood of C in I . For x ∈ I , let d(x, C) := minx̂∈C |x − x̂|. Definition 1.1. We say f ∈ C2(I, I ) is in the class M if the following hold for some δ0 > 0: (a) Outside of Cδ0 : there exist λ0 > 0,M0 ∈ Z+ and 0 < c0 ≤ 1 such that (i) for all n ≥ M0, if x, f (x), · · · , f n−1(x) ∈ Cδ0 , then |(f n)′(x)| ≥ e0; (ii) if x, f (x), · · · , f n−1(x) ∈ Cδ0 and f (x) ∈ Cδ0 , any n, then |(f n)′(x)| ≥ c0e 0. (b) Inside Cδ0 : (i) f ′′(x) = 0 for all x ∈ Cδ0 ; (ii) for all x̂ ∈ C and n > 0, d(f (x̂), C) ≥ δ0; (iii) for all x ∈ Cδ0 \ C, there exists p0(x) > 0 such that f j (x) ∈ Cδ0 for all j < p0(x) and |(f p0(x))′(x)| ≥ c−1 0 e 1 300. Nonuniformly Expanding 1D Maps Remark 1. The maps in M are among the simplest with nonuniform expansion: The phase space is divided into two regions, Cδ0 and I \ Cδ0 . Condition (a) in Definition 1.1 says that on I \ Cδ0 , f is essentially uniformly expanding. (b)(iii) says that for x ∈ Cδ0 \ C, even though |f ′(x)| is small, the orbit of x does not return to Cδ0 again until its derivative has regained a definite amount of exponential growth; i.e., if n is the first return time of x ∈ Cδ0 to Cδ0 , then |(f n)′(x)| ≥ e 1 30. (To see this, use (b)(iii) followed by (a)(ii).) Remark 2. We identify two properties of the critical orbits of f ∈ M that will serve as the basis of the generalization in Sect. 2. Let x̂ ∈ C. (1) d(f (x̂), C) ≥ δ0 for all n > 0, i.e., (b)(ii) in Definition 1.1. (This condition is redundant and is included solely for emphasis; it follows from (b)(iii) together with the observation that p0(x) → ∞ as d(x, C) → 0.) (2) |(f n)′(f x̂)| ≥ c′ 0eλn for all n > 0, where c′ 0 = (max |f ′|)−M0 . This follows from (b)(ii) and (a)(i). We record for future use the following important fact about the behavior of f ∈ M outside of Cδ for arbitrary δ < δ0: Lemma 1.1. There exists c′′ 0 > 0 depending only on f such that for all δ < δ0 and n > 0: (a) if x, f (x), . . . , f n−1(x) ∈ Cδ , then |(f n)′(x)| ≥ c′′ 0δe 1 30; (b) if x, f (x), . . . , f n−1(x) ∈ Cδ and f (x) ∈ Cδ0 , then |(f n)′(x)| ≥ c0e 1 30. Proof. Let x be such that f (x) ∈ Cδ for i ∈ [0, n). We divide [0, n] into maximal time intervals [i, i + k] such that f i+j (x) ∈ Cδ0 for 0 < j < k, and estimate |(f k)′(f i(x))| as follows: Case 1. f (x), f i+k(x) ∈ Cδ0 . |(f k)′(f i(x))| ≥ e 1 3λ0k by Definition 1.1(a)(ii) and (b)(iii). Case 2. f (x) ∈ Cδ0 , f i+k(x) ∈ Cδ0 . The estimate is given by Definition 1.1(a)(ii). Case 3. f (x), f i+k(x) ∈ Cδ0 . If k ≥ M0, then |(f k)′(f i(x))| > e0 by Definition 1.1(a)(i). If k < M0, we let k̂ be the smallest integer > k such that f i+k̂(x) ∈ Cδ0 . Using Definition 1.1(a)(i) for k̂ ≥ M0 and Definition 1.1(a)(ii) for k̂ < M0, we conclude that |(f k)′(f i(x))| > c0(max |f ′(x)|)−M0eλ0k . Case 4. f (x) ∈ Cδ0 , f i+k(x) ∈ Cδ0 . As in Case 3, with extra factor (miny∈Cδ0 |f ′′ (y)|)δ. Cases 3 and 4 are relevant only for part (a); each appears at most once in the estimate on |(f n)′(x)|. In the interest of carrying as few constants around as possible, we write c1 = min{c0, c′ 0, c′′ 0}. 1.2. Examples. Example 1. Let f ∈ C3(I, I ) be such that (i) S(f ) < 0, where S(f ) denotes the Schwarzian derivative of f ,1 1 We have elected to replace this condition by an explicit description of the dynamics in Definition 1.1 because (1) that is exactly what is used and (2) we have found that maps that arise in applications often do not have negative Schwarzian derivative. Q. Wang, L.-S. Young (ii) f ′′(x̂) = 0 for all x̂ ∈ C, (iii) if f (x) = x, then |(f n)′(x)| > 1, and (iv) for all x̂ ∈ C, infn>0d(f (x̂), C) > 0. Then f ∈ M. For a proof of this fact, see Lemma 2.5 of [WY1]. We note that (i) and (ii) above are satisfied by all members of the quadratic family fa(x) = 1 − ax2, a ∈ (0, 2], and (iii) and (iv) are satisfied by an uncountable number of a including a = 2. Example 2. Another situation where maps in M arise naturally is through scaling. The following is a slight generalization of Lemma 5.3 in [WY3] and has the same proof: Let fa : S1 → S1 be given by fa(θ) = θ + a + L (θ), where a, L ∈ R and : S1 → S1 is an arbitrary function with nondegenerate critical points (and the right side is to be read mod 1). Then there exists L0 > 0 such that for all L ≥ L0, there exists an O( 1 L)-dense set of a for which fa ∈ M. References: Maps of the type in Example 1 are introduced and studied in [M]. Maps of the type in Example 2 appear naturally in [WY3] and [WY4]. 2. The Class G: 3 Basic Properties Condition (b)(ii) in Definition 1.1 severely limits the scope ofM as a subset ofC2(I, I ). We now introduce in a neighborhood of each f0 ∈ M an admissible set of perturbations G(f0). Our set of “good maps” G is then defined to be ⋃ f0∈M G(f0). Throughout this section, let f0 ∈ M be fixed, and let δ0, λ0, c1 etc. be the constants in Sect. 1.1 associated with f0. 2.1. Definition of G(f0) and basic properties. For λ, α, ε > 0 and f ∈ C2(I, I ), we say f ∈ G(f0; λ, α, ε) if ‖f − f0‖C2 < ε and the following hold for all x̂ ∈ C = C(f ) and n > 0: (G1) d(f (x̂), C) > min{ 2δ0, e−αn}; (G2) |(f n)′(f (x̂))| ≥ c1e. Note that with λ < λ0, (G1) and (G2) are relaxations of the conditions on critical orbits for f0 (see Remark 2 in Sect. 1.1). The main result of this section is Proposition 2.1. Given f0 ∈ M, λ < 4λ0 and α < 1 100λ, there exists δ = δ(f0, λ, α) and ε = ε(f0, λ, α, δ) > 0 such that (P1)–(P3) below hold for all f ∈ G(f0; λ, α, ε). Here δ < 2δ0 is an auxiliary constant. For simplicity, we assume ε is small enough that d(f (x̂), C) > 2δ0 for all x̂ ∈ C and 1 ≤ n ≤ n0, where n0 is a large integer satisfying e−αn0 << δ. Consequently, (G1) can be violated only when f (x̂) ∈ Cδ . Precise requirements on δ and ε will become clear in the proofs. In general, ε << δ << 1. The arguments are perturbative; some of them will require that ε be taken very close to 0. The set G(f0) is defined to be the union of G(f0; λ, α, ε) as (λ, α, ε) ranges over all triples satisfying the conditions in Proposition 2.1. We now state (P1)–(P3), introducing some useful language along the way. Nonuniformly Expanding 1D Maps (P1) Outside of Cδ . (i) If x, f (x), . . . , f n−1(x) ∈ Cδ , then |(f n)′(x)| ≥ c1δe 14λ0n; (ii) if x, f (x), . . . , f n−1(x) ∈ Cδ and f (x) ∈ Cδ0 , then |(f n)′(x)| ≥ c1e 1 40. Let x̂ ∈ C, and let Cδ(x̂) := (x̂ − δ, x̂ + δ). For x ∈ Cδ(x̂) \ {x̂}, we define p(x), the bound period ofx, to be the largest integer such that |f i(x)−f i(x̂)| ≤ e−2αi ∀i < p(x). (P2) Partial derivative recovery for x ∈ Cδ \ C. For x ∈ Cδ(x̂) \ {x̂}, (i) 1 3 ln(max |f ′|) log 1 |x−x̂| ≤ p(x) ≤ 3 λ log 1 |x−x̂| ; (ii) |(f p(x))′(x)| > eλ3 . (P2) leads to the following general description of orbits: Decomposition into “bound” and “free” states. For x ∈ I such that f (x) ∈ C for all i ≥ 0 (for example, x = f (x̂) for x̂ ∈ C), let t1 < t1 + p1 ≤ t2 < t2 + p2 ≤ · · · be defined as follows: t1 is the smallest j ≥ 0 such that f j (x) ∈ Cδ . For k ≥ 1, let pk be the bound period of f tk (x), and let tk+1 be the smallest j ≥ tk + pk such that f j (x) ∈ Cδ . (Note that an orbit may return to Cδ during its bound periods, i.e. ti are not the only return times toCδ .) This decomposes the orbit of x into segments corresponding to time intervals (tk, tk + pk) and [tk + pk, tk+1], during which we describe the orbit of x as being in “bound” and “free”states respectively; tk are called times of free returns. (P3) is about comparisons of derivatives for nearby orbits. To state what it means for two points to be close to each other, we introduce a partition P on I . First let P0 = {Iμj } be the following partition on (−δ, δ):Assume δ = e−μ∗ for someμ∗ ∈ Z+. Forμ ≥ μ∗, let Iμ = (e−(μ+1), e−μ); for μ ≤ −μ∗, let Iμ be the reflection of I−μ about 0. Each Iμ is further subdivided into 1 μ2 subintervals of equal length called Iμj . For x̂ ∈ C, let P x̂ 0 be the partition on Cδ(x̂) obtained by shifting the center of P0 from 0 to x̂. The partition P is defined to be P x̂ 0 on Cδ(x̂); on I \ Cδ , its elements are intervals of length ≈ δ. The following shorthand is used: We refer to π ∈ P corresponding to (translated) Iμj intervals in P x̂ 0 simply as “Iμj ”. For π ∈ P , π+ denotes the union of π and the two elements of P adjacent to it. For an interval γ ⊂ I , we say γ ≈ π if π ⊂ γ ⊂ π+. For practical purposes, π+ intersecting ∂Cδ can be treated as “inside Cδ” or “outside Cδ”. For γ ⊂ I+ μj , we define the bound period of γ to be p(γ ) = minx∈I+ μj {p(x)}. For x, y ∈ I , [x, y] denotes the segment connecting x and y. We say x and y in I have the same itinerary (with respect to P) through time n− 1 if there exist t1 < t1 + p1 ≤ t2 < t2 + p2 ≤ · · · ≤ n such that for every k, f tk [x, y] ⊂ π+ for some π ⊂ Cδ , pk = p(f tk [x, y]), and for all i ∈ [0, n) \ ∪k[tk, tk + pk), f [x, y] ⊂ π+ for some π ∈ P with π ∩ Cδ = ∅. (P3) Distortion estimate. There exists K0 > 1 (depending only on f0 and on λ) such that if x and y have the same itinerary through time n− 1, then ∣ ∣ ∣ ∣ (f n)′(x) (f n)′(y) ∣ ∣ ∣ ∣ ≤ K0. (P1)–(P3) are proved in the next subsection. We finish by recording the following corollary of Proposition 2.1. 2 In particular, if π is one of the outermost Iμj in Cδ , then π+ contains an interval of length δ. Q. Wang, L.-S. Young Corollary 2.1. There exists K1 (depending only on f0 and on λ) such that for all x ∈ I with f (x) ∈ C for all 0 ≤ i < n, |(f n)′(x)| > K−1 1 d(f j (x), C) e 1 4, where j is the time of the last free return before n. The factor d(f j (x), C) may be replaced by δ if f (x) is free. Proof. Let 0 ≤ t1 < t1 + p1 ≤ t2 < t2 + p2 ≤ · · · be as in the paragraph following (P2). The derivatives on time intervals [tk, tk + pk) and [tk + pk, tk+1) are given by (P2)(ii) and (P1)(ii) respectively, provided these intervals are completed before time n. We assume δ is sufficiently small so that the constant c1 in (P1)(ii) is absorbed into the exponential estimate from the proceeding bound period. If f (x) is in a bound period initiated at time j , then |(f (n−j))′(f j (x))| ≥ |f ′(f j (x))|K−1 0 c1eλ(n−j−1); see (G2) and (P3). If tk + pk ≤ n < tk+1 for some k, then the derivative between time tk + pk and n is given by (P1)(i). Remarks on the use of constants. In this article, K,K1,K2, . . . , are reserved for use as system constants, which in Part I are constants that are allowed to depend only on (1) f0, by which we included also the constants in Sect. 1.1 associated with f0, and (2) our choice of λ. The more important of these constants, such as K0 in (P3), carry a subscript; all others are referred to by the generic name K . The value of K , therefore, may vary from expression to expression. Notation. Where no ambiguity arises, i.e. when only one map f is involved, we will sometimes write xi = f (x) for i = 1, 2, . . . . 2.2. Proofs of (P1)–(P3). Proof of (P1). First we deduce from Lemma 1.1(a) that there exists N = N(δ) such that for all y ∈ I , if y, f0(y), . . . , f N−1 0 (y) ∈ C 2 δ(f0), then |(f N 0 ) ′(y)| > e 1 3.5λ0N . We then choose ε small enough that f is sufficiently close to f0 for N iterates in the sense below: (i) if x and n are as in (P1) and n ≤ N , then |(f n)′(x) − (f n 0 )′(x)| is small enough that the conclusions of (P1) follow from Lemma 1.1; (ii) if f (y) ∈ Cδ for 0 ≤ i < N , then |(f N)′(y)| > e 4λ0N . If n in (P1) is > N , we let k be such that kN ≤ n < (k + 1)N , and estimate |(f n)′(x)| by the chain rule, comparing (f N)′(f iN (x)) with (f N 0 ) ′(f iN (x)) for i ≤ k using (ii) above, and (f n−kN )′(f (x)) with (f n−kN 0 ) ′(f (x)) using (i). Lemma 2.1. The following holds if δ and ε are sufficiently small and suitably related: Let x̂ ∈ C, and let x ∈ Cδ(x̂). Then for all y ∈ [x̂, x] and k < p(x), 1 2 ≤ (f k)′(y1) (f k)′(x̂1) ≤ 2. Proof. First, log (f k)′(y1) (f k)′(x̂1) ≤ k ∑ j=1 |f ′(yj )− f ′(x̂j )| |f ′(x̂j )| ≤ K k ∑ j=1 |yj − x̂j | d(x̂j , C) . Nonuniformly Expanding 1D Maps We choose h0 large enough that (i) ∑∞ i>h0 e−αj << 1 and (ii) e−αh0 < 2δ0 (so that d(x̂j , C) > e −αj for j > h0). Next we choose δ small enough that δ ∑h0 j=1 2 δ0 (max |f ′|)j << 1. Finally, let ε be small enough that d(x̂j , C) > 2δ0 for all j ≤ h0. Then