Nonuniformly Expanding 1D Maps With Logarithmic Singularity

Let fa,L : R→ R be such that (1) fa,L : θ 7→ θ + a + L ln |Φ(θ)| where a ∈ [0, 1], L ∈ R are real parameters and Φ(θ) is such that Φ(θ + 1) = Φ(θ). We assume that Φ(θ) is a Morse function and the graph of y = Φ(θ) is transversal to the θ-axis. The functions fa,L induce a two parameter family of 1D maps from S1 to S1 where S1 = R/Z is the unit circle. In this paper we prove that there exists an L0 > 0 sufficiently large, so that for every L satisfying |L| > L0, there exists a set ∆(L) of positive measure for a, such that for a ∈ ∆(L), fa.L admits an invariant measure that is absolutely continuous with respect to the Lebesgue measure. We also prove that |∆(L)| → 1 as |L| → ∞. If Φ(θ) 6= 0 for all θ ∈ S1, then this result (minus the asymptotic estimate on |∆(L)|) follows from combining the theory of [WY1] on multi-modal 1D maps with the proof of [WY2] on the existence of multi-modal Misiuriewicz maps. When Φ(θ) = 0 is allowed, however, no previous theory on 1D maps apply and the result of this paper is new. Our study of fa,L in (1) is motivated by the recent studies of [WO], [WOk] and [W] on homoclinic tangles and strange attractor in periodically perturbed differential equations. When a homoclinic solution of a dissipative saddle is periodically perturbed, the perturbation either pulls the stable and the unstable manifold of the saddle fix point completely apart, or it creates chaos through homoclinic intersections. In both cases, the separatrix map induced by the solutions of the perturbed equation in the extended phase space is a family of 2D maps with a singular 1D limit in the form of (1) (with the absolute value sign around Φ(θ) removed). Let μ be a small parameter representing the magnitude of the perturbation and ω be the forcing frequency. We have a ∼ ω lnμ−1 mod(1), L ∼ ω; and Φ(θ) is the classical Melnikov function (See [WO] and [WOk]). When we start with two unperturbed homoclinic loops and assume symmetry, then the separatrix maps are a family of 2D maps, the 1D singular limit of which is precisely fa,L in (1) (See [W]). If the stable and unstable manifolds of the perturbed saddle are pulled completely apart by the forcing function, then Φ(θ) 6= 0 for all θ. In this case we obtain strange attractors, to which the theory of rank one maps developed in [WY3] apply. If the stable and unstable manifold intersect, then Φ(θ) = 0 is allowed and the strange attractors are associated to homoclinic intersections. For the modern theory of chaos and dynamical systems, this is a case of historical and practical importance; see [GM], [SSTC1], [SSTC2]. To this case, unfortunately, the theory of rank one maps in [WY3] does not apply because of the existence of logarithmal singularities in fa,L. Our ultimate goal is to develop a theory that can be applied to the separatrix maps allowing Φ(θ) = 0. This paper is the first step, in which we develop a 1D theory. We now present our main result in precise terms. For f = fa,L, let C(f) = {f ′(θ) = 0} be the set of critical points and S(f) = {Φ(θ) = 0} be the set of singular points. The distances from θ ∈ S1 to C(f) and S(f) are denoted as dC(θ) and dS(θ) respectively. Assumptions on Φ(θ): We assume that Φ : R → R is C2 satisfying (i) Φ(θ) = Φ(θ + 1); (ii) Φ′(θ) 6= 0 on {Φ(θ) = 0}; and (iii) Φ′′(θ) 6= 0 on {Φ′(θ) = 0}. We also need a few constructive parameters and they are λ, α, N0, σ and δ: