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Induced expansion for quadratic polynomials. (English) Zbl 0867.58048
The present paper deals with a class $$\mathcal F$$ of maps defined as follows. A map $$f:[-1,1]\to [-1,1]$$ is said to be unimodal if $$f(-1)=-1$$ and there is an orientation-reversing diffeomorphism $$h:[0,1]\to [-1,a]$$ (with $$0<a<1$$) such that $$f(x)=h(x^2)$$. Further, let $$\eta>0$$. We say that a unimodal map $$f(x)=h(x^2)$$ belongs to the class $${\mathcal F}_\eta$$ if $$h^{-1}$$ has a conformal extension which maps the upper half-plane into the lower half-plane, and $$h$$ has a real analytic extension to some open interval $$U\supset [0,1]$$ as a diffeomorphism onto $$(-1-\eta,1+\eta)$$. Finally we define $${\mathcal F}=\bigcup_{\eta>0}{\mathcal F}_\eta$$. All maps from $${\mathcal F}$$ have non-positive Schwarzian derivative. In particular, $$\mathcal F$$ includes the family of quadratic polynomials $$f_a(x)=a-(a+1)x^2$$.
The main results of the paper are Theorems 1 and 2 below.
Theorem 1. Let $$f\in {\mathcal F}_\eta$$ be non-renormalizable (that is, there is no proper subinterval $$I$$ of $$[-1,1]$$ containing 0 with the properties $$f^n(I)\subset I$$ and $$f^n(\partial I)\subset \partial I$$ for some $$n>1$$). Then $$f$$ is expansion inducing, that is, there are an open, dense and having full measure subset $$O$$ of $$(-q,q)$$ and a map $$F:O\to (1-q,q)$$ satisfying the following properties (here $$q$$ is the only fixed point of $$f$$ in $$[0,1]$$):
(i) If $$O_i$$ is a connected component of $$O$$ then $$F|_{O_i}=f^{n_i}|_{O_i}$$ for some $$n_i$$ and $$F(O_i)=(1-q,q)$$.
(ii) $$F$$ is expanding and has bounded distortion, which means that there are constants $$K>1$$ and $$D>0$$ such that $$|F'(x)|>K$$ and $$|(\log F')'(x)|<D$$ for any $$x\in O$$. Moreover, $$D$$ depends only on $$\eta$$.
Theorem 2. Let $$f\in {\mathcal F}_\eta$$ be renormalizable and let $$I\ni 0$$ be a maximal subinterval of $$[-1,1]$$ such that $$f^n(I)\subset I$$ and $$f^n(\partial I)\subset \partial I$$ for some $$n>1$$. Define a point $$x\in [-1,1]$$ to be almost parabolic with period $$m$$ and depth $$k$$ provided that the derivative of $$f^m$$ at $$x$$ is one, $$f^m$$ is monotone between $$x$$ and $$0$$, and $$k$$ consecutive images $$f^m(0),\dots,f^{km}(0)$$ are between $$x$$ and $$0$$, and denote by $$k(n)$$ the maximum of depths of almost parabolic points with periods less than $$n$$. Specify a number $$D>0$$. Then, for every given $$k$$, there is a number $$N(\eta,D,k)$$ not depending on $$f$$ so that if $$n>N(\eta,D,k)$$ and $$k(n)\leq k$$, then $$f^n|_I$$ is conjugate to a map from $${\mathcal F}_D$$ via a linear transformation.
Theorem 1 implies that if $$f\in {\mathcal F}$$ is non-renormalizable then the $$\omega$$-limit set of almost all points from $$[-1,1]$$ is the whole interval $$[f^2(0),f(0)]$$; this result was also previously proved (in the more general setting of maps with non-positive Schwarzian derivative whose only critical point $$c$$ satisfies $$f''(c)\neq 0$$) by M. Lyubich [Ann. Math., II. Ser. 140, No. 2, 347-404 (1994; Zbl 0821.58014)]. Theorem 2 is used by the second author [‘Hyperbolicity is dense in the real quadratic family’, to appear in Ann. Math.], where it is shown that the set of parameters $$a$$ such that the polynomial $$f_a(x)=a-(a+1)x^2$$ has a hyperbolic periodic attractor is dense in $$[-1,1]$$. Both Theorems 1 and 2 are consequences of a technical Theorem C, where, roughly speaking, so-called “decaying of box geometry” is demonstrated for a special type of maps called box mappings. The proof uses a mixture of real and complex tools.
While the significance of the results of the paper is obvious, in the reviewer’s opinion they are presented in a somewhat involved way (besides, there are some misprints: in the paper it is written “upper half-plane” instead of “lower half-plane” in the definition of $${\mathcal F}_\eta$$, and $$1/2$$ instead of $$0$$ in Theorem 2). Hence an interested but non-truly specialist reader may also wish to check the introduction to M. Jakobson and G. Świątek [Ergodic Theory Dyn. Syst. 14, No. 4, 721-755 (1994; Zbl 0830.58019)] (which is generalized by the paper under review) and Chapter V from the monograph by W. de Melo and S. van Strien [‘One-dimensional dynamics’. Berlin: Springer-Verlag (1993; Zbl 0791.58003)].

##### MSC:
 37D99 Dynamical systems with hyperbolic behavior 37F99 Dynamical systems over complex numbers 37A99 Ergodic theory 37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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