Induced expansion for quadratic polynomials.

*(English)*Zbl 0867.58048The 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)].

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)].

Reviewer: V.Jiménez López (Murcia)

##### 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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