Posted by: matheuscmss | May 29, 2016

## Yoccoz proof of Jakobson theorem I

Almost twenty years ago, Jean-Christophe Yoccoz gave several lectures (at ETH and Collège de France, for instance) on a proof of Jakobson theorem (based on the so-called Yoccoz puzzles), and he distributed hard copies of his lecture notes whenever they were requested (for example, he gave me such a copy when I became his post-doctoral student in 2007).

More recently, he gave a couple of lectures at Collège de France aiming to generalize his proof of Jakobson theorem to obtain some results improving upon the Wang-Young theory of rank one attractors.

Of course, his lectures (whose videos are available at his website) started by recalling several elements of his proof of Jakobson theorem and I decided to take the opportunity to write a series of posts about Jean-Christophe’s proof of Jakobson theorem (based on his hand-written lecture notes from 1997).

For the first installment of this series, we’ll review some aspects of the dynamics of the one-dimensional quadratic maps and, after that, we’ll state Jakobson’s theorem, describe the three main steps of Yoccoz proof of Jakobson theorem and implement the first step of the strategy.

1. Introduction

The dynamics of one-dimensional affine maps ${F_{a,b}(z)=az+b}$ is fairly easy understand. The change of variables provided by the translation ${T_c(z)=z+c}$ transforms ${F_{a,b}}$ into ${(T_c^{-1}\circ F_{a,b}\circ T_c)(z) = az + ((a-1)c+b)}$. If ${a\neq 1}$, the choice ${c=-b/(a-1)}$ shows that the affine map ${F_{a,b}}$ is conjugated to its linear part ${F_{a,0}(z)=az}$. If ${a=1}$, ${F_{a,b}(z)=z+b}$ is a translation.

In other words, the dynamics of polynomial maps ${P:\mathbb{C}\rightarrow\mathbb{C}}$ is not very interesting when ${\textrm{deg}(P)=1}$. On the other hand, we shall see below that quadratic maps are already sufficiently complicated to produce all kinds of rich dynamical behaviors.

The quadratic family is ${P_c(z)=z^2+c}$ where ${z\in\mathbb{C}}$ and ${c\in\mathbb{C}}$.

Remark 1 This family is sometimes presented in the literature as ${c-z^2}$ or ${cz(1-z)}$. As it turns out, these presentations are equivalent to each other: indeed, the affine change of variables ${h(z)=Az+B}$ converts ${P_c}$ into

$\displaystyle (h^{-1}\circ P_c\circ h)(z) = Az^2+2Bz+\frac{B^2+c-B}{A}$

The dynamics of ${P_c}$ near infinity is easy to understand: in fact, since ${P_c(z)\approx z^2}$ for ${|z|\gg 1}$, one has that ${\infty}$ attracts the orbit ${\{P_c^n(z)\}_{n\in\mathbb{N}}}$ of any ${z\in\mathbb{C}}$ with ${|z|\gg 1}$.

This means that the interesting dynamics of ${P_c}$ occurs in the filled-in Julia set:

$\displaystyle K(c):=\{z\in\mathbb{C}: P_c^n(z)\not\rightarrow\infty \textrm{ as } n\rightarrow\infty\}$

Note that ${K(c)}$ is totally invariant, that is, ${P_c(K(c)) = K(c) = P_c^{-1}(K(c))}$. Also, ${K(c)}$ is a compact set because ${|z|>R}$ implies that ${|P_c(z)|>R|z|-|c|>(R-\frac{|c|}{R})|z|}$ and ${R-\frac{|c|}{R}>1}$ whenever ${R>\frac{1+\sqrt{1+4|c|}}{2}:=R_c}$, so that

$\displaystyle K(c)=\bigcap\limits_{n\in\mathbb{N}} P_c^{-n}(\mathbb{D}_{R_c}(0))$

(where ${\mathbb{D}_R(0):=\{z\in\mathbb{C}: |z|\leq R\}}$).

Moreover, ${K(c)\neq\emptyset}$ because it contains all periodic points of ${P_c}$ (i.e., all solutions of the algebraic equations ${P_c^n(z)=z}$, ${n\in\mathbb{N}}$).

Remark 2 ${K(c)}$ is a full compact set, i.e., ${\mathbb{C}-K(c)}$ is connected: indeed, this happens because the maximum principle implies that a bounded open set ${U\subset \mathbb{C}}$ with boundary ${\partial U\subset K(c)}$ must be completely contained in ${K(c)}$ (i.e., ${U\subset K(c)}$).

The dynamics of ${P_c}$ on ${K(c)}$ is influenced by the behaviour of the orbit ${\{P_c^n(0)\}_{n\in\mathbb{N}}}$ of the critical point ${0\in\mathbb{C}}$. More precisely, let us consider the Mandelbrot set

$\displaystyle M:=\{c\in\mathbb{C}:P_c^n(0)\not\rightarrow\infty\}:=\{c\in\mathbb{C}: 0\in K(c)\}$

Indeed, we shall see in a moment that there is a substantial difference between the dynamics of ${P_c}$ on ${K(c)}$ depending on whether ${c\notin M}$ or ${c\in M}$.

Remark 3 If ${|c|>2}$, then one can check by induction that ${|P_c^n(0)|\geq |c|(|c|-1)^{2^{n-1}}}$ for all ${n\geq 1}$. Thus, the Mandelbrot set is the compact set

$\displaystyle M=\bigcap\limits_{n\in\mathbb{N}} \{c\in\mathbb{C}: |P_c^n(0)|\leq 2\}$

Also, the maximum principle implies that ${M}$ is full (i.e., ${\mathbb{C}-M}$ is connected).Furthermore, the Mandelbrot set is symmetric with respect to the real axis: ${c\in M}$ if and only if ${\overline{c}\in M}$.

Moreover, ${M}$ intersects the real axis at the interval ${[-2,1/4]}$: indeed, we already know that ${M\cap\mathbb{R} \subset [-2,2]}$; on the other hand, the critical orbit ${\{P_c^n(0)\}_{n\in\mathbb{N}}}$ is trapped between the fixed point ${\beta_c=\frac{1+\sqrt{1-4c}}{2}}$ (solving the equation ${z^2+c=z}$) and its preimage ${-\beta_c}$ under ${P_c}$ for ${c\in[-2,1/4]}$, so that ${[-2, 1/4]\subset M\cap\mathbb{R}}$; finally, the fixed point ${\frac{1+\sqrt{1-4c}}{2}}$ moves away from the real axis for ${c>1/4}$ and this allows for the critical orbit ${\{P_c^n(0)\}_{n\in\mathbb{N}}}$ to escape to infinity, so that ${(1/4,2]\cap M=\emptyset}$.

Let us analyze the dynamics of ${P_c}$ on ${K(c)}$. For this sake, let us take ${V_0}$ a sufficiently large open disk centered at the origin ${0\in\mathbb{C}}$ (and containing the critical value ${c=P_c(0)}$). The preimage ${V_1=P_c^{-1}(V_0)}$ is a topological open disk whose closure ${\overline{V_1}}$ is contained in ${V_0}$, and ${P_c:V_1-\{0\}\rightarrow V_0-\{c\}}$ is a degree two covering map.

By induction, we can define inductively ${V_{n}=P_c^{-1}(V_{n-1})}$. We have two possibilities:

• (a) if the critical value ${c}$ belongs to ${V_n}$, then ${V_{n+1}=P_c^{-1}(V_n)}$ is a topological disk whose closure is contained in ${V_n}$, and ${P_c:V_{n+1}-\{0\}\rightarrow V_n-\{c\}}$ is a degree two covering map;
• (b) if ${c\in V_{n-1}-V_n}$ for some ${n\in\mathbb{N}}$, then ${P_c^{-1}(V_n)}$ is the union of two topological disks ${U_0}$ and ${U_1}$ whose closures ${\overline{U}_0, \overline{U}_1\subset V_n}$ are disjoint, and ${P_c|_{U_0}}$ and ${P_c|_{U_1}}$ are univalent maps with images ${V_n}$; in particular, by Schwarz lemma, ${P_c^{-1}:V_n\rightarrow U_j}$, ${j=0, 1}$, are (uniform) contractions with respect to the Poincaré (hyperbolic) metrics on ${U_j}$ and ${V_n}$, the filled-in Julia set

$\displaystyle K(c)=\{z\in U_0\cup U_1: P_c^n(z)\in U_0\cup U_1 \, \forall \, n\in\mathbb{N}\}$

is a Cantor set, and the dynamics of ${P_c}$ on ${K(c)}$ is conjugated to a full unilateral two-shift ${\sigma:\{0,1\}^{\mathbb{N}}\rightarrow\{0,1\}^{\mathbb{N}}}$, ${\sigma((x_n)_{n\in\mathbb{N}}) = (x_{n+1})_{n\in\mathbb{N}}}$, via ${H:K(c)\rightarrow\{0,1\}^{\mathbb{N}}}$, ${H(z) = (x_n)_{n\in\mathbb{N}}}$ where ${P_c^n(z)\in U_{x_n}}$.

Remark 4 Observe that, in item (b) above, we got analytical information (${P_c:V_n\rightarrow U_j}$ is an uniform contraction) from topological information (${U_j}$ is an open topological disk with ${\overline{U_j}\subset V_n}$) thanks to some tools from Complex Analysis (namely, Schwarz lemma). This type of argument is a recurrent theme in one-dimensional dynamics.

Note that the situation in item (b) occurs if and only if the critical orbit ${\{P_c^n(0)\}_{n\in\mathbb{N}}}$ escapes to infinity, i.e., ${c\notin M}$. Otherwise, the sequence of nested topological disks ${V_n}$ are defined for all ${n\in\mathbb{N}}$ and ${K(c)=\bigcap\limits_{n\in\mathbb{N}} V_n}$.

In summary, the dynamics of ${P_c}$ on ${K(c)}$ falls into the classical realm of hyperbolic dynamical systems when ${c\notin M}$ (cf. item (b)).

From now on, we shall focus on the discussion of ${P_c|_{K(c)}}$ for ${c\in M}$. Actually, we will talk exclusively about real parameters ${c\in M\cap\mathbb{R} = [-2, 1/4]}$ in what follows.

3. Statement of Jakobson’s theorem

Suppose that ${P_c}$ has an attracting periodic orbit ${O(z_0)=\{z_0,\dots, P_c^{m-1}(z_0)\}}$ of period ${m}$, i.e., ${P_c^m(z_0)=z_0}$ and ${|(P_c^m)'(z_0)|<1}$.

It is possible to show that the basin of attraction

$\displaystyle W=W(O(z_0))=\{z\in\mathbb{C}: \lim\limits_{n\rightarrow\infty} d(P_c^n(z),O(z_0)) = 0\}$

must contain the critical point ${0\in\mathbb{C}}$. Thus, given ${c\in\mathbb{C}}$, there exists at most one attracting periodic orbit ${O(z_0)\subset\mathbb{C}}$ of ${P_c}$.

Remark 5 If ${P_c}$ has an attracting periodic orbit ${O(z_0)}$, then ${P_c}$ is hyperbolic in ${K(c)}$. Indeed, this follows from the same argument in item (b) above: the basin of attraction ${W(O(z_0))}$ is the interior of ${K(c)}$ and the boundary ${J(c):=\partial K(c)}$ (called Julia set in the literature) is disjoint from the closure of the critical orbit ${T=\{P_c^n(0)\}_{n\in\mathbb{N}}}$; thus, ${J(c)}$ is contained in the open connected set ${S=\mathbb{C}-T}$ with ${P_c^{-1}(S)\subset S}$, so that the univalent map ${P_c^{-1}:S\rightarrow R}$ from ${S}$ to a component ${R}$ of ${P_c^{-1}(S)}$ is a local contraction with respect to the Poincaré metric ${\|.\|_P}$ on ${S}$; since ${J(c)}$ is a compact subset of ${S}$, it follows that there exists ${\lambda>1}$ such that ${\|D_zP_c(v)\|_P\geq \lambda\|v\|_P}$ for all ${z\in J(c)}$ and ${v\in T_z\mathbb{C}}$, i.e., ${P_c}$ is uniformly expanding on ${J(c)}$.

Denote by ${A}$ the set of parameters ${c\in M}$ such that ${P_c}$ has an attracting periodic point. The hyperbolicity of ${P_c}$ on ${K(c)}$ for ${c\in A}$ permits to show that ${A}$ is contained in the interior of the Mandelbrot set ${M}$.

The so-called hyperbolicity conjecture of Douady and Hubbard in 1982 asserts that ${A}$ coincides with the interior of ${M}$. This conjecture is one of the main open problems in this subject (despite partial important progress in its direction).

Nevertheless, for real parameters ${c\in M\cap\mathbb{R} = [-2, 1/4]}$, we have a better understanding of the dynamics of ${P_c}$ for most choices of ${c}$.

More precisely, it was conjectured by Fatou that ${A\cap\mathbb{R}}$ is dense in ${M\cap\mathbb{R}=[-2, 1/4]}$. This conjecture was independently established by Lyubich and Graczyk-Swiatek in 1997: they showed that hyperbolicity of ${P_c}$ is open and dense property in real parameter space ${c\in\mathbb{R}}$. In particular, this gives an extremely satisfactory description of the dynamics of ${P_c}$ for a topologically large set of parameters ${c\in\mathbb{R}}$.

Of course, it is natural to ask for a measure-theoretical counterpart of this result: can we describe the dynamics of ${P_c}$ for Lebesgue almost every ${c\in\mathbb{R}}$?

It might be tempting to conjecture that ${A\cap\mathbb{R}}$ has full Lebesgue measure on ${M\cap\mathbb{R}=[-2, 1/4]}$. Jakobson showed already in 1981 that this naive conjecture is false: ${A\cap\mathbb{R}}$ does not have full Lebesgue measure on ${M\cap\mathbb{R}}$.

Theorem 1 (Jakobson) Let ${\Lambda\subset [-2, 1/4]}$ be the subset of parameters ${c}$ such that

• there exists ${\lambda(c)>0}$ such that ${\lim\limits_{n\rightarrow\infty}\frac{1}{n}\log |(P_c^n)'(x)|=\lambda(c)}$ for Lebesgue almost every ${x\in [-\beta_c, \beta_c]}$ (where ${\beta(c)=\frac{1+\sqrt{1-4c}}{2}}$);
• ${P_c}$ preserves a probability measure ${\mu}$ which is absolutely continuous with respect to the Lebesgue measure on ${[-\beta_c, \beta_c]}$.

Then, the Lebesgue measure of ${\Lambda}$ is positive and, in fact, ${\Lambda}$ has density one near ${-2}$:

$\displaystyle \lim\limits_{\varepsilon\rightarrow 0}\frac{Leb(\Lambda\cap [-2,-2+\varepsilon])}{\varepsilon}=1$

(where ${Leb}$ stands for the Lebesgue measure).

Nevertheless, Lyubich showed in 2002 that ${(A\cap\mathbb{R})\cup\Lambda}$ has full Lebesgue measure on ${M\cap[-2, 1/4]}$, so that we get a satisfactory description of the dynamics of ${P_c}$ for Lebesgue almost every parameters ${c\in\mathbb{R}}$.

4. Overview of Yoccoz proof of Jakobson theorem

In 1997 and 1998, Jean-Christophe Yoccoz gave a series of lectures at Collège de France where he explained a proof of Jakobson theorem based on some combinatorial objects for the study of ${P_c}$ called Yoccoz puzzles (see, e.g., the first section of this paper here by Milnor).

Our goal in this series of posts is to describe Yoccoz’s proof of Jakobson theorem by following Yoccoz’s lecture notes (distributed by him to those who asked for them) recently submitted for publication (after this announcement here).

Very roughly speaking, Yoccoz proof of Jakobson’s theorem has three steps. First, one introduces the notion of regular parameters ${c\in[-2,0)}$ and one shows that ${P_c}$ satisfies the conclusion of Jakobson theorem whenever ${c}$ is a regular parameter. Unfortunately, it is not so easy to estimate the Lebesgue measure of the set ${\mathcal{R}}$ of regular parameters. For this reason, one introduces a notion of strongly regular parameters and one proves that strongly regular parameters are regular (thus justifying the terminology). Finally, one transfers the estimates on “Yoccoz puzzles” in the phase space to the parameter space (via certain analogs of Yoccoz puzzles in parameter space called “Yoccoz parapuzzles”): by studying how the real traces of Yoccoz puzzles move with the parameter ${c}$, one completes the proof of Jakobson theorem by showing that

$\displaystyle \lim\limits_{\varepsilon\rightarrow 0}\frac{Leb(\{c\in[-2,-2+\varepsilon]: c \textrm{ is strongly regular}\})}{\varepsilon} = 1$

In the remainder of this post, we will implement the first step of this strategy (i.e., the introduction of regular parameters and the derivation of the conclusion of Jakobson theorem for regular parameters via arguments from Thermodynamical Formalism).

4.1. Preliminaries

${P_c}$ has two real fixed points ${\beta=\frac{1+\sqrt{1-4c}}{2}}$ and ${\alpha=\frac{1-\sqrt{1-4c}}{2}}$ for ${c<1/4}$. Note that ${\beta}$ is repulsive for all ${c<1/4}$, while ${\alpha}$ is attractive, resp. repulsive for ${-3/4, resp. ${c<-3/4}$. Moreover, the real filled-in Julia set ${K_{\mathbb{R}}(c):= K(c)\cap\mathbb{R}}$ equals ${[-\beta,\beta]}$ for all ${c\in M\cap\mathbb{R}=[-2, 1/4]}$.

The dynamical features of ${P_c}$ are determined by the (recurrence) properties of the critical orbit ${\{P_c^n(0)\}_{n\in\mathbb{N}}}$. For this reason, we consider ${c\in[-2,0)}$ and we will analyze the returns of the critical orbit to the central interval

$\displaystyle A:=[\alpha, -\alpha]$

and its neighborhood ${\widehat{A}:=(\alpha^{(1)}, -\alpha^{(1)})}$ where ${P_c^{-1}(\alpha):=\{\alpha^{(1)}, -\alpha^{(1)}\}}$, ${-\beta<\alpha^{(1)}<\alpha}$. (Observe that these definitions make sense because the fixed points ${\alpha}$ and ${\beta}$ satisfy ${-\beta<\alpha<0}$ for ${c\in[-2,0)}$.)

Definition 2 A point ${x\in A}$ is ${n}$-regular if there exists ${0 and an open interval ${\widehat{J}\ni x}$ such that ${P_c^m(x)\in A}$ and ${P_c^m|_{\widehat{J}}: \widehat{J}\rightarrow \widehat{A}}$ is a diffeomorphism.

Definition 3 A parameter ${c\in[-2,0)}$ is regular if most points in ${[-\beta,\beta]}$ are regular, i.e., there are constants ${C>0}$ and ${\theta>0}$ such that

$\displaystyle Leb(\{x\in A: x \textrm{ is not } n\textrm{-regular}\})\leq Ce^{-\theta n}$

for all ${n\in\mathbb{N}}$. The set of regular parameters is denoted by ${\mathcal{R}}$.

As we already said, our current goal is to show the following result:

Theorem 4 If ${c\in\mathcal{R}}$ is a regular parameter, then there exists ${\lambda(c)>0}$ such that

$\displaystyle \lim\limits_{n\rightarrow\infty}\frac{1}{n}\log|(P_c^n)'(x)|=\lambda(c)$

for Lebesgue a.e. ${x\in[-\beta,\beta]}$, and ${P_c}$ admits an invariant probability measure which is absolutely continuous with respect to the Lebesgue measure.

4.2. The “tent map” parameter ${c=-2}$

Before proving Theorem 4, let us warm-up by showing that the parameter ${c=-2}$ is regular and, moreover, ${P_{-2}}$ satisfies the conclusions of this theorem.

The polynomial ${P_{-2}(x)=x^2-2}$ is conjugated to the tent map

$\displaystyle F(y)=\left\{\begin{array}{cc} 2y & \textrm{if } 0\leq y<1/2 \\ 2(1-y) & \textrm{if } 1/2

This conjugation can be seen by interpreting ${P_{-2}}$ as a Chebyshev polynomial:

$\displaystyle P_{-2}(2\cos\theta) = 2\cos(2\theta).$

The hyperbolic features displayed by ${P_{-2}}$ (described below) are related to the fact that the critical point ${0}$ is pre-periodic: ${P_{-2}(0)=-2=-\beta}$ and ${P_{-2}^n(0)=\beta}$ for all ${n>1}$, so that the critical orbit does not enter the central interval ${A=[\alpha,-\alpha]=[-1,1]}$ (and, thus, it is not recurrent).

The relationship between ${P_{-2}}$ and the tent map allows us to organize the returns of the orbits of ${P_{-2}}$ to ${A=[-1,1]}$ as follows. Let ${\alpha:=\alpha^{(0)}:=\widetilde{\alpha}^{(1)}:=-1}$ and define recursively ${\alpha^{(n)}<0}$ and ${\widetilde{\alpha}^{(n)}<0}$ as

$\displaystyle P_{-2}(\alpha^{(n)}) = -\alpha^{(n-1)} \quad \textrm{and} \quad P_{-2}(\widetilde{\alpha}^{(n)}) = \alpha^{(n-1)} \quad \forall \, n>0.$

Note that the interpretation of ${P_{-2}}$ as a Chebyshev polynomial says that

$\displaystyle \alpha^{(n)}=-2\cos\frac{\pi}{3\cdot 2^n} \quad \textrm{and} \quad \widetilde{\alpha}^{(n+1)} = -2\sin\frac{\pi}{3\cdot 2^{n+1}}$

In particular, ${(\alpha^{(n)})_{n\in\mathbb{N}}}$ is a decreasing sequence converging to ${-2}$ and ${(\widetilde{\alpha}^{(n)})_{n\in\mathbb{N}}}$ is an increasing sequence converging to ${0}$.

In terms of these sequences, the return map ${R}$ of ${P_{-2}}$ to ${A}$ is not difficult to describe:

• recall that the critical point ${0}$ does not return to ${A}$ (because ${P_{-2}^n(0)=\beta=2}$ for all ${n>1}$);
• the points ${\pm\alpha}$ at ${\partial A}$ return after one iteration: ${P_{-2}(\pm\alpha)=\alpha}$;
• for ${n>1}$, the return map ${R}$ on the intervals ${(\widetilde{\alpha}^{(n-1)},\widetilde{\alpha}^{(n)}]}$ and ${[-\widetilde{\alpha}^{(n)}, -\widetilde{\alpha}^{(n-1)})}$ is precisely ${P_{-2}^n}$; moreover, ${R=P_{-2}^n}$ is an orientation-preserving, resp. orientation-reversing, diffeomorphism from ${I_n^-:=[\widetilde{\alpha}^{(n-1)},\widetilde{\alpha}^{(n)}]}$, resp. ${I_n^+:=[-\widetilde{\alpha}^{(n)}, -\widetilde{\alpha}^{(n-1)}]}$, onto ${A}$, and ${P_{-2}^n(\pm\widetilde{\alpha}^{(n)})=-\alpha}$, ${P_{-2}^n(\widetilde{\alpha}^{(n-1)})=\alpha}$;
• for ${n>1}$, ${P_{-2}^n}$ extends into diffeomorphisms from neighborhoods of ${I_n^{\pm}}$ to ${(-\beta,\beta) (\supset\widehat{A}:=[\alpha^{(1)},-\alpha^{(1)}])}$.

The graph of the return map ${R}$ of ${P_{-2}}$ to ${A}$ looks like two copies of the Gauss map ${x\in (0,1]\mapsto\{1/x\}\in [0,1]}$ (where ${\{.\}}$ denotes the fractional part): it is an instructive exercise to draw the graph of ${R}$.

At this point, we are ready to establish the regularity of ${c=-2}$.

Proposition 5 The parameter ${c=-2}$ is regular.

Proof: From the previous description of the return map ${R}$, we see that, for ${n>1}$, the set of ${n}$-regular points is precisely

$\displaystyle [\alpha, \widetilde{\alpha}^{(n)}]\cup[-\widetilde{\alpha}^{(n)}, -\alpha]$

Therefore,

$\displaystyle \{x\in A: x \textrm{ is not } n\textrm{-regular}\} = (\widetilde{\alpha}^{(n)}, -\widetilde{\alpha}^{(n)}),$

so that ${c=-2}$ is a regular parameter because, for ${\theta=\log 2}$ and some constant ${C>0}$, one has

$\displaystyle Leb(\{x\in A: x \textrm{ is not } n\textrm{-regular}\}) = 2|\widetilde{\alpha}^{(n)}| = 4\sin\frac{\pi}{3\cdot 2^n}\leq Ce^{-\theta n}$

for all ${n\in\mathbb{N}}$. $\Box$

Next, we show that ${P_{-2}}$ satisfies the conclusion of Jakobson theorem.

Proposition 6 ${P_{-2}}$ preserves the probability measure ${\mu=\frac{1}{\pi\sqrt{4-x^2}}dx}$ on ${[-2,2]}$ and

$\displaystyle \lim\limits_{n\rightarrow\infty} \frac{1}{n}\log|(P_{-2}^n)'(x)|=\log 2$

for Lebesgue a.e. ${x\in [-2,2]}$.

Proof: The reader can easily check (either by direct computation or by inspection of the conjugation equation between ${P_{-2}}$ and the tent map) that

$\displaystyle |P_{-2}'(x)| = 2\frac{h(x)}{h(P_{-2}(x))} \quad \textrm{for all } x\in(-2,2)-\{0\},$

where ${h(x):=1/\sqrt{4-x^2}}$ and, hence,

$\displaystyle |(P_{-2}^n)'(x)|=2^n\frac{h(x)}{h(P_{-2}^n(x))} \ \ \ \ \ (1)$

for all ${n\in\mathbb{N}}$ and ${x\in(-2,2)}$ with ${P_{-2}^n(x)\in(-2,2)}$). It follows that ${P_{-2}}$ preserves the probability measure ${\mu:=\frac{1}{\pi}h(x)dx}$ on ${[-2,2]}$.

So, our task is reduced to compute the Lyapunov exponent of ${P_{-2}}$. Since Lebesgue a.e. ${x\in [-2,2]}$ eventually enters the central interval ${A=[-1,1]}$ and the normalized restriction ${\mu_A:=\frac{1}{\mu(A)}\mu|_{A} = 3\mu|_{A}}$ is absolutely continuous with respect to the Lebesgue measure, it suffices to prove that

$\displaystyle \lim\limits_{n\rightarrow\infty} \frac{1}{n}\log|(P_{-2}^n)'(x)| = \log 2$

for ${\mu_{A}}$-a.e. ${x\in A}$.

Denote by ${N(x)=\min\{n>0: P_{-2}^n(x)\in A\}}$ the first return time of ${x}$, so that ${R(x)=P_{-2}^{N(x)}(x)}$ (by definition).

Note that ${\mu_A}$ is an ergodic ${R}$-invariant probability measure such that

$\displaystyle \begin{array}{rcl} \mu_A(\{x\in A: N(x)=n\}) &=& 3\int_{I_n^-\cup I_n^+}\frac{dx}{\pi\sqrt{4-x^2}} = \frac{6}{\pi}\int_{-\widetilde{\alpha}^{(n)}}^{-\widetilde{\alpha}^{(n-1)}}\frac{dx}{\sqrt{4-x^2}}\\ &=& \frac{6}{\pi}\int_{\pi/3\cdot 2^n}^{\pi/3\cdot 2^{n-1}}d\theta = 2^{1-n} \end{array}$

Therefore, ${N}$ is a ${\mu_A}$-integrable function, so that Birkhoff’s theorem says that

$\displaystyle \lim\limits_{k\rightarrow\infty}\frac{N_k(x)}{k}\rightarrow\int_A N(x)d\mu_A = \sum\limits_{n=2}^{\infty} n 2^{1-n} = 3 \ \ \ \ \ (2)$

for ${\mu_A}$-a.e. ${x}$, where ${N_k(x):=\sum\limits_{j=0}^{k-1} N(R^j(x))}$.

This asymptotic information on the return times permits to compute the Lyapunov exponent of ${P_{-2}}$ as follows.

Since the ${P_{-2}}$-orbit of a point ${x\in[-2,2]}$ is confined to ${[-2,2]}$ and ${\sup\limits_{y\in[-2,2]}|P_{-2}(y)|=4}$, we see from the chain rule that

$\displaystyle |(P_{-2}^{N_{k+1}(x)})'(x)|+(n-N_{k+1}(x))\log 4\leq |(P_{-2}^n)'(x)|\leq |(P_{-2}^{N_k(x)})'(x)| + (n-N_k(x))\log 4$

for all ${N_k(x)\leq n\leq N_{k+1}(x)}$. By combining this estimate with (2), we obtain

$\displaystyle \lim\limits_{n\rightarrow\infty}\frac{1}{n}|(P_{-2}^n)'(x)| = \lim\limits_{k\rightarrow\infty} \frac{1}{N_k(x)}|(P_{-2}^{N_k(x)})'(x)|$

for ${\mu_A}$-a.e. ${x}$.

In other words, it suffices to compute the Lyapunov exponent along the subsequence of return times. As it turns out, this task is not difficult. Indeed, from (1), we have

$\displaystyle \log|(P_{-2}^{N_k(x)})'(x)| = N_k(x)\cdot\log 2 + \log\frac{h(x)}{h(P_{-2}^{N_k(x)}(x))}$

Since ${h(z)=1/\sqrt{4-z^2}\in [1/2, 1/\sqrt{3}]}$ for ${z\in A=[-1,1]}$, we conclude that

$\displaystyle \frac{1}{N_k(x)}\log|(P_{-2}^{N_k(x)})'(x)|\rightarrow \log 2$

as ${N_k(x)\rightarrow\infty}$. This completes the proof of the proposition. $\Box$

4.3. Dynamics of ${P_c}$ when ${c}$ is regular

We shall use the same ideas from the previous subsection to show that ${P_c}$ satisfies the conclusion of Theorem 4 when ${c}$ is regular. More precisely, we will use the returns to ${A}$ of certain “maximal” intervals of regular points to describe the return map ${T: A\rightarrow A}$ of ${P_c}$ as an uniformly expanding countable Markov map. As it turns out, one can easily construct a ${T}$-invariant absolutely continuous probability ${\mu_T}$ (via the classical approach of thermodynamical formalism [i.e., analysis of Ruelle transfer operator]), and our regularity assumption on ${c}$ permits to convert ${\mu_T}$ into a ${\mu_{P_c}}$-invariant absolutely continuous probability measure (by a standard summation procedure).

Definition 7 An interval ${J}$ is regular of order ${n>0}$ if there exists a neighborhood ${\widehat{J}}$ of ${J}$ such that ${P_c^n|_{\widehat{J}}}$ is a diffeomorphism from ${\widehat{J}}$ to ${\widehat{A}}$ with ${P_c^n(J)=A}$. We denote by ${g_J=(P_c^n|_{\widehat{J}})^{-1}}$ the inverse branch of the restriction of ${P_c^n}$ to ${\widehat{J}}$.

Remark 6 By definition, all points in a regular interval ${J}$ of order ${n}$ are ${n}$-regular.

As we are going to see in a moment, the definition of regular interval ${J}$ of order ${n>0}$ was setup in a such a way that the “margin” provided by ${\widehat{A}-A}$ ensures good distortion and expansion properties of ${P_c^n|_J}$.

Remark 7 In a certain sense, the previous paragraph is the real analog of Koebe distortion theorem and Schwarz lemma ensuring good properties for univalent maps in terms of the modulus of certain annuli.

4.3.1. Distortion estimates

The Schwarzian derivative ${Sf(x):=D^2\log|Df|(x) - \frac{1}{2}(D\log|Df|(x))^2}$ measures how far is a ${C^3}$ diffeomorphism ${f}$ from a Möbius transformation, and, for this reason, it is extremely efficient for studying distortion effects on derivative.

In fact, the composition rule ${S(f\circ g) = (Sf\circ g)\cdot D^2g + Sg}$ reveals that the iterates ${P_c^n|_J}$ have negative Schwarzian derivative on any regular interval ${J}$ of order ${n>0}$ (because ${P_c}$ has negative Schwarzian derivative).

In particular, if ${g_J:=(P_c^n|_{\widehat{J}})^{-1}}$ is the inverse branch of the restriction of ${P_c^n}$ to the neighborhood ${\widehat{J}}$ a regular interval of order ${n>0}$, then ${g_J}$ has positive Schwarzian derivative on ${\widehat{A}=P_c^n(\widehat{J})}$.

Since a ${C^3}$ diffeomorphism ${h:I\rightarrow h(I)\subset\mathbb{R}}$ with positive Schwarzian derivative satisfies the estimate ${|D\log |Dh|(x)|\leq 2/d(\partial I, x)}$ (see, e.g., Exercise 6.4 at page 166 of this book of de Faria-de Melo), we conclude:

Lemma 8 (Bounded distortion estimate) For any regular interval ${J}$, the inverse branch ${g_J}$ satisfies

$\displaystyle |D\log|Dg_J|(x)|\leq C_0:=\frac{2}{|\alpha^{(1)}-\alpha|} = \frac{2}{d(\partial\widehat{A}, \partial A)}$

for all ${x\in A}$. In particular,

$\displaystyle \max\limits_{x\in A} |Dg_J(x)|\leq C_1\min\limits_{x\in A} |Dg_J(x)|$

and, for all measurable subset ${E\subset A}$,

$\displaystyle \frac{1}{C_1} \frac{Leb(E)}{Leb(A)}\leq\frac{Leb(g_J(E))}{Leb(J)}\leq C_1 \frac{Leb(E)}{Leb(A)}$

where ${C_1:=e^{C_0 Leb(A)}}$.

4.3.2. Expansion estimates

Let ${\mathcal{J}}$ be the countable family of regular intervals ${J\subset A}$ of positive order ${n>0}$ which are maximal for these properties.

Given ${\underline{J}=(J_1,\dots, J_m)\in\mathcal{J}^m}$, the composition ${g_{J_1}\circ\dots\circ g_{J_m}}$ is the inverse branch associated to a regular interval ${\underline{J}\subset A}$. Conversely, all regular intervals contained in ${A}$ are obtained in this way.

The bounded distortion estimates above allow to deduce uniform expansion estimates for the first return map ${T:A\rightarrow A}$ of ${P_c}$ on regular intervals.

Lemma 9 (Expansion estimates) Let ${\underline{J}=(J_1,\dots, J_m)\in\mathcal{J}^m}$ be a regular interval of positive order contained in ${A}$ and denote ${\underline{J}'=(J_1,\dots, J_{m-1})}$. Then,

$\displaystyle \left(1-\frac{Leb(J_m)}{Leb(A)}\right)\leq C_1\left(1-\frac{Leb(\underline{J})}{Leb(\underline{J}')}\right), \quad Leb(\underline{J})\leq (1-c_2)^m Leb(A)$

and

$\displaystyle |DT^m(x)|\geq \frac{1}{C_1}\left(\frac{1}{1-c_2}\right)^m \quad \forall\, x\in\underline{J},$

where

$\displaystyle 0

Proof: The first inequality follows from Lemma 8 by setting ${E:=A-J_m}$ and ${J=\underline{J}'}$. The second estimate follows from the first inequality after taking into account the definition of ${c_2}$. Finally, the third estimate follows from the second inequality, the distortion estimate in Lemma 8 and the fact that ${T^m(\underline{J})=A}$ (so that ${\int_J |DT^m(x)| dx= Leb(A)}$, and, hence, ${|DT^m(x_{\underline{J}})|=Leb(A)/Leb(\underline{J})}$ for some ${x_{\underline{J}}\in\underline{J}}$). $\Box$

4.3.3. Thermodynamical formalism for regular parameters ${c\in\mathcal{R}}$

Let ${c\in\mathcal{R}}$ be a regular parameter. By definition,

$\displaystyle Leb(\{x\in A: x \textrm{ is not } n\textrm{-regular}\})\leq Ce^{-\theta n}$

for some constants ${C, \theta>0}$ and all ${n\in\mathbb{N}}$. In particular, if

$\displaystyle W:=\bigcup\limits_{J\in\mathcal{J}} \textrm{int}(J),$

then ${Leb(A-W)=0}$. Thus, most of the dynamics of the return map ${T:A\rightarrow A}$ is described by ${T: W\rightarrow A}$,

$\displaystyle T(x) = P_c^{N(x)}(x)$

where ${N(x)}$ is the order of ${J\ni x}$.

Note that ${T:W\rightarrow A}$ is a countable uniformly expanding Markov map (thanks to Lemmas 8 and 9). Using this fact, we will construct a ${T}$-invariant absolutely continuous probability measure.

For this sake, we consider the dual ${\mathcal{L}}$ of the pullback operator ${\nu\mapsto T_*\nu}$ acting on finite absolutely continuous measures ${\nu=h(x)dx}$ on ${A}$. More concretely, the (Ruelle operator) ${\mathcal{L}}$ acts on densities by the formula

$\displaystyle \mathcal{L}h(x)=\sum\limits_{J\in\mathcal{J}} |Dg_J(x)|\cdot h(g_J(x))$

derived from the change of variables formula and the definition ${T_* (h dx) = (\mathcal{L}h)dx}$.

We will take advantage of tools from Complex Analysis in order to study ${\mathcal{L}}$.

More precisely, we want to use the standard procedure (from the usual proof of Bogolyubov-Krylov theorem) of building invariant measures as the limit of Cesaro means

$\displaystyle \frac{1}{m}\sum\limits_{k=0}^{m-1}(T^k)_*dx = \left(\frac{1}{m}\sum\limits_{k=0}^{m-1} (\mathcal{L}^k \textbf{1})\right) dx,$

where ${\textbf{1}}$ is the constant function on ${A}$ taking the value one. Thus, our task is to show the convergence of the sequence ${\frac{1}{m}\sum\limits_{k=0}^{m-1} \mathcal{L}^m \textbf{1}}$ (and this amounts to establish adequate estimates on ${\mathcal{L}^m\textbf{1}}$).

In this direction, let us show that ${\mathcal{L}^m\textbf{1}}$ admits an holomorphic extension. Let ${U}$ be the simply connected domain ${U:=(\mathbb{C}-\mathbb{R})\cup\widehat{A}}$. Since the critical values of ${P_c^n}$ are all real (for all ${n\in\mathbb{N}}$ and ${c\in\mathbb{R}}$), the inverse branches ${g_J}$ associated to regular intervals ${J}$ extend to univalent maps ${g_J:U\rightarrow\mathbb{C}}$ satisfying

$\displaystyle g_J(\mathbb{C}-\mathbb{R})\subset \mathbb{C}-\mathbb{R} \quad \textrm{ and } \quad g_J(\widehat{A})=\widehat{J}$

Given a regular interval, let ${\varepsilon_J\in\{-1,+1\}}$ be the sign of ${Dg_J}$ on ${\widehat{A}}$ and define

$\displaystyle \widetilde{g}_J(z):=\varepsilon_J \frac{Leb(A)}{Leb(J)}(g_J(z)-g_J(0))$

The family ${(\widetilde{g}_J)_{J \textrm{ regular}}}$ is normal on ${U}$ because ${\widetilde{g}_J}$ are univalent functions on ${U}$ such that ${\widetilde{g}_J(0)=0}$ and ${|D\widetilde{g}_J(0)|\leq C_1}$ is uniformly bounded (by the distortion estimate in Lemma 8).

Since ${\sum\limits_{\underline{J}\in\mathcal{J}^k} Leb(\underline{J}) = Leb(A)}$ for all ${k\in\mathbb{N}}$, the series

$\displaystyle h_k:= \sum\limits_{\underline{J}\in\mathcal{J}^k}\varepsilon_{\underline{J}} Dg_{\underline{J}} = \sum\limits_{\underline{J}\in\mathcal{J}^k}\varepsilon_{\underline{J}} \frac{Leb(\underline{J})}{Leb(A)} D\widetilde{g}_{\underline{J}}$

defines a holomorphic extension to ${U}$ of the functions

$\displaystyle \mathcal{L}^k\textbf{1}:=\sum\limits_{\underline{J}\in\mathcal{J}^k} |Dg_{\underline{J}}|$

Moreover, the distortion estimates in Lemma 8 say that

$\displaystyle 1/C_1\leq h_k(x)\leq C_1 \ \ \ \ \ (3)$

for all ${x\in A}$ and ${k\in\mathbb{N}}$.

Therefore, the family ${H_m:=\frac{1}{m}\sum\limits_{k=0}^{m-1}\mathcal{L}^k\textbf{1}}$, ${m\in\mathbb{N}}$, is normal (because ${H_m}$ belongs to the closed convex hull of the normal family ${(h_k)_{k\in\mathbb{N}}}$) and we can extract a subsequence of ${H_m}$ converging uniformly on compact subsets of ${U}$ to a holomorphic function ${h_T}$ on ${U}$ such that

$\displaystyle 1/C_1\leq h_T(x)\leq C_1 \ \ \ \ \ (4)$

for all ${x\in A}$.

Lemma 10 ${\mu_T:=h_T dx}$ is an ergodic ${T}$-invariant measure equivalent to Lebesgue measure with total mass ${Leb(A)}$.

Proof: By (4), ${\mu_T}$ is a measure equivalent to the Lebesgue measure. Since ${\mathcal{L}}$ preserves the total mass,

$\displaystyle \mathcal{L}H_m = H_m+\frac{1}{m}(h_m-1),$

and ${|h_m-1|\leq 1+C_1}$ is uniformly bounded (thanks to (3)), we obtain that ${\mu_T}$ is a ${T}$-invariant measure with total mass ${\mu_T(A)=Leb(A)}$.

Finally, the ergodicity of ${\mu_T}$ is proved by the following standard argument (exploiting the good distortion and expansion properties of ${T}$). Suppose that ${E\subset A}$ is a ${T}$-invariant subset of positive ${\mu_T}$, or equivalently Lebesgue, measure. Fix ${x_0\in E}$ a Lebesgue density point and, for each ${m\in\mathbb{N}}$, denote by ${J^{(m)}\in\mathcal{J}^m}$ the regular interval containing ${x_0}$. By Lemma 9, we know that ${Leb(J^{(m)})\rightarrow 0}$ (exponentially fast) as ${m\rightarrow\infty}$. Hence, given ${\varepsilon>0}$, there exists ${m(\varepsilon)\in\mathbb{N}}$ such that

$\displaystyle Leb(J^{(m)}\cap E)\geq (1-\varepsilon)Leb(E)$

for all ${m\geq m(\varepsilon)}$. By plugging this information into the distortion estimate in Lemma 8 for ${g_J=g_{J^{(m)}}}$, ${m=m(\varepsilon)}$, and by using the ${T}$-invariance of ${E}$, we get

$\displaystyle Leb(E)\geq (1-C_1\varepsilon)Leb(A).$

Since ${\varepsilon>0}$ is arbitrary, we conclude that ${Leb(E)=Leb(A)}$ as desired. $\Box$

4.3.4. Lyapunov exponent of ${P_c}$ for ${c\in\mathcal{R}}$

For any ${c\in\mathcal{R}}$, the return time function ${N:W\rightarrow A}$ and the logarithm ${\log|DT|}$ of the derivative of the return function are ${L^p}$-integrable with respect to ${\mu_T}$ for all ${1\leq p<\infty}$. Indeed, the regularity of the parameter ${c}$ means that

$\displaystyle Leb(\{x\in A: N(x)\geq n\})\leq Ce^{-\theta n}$

and hence the ${L^p(\mu_T)}$-integrability of ${N}$ follows from the fact that ${\mu_T}$ is equivalent to the Lebesgue measure (cf. Lemma 10). Also, the expansion estimate in Lemma 9 says that ${\log|DT|}$ is bounded from below (by ${-\log C_1}$), while the fact that ${|P_c'(x)|\leq 4}$ for all ${x\in[-\beta,\beta]}$ says that ${\log|DT(x)|\leq N(x)\log 4}$, so that the ${L^p(\mu_T)}$-integrability of ${\log|DT|}$ follows from the corresponding fact for ${N}$.

By Birkhoff theorem, we have ${\mu_T}$ (i.e., Lebesgue) almost everywhere convergence of the Birkhoff sums

$\displaystyle \lim\limits_{m\rightarrow\infty}\frac{1}{m}\sum\limits_{k=0}^{m-1} N(T^k(x)) = \frac{1}{\mu_T(A)}\int N d\mu_T := N_T \ \ \ \ \ (5)$

and

$\displaystyle \lim\limits_{m\rightarrow\infty}\frac{1}{m}\sum\limits_{k=0}^{m-1} \log|DT(T^k(x))| = \lambda_T$

where ${N_m(x):=\sum\limits_{k=0}^{m-1} N(T^k(x))}$.

Note that ${N_T\geq 2}$ (because ${N(x)\geq 2}$ for all ${x\in \textrm{int}(A)}$), ${\lambda_T\geq \log(1/(1-c_2))>0}$ (by Lemma 9) and

$\displaystyle \lim_{m\rightarrow\infty}\frac{|(P_c^{N_m(x)})'(x)|}{N_m(x)} =\lim\limits_{m\rightarrow\infty}\frac{m}{N_m(x)}\frac{1}{m}\sum\limits_{k=0}^{m-1} \log|DT(T^k(x))| = \frac{\lambda_T}{N_T}:=\lambda_P>0 \ \ \ \ \ (6)$

Let us compute the Lyapunov exponent of ${P_c}$ using (6) and the same argument from Section 4.

Proposition 11 Let ${c\in\mathcal{R}}$ be a regular parameter. Then, for Lebesgue almost every ${x\in[-\beta,\beta]}$, one has

$\displaystyle \lim\limits_{n\rightarrow\infty}\frac{1}{n}\log|(P_c^n)'(x)| = \lambda_P>0$

Proof: Recall that, for every ${x\in A}$ and ${N_m(x)\leq n\leq N_{m+1}(x)}$, we have

$\displaystyle |(P_c^{N_{m+1}(x)})'(x)|+(n-N_{m+1}(x))\log 4\leq |(P_c^n)'(x)|\leq |(P_c^{N_m(x)})'(x)| + (n-N_m(x))\log 4$

Since (5) implies that ${N_{m+1}(x)/N_m(x)\rightarrow 1}$ as ${m\rightarrow\infty}$, the desired proposition is a consequence of the estimate above and (6). $\Box$

4.3.5. Absolutely continuous invariant measures for ${P_c}$, ${c\in\mathcal{R}}$

Let us use the invariant measure ${\mu_T}$ of the acceleration ${T}$ of ${P_c}$ to build an invariant measure ${\mu_P}$ of ${P_c}$ on ${[-\beta,\beta]}$.

The basic idea is very simple: we want to use the iterates of ${P_c}$ between the initial time and the return time to produce ${\mu_P}$ from ${\mu_T}$. More concretely, given a continuous function ${\varphi}$ on ${[-\beta,\beta]}$, let

$\displaystyle \int \varphi d\mu_P:=\int B\varphi(x) d\mu_T(x) \ \ \ \ \ (7)$

where ${B\varphi(x)}$ is the Birkhoff sum ${B\varphi(x) = \sum\limits_{k=0}^{N(x)-1} \varphi(P_c^k(x))}$.

Remark 8 ${\mu_P}$ is well-defined because ${|S_{N(x)}\varphi(x)|\leq N(x)\|\varphi\|_{L^{\infty}}}$ and ${N\in L^p(\mu_T)}$ for all ${1\leq p<\infty}$.

Note that ${\mu_P}$ has total mass ${N_T \mu_T(A) = N_T Leb(A)}$ (by Lemma 10). Moreover, ${\mu_P}$ is supported on

$\displaystyle \bigcup\limits_{n\in\mathbb{N}} P_c^n(A),$

that is

$\displaystyle \bigcup\limits_{n\in\mathbb{N}} P_c^n(A) = P_c(A)\cup P_c^2(A) = [P_c(0), P_c^2(0)]$

for ${c\in\mathcal{R}}$. Furthermore, ${\mu_P}$ is ${P_c}$-invariant is a consequence of the ${T}$-invariance of ${\mu_T}$ because ${B(\varphi\circ P_c)}$ differs from ${B\varphi}$ by a coboundary:

$\displaystyle B(\varphi\circ P_c) = B\varphi +(\varphi\circ T - \varphi)$

Also, ${\mu_P}$ is ${P_c}$-ergodic: any ${P_c}$-invariant function ${\varphi}$ restricts to a ${T}$-invariant function on ${A}$; hence, the ergodicity of ${\mu_T}$ implies that ${\varphi|_A}$ is almost everywhere constant, and so ${\varphi}$ is almost everywhere constant on ${[P_c(0), P_c^2(0)]}$.

Finally, the formula (7) defining ${\mu_P}$ shows that it is absolutely continuous with respect to the Lebesgue measure and its density ${ h_P=d\mu_P/dx}$ is given by

$\displaystyle h_P = \sum\limits_{J\in\mathcal{J}} \sum\limits_{0\leq n <\textrm{order of } J} \chi_{P_c^n(J)} |D((P_c^n|_J)^{-1})| h_T\circ (P_c^n|_J)^{-1}$

This completes the verification of the conclusion of Jakobson theorem for regular parameters ${c\in\mathcal{R}}$.

Closing this post, let us observe that the density of ${\mu_P}$ is not square-integrable.

Remark 9 By inspecting the terms ${n=0}$, ${1}$ and ${2}$ in the definition of ${h_P}$, we see that the distortion estimates (4) imply that ${h_P(x)\geq 1/C_1}$ for almost every ${x\in A}$,

$\displaystyle h_P(x)\geq \frac{1}{C_1}\cdot\frac{1}{2\sqrt{x-P_c(0)}}$

for almost every ${x\in[P_c(0),\alpha]}$ and

$\displaystyle h_P(x)\geq \frac{1}{C_1}\cdot\frac{1}{8\sqrt{P_c^2(0)-x}}$

for almost every ${x\in[-\alpha, P_c^2(0)]}$. Therefore, ${h_P}$ is bounded away from zero but ${h_P}$ is not ${L^2}$ in ${[P_c(0), P_c^2(0)]}$.

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