# Collet–Eckmann maps in the unicritical family

Magnus Aspenberg, Mats Bylund and Weiwei Cui

###### Abstract

In this paper we study perturbations of complex unicritical polynomials
satisfying the Collet–Eckmann condition. We show that Collet–Eckmann
parameters are Lebesgue density points of the complement of the Mandelbrot set
(i.e. the connectedness locus).

##  1 Introduction

We consider the unicritical family

$f_{c}(z)=z^{d}+c,$

where $d\geq 2$ is an integer and $c\in{\mathbb{C}}$.

A non-hyperbolic unicritical map $f_{c}$ satisfies the _Collet–Eckmann_
condition (CE) if there exist $C>0$ and $\gamma>0$ such that

$|Df_{c}^{n}(c)|\geq Ce^{\gamma n}~{}\,~{}\text{for all}~{}\,n>0.$ (1.1)

Recall that a map $f$ in this family is hyperbolic if
$\omega(0)\cap{\mathcal{J}}(f)=\emptyset$, where ${\mathcal{J}}(f)$ is the
Julia set of $f$. For $d\geq 2$, the set $\mathcal{M}_{d}$ is defined as the
set of parameters $c$ such that the Julia set ${\mathcal{J}}(f_{c})$ of
$f_{c}$ is connected (or equivalently, the critical orbit $\\{f^{n}_{c}(0)\\}$
is bounded). When $d=2$, the set $\mathcal{M}_{2}={\mathcal{M}}$ is the well
known _Mandelbrot set_ (but we also call ${\mathcal{M}}_{d}$ a Mandelbrot
set).

Our paper is about perturbations of CE-maps in the unicritical family. It has
a quite long history. In [24], J. Rivera-Letelier proved that Misiurewicz maps
(i.e. critically non-recurrent maps) are Lebesgue density points of
${\mathbb{C}}\setminus{\mathcal{M}}_{d}$. It was generalised by the first
author in [1] to rational maps. However, even if Misiurewicz maps behave quite
nicely under perturbations, they are rare in the parameter space ([2]).

In [6], A. Avila, M. Lyubich and W. Shen proved that the Lebesgue density of
${\mathcal{M}}_{d}$ at at most finitely renormalizable maps in the unicritical
family, is strictly less than one (Theorem 1.4 and Remark 6.1 in [6]). Since
Collet–Eckmann maps are at most finitely renormalizable, the result applies to
them and consequently CE-maps cannot be density points of ${\mathcal{M}}_{d}$.

In a series of quite recent papers [12, 15, 16], J. Graczyk and G. Świa̧tek
made careful estimates of the geometry of the Julia set around typical
Collet–Eckmann maps with respect to harmonic measure on
$\partial{\mathcal{M}}_{d}$. These maps are typically critically slowly
recurrent, i.e. not Misiurewicz maps, but the speed of recurrence is slow (see
[14]). Similar to the Misiurewicz case (which is easier), the goal is to get
some kind of similarity between phase and parameter space. One cannot expect a
complete (conformal) similarity between them since the Mandelbrot set contains
an interior and the Julia set for a CE-map is a dendrite. However, at small
scales this similarity becomes asymptotically conformal. The result is
obtained by introducing a powerful tool called the similarity map, which
transfers the geometry around the critical value $c$ between the Julia set and
the Mandelbrot set ${\mathcal{M}}_{d}$ in an intriguing way. In particular it
follows that typical Collet–Eckmann maps, with respect to harmonic measure,
are density points of ${\mathbb{C}}\setminus{\mathcal{M}}_{d}$. The slow
recurrence condition is also imposed on the Collet–Eckmann maps in [4], where
we proved a corresponding result for rational maps. We also mention an
unpublished work by M. Benedicks and J. Graczyk, where they prove
corresponding results for one-dimensional families (with some regularity)
around typical CE-maps with respect to harmonic measure in
$\partial{\mathcal{M}}_{d}$. For more general CE-maps, the dynamical constants
can be drastically different. This yields quite wild dynamics, and crucial
parts of the traditional methods break down.

In [13] it is shown that at most finitely renormalizable maps are weak density
points of ${\mathbb{C}}\setminus{\mathcal{M}}_{d}$ (again, this applies to CE-
maps). The combinatorics for such maps seem to allow for the existence of deep
central returns in the puzzle of the induced map at infinitely many small
scales. Seemingly, this implies that this weak density cannot be replaced by
strong density. We claim that this is not the case. Our main result is the
following.

######  Theorem 1.1.

In the unicritical family, Collet–Eckmann maps are Lebesgue density points of
${\mathbb{C}}\setminus{\mathcal{M}}_{d}$.

######  Remark 1.2.

One should compare the above theorem with a corresponding result in the real
setting. M. Tsujii proved that (slowly recurrent) CE-maps with all periodic
cycles repelling are Lebesgue density points of CE-maps [25] in families of
real polynomials, including the unicritical family, restricted to
${\mathcal{M}}_{d}\cap{\mathbb{R}}$; see also [10].

It will be quite clear from the proof that the same methods used in this paper
can be applied to get the same conclusion as in Theorem 1.1 for rational CE-
maps with precisely one critical point in the Julia set.

Recall that a rational map satisfies the Collet–Eckmann condition if the map
has no parabolic cycles and (1.1) is satisfied for each critical point in the
Julia set (usually one requires the map to be non-hyperbolic). Let
$\mathcal{R}_{d}$ be the space of rational maps of degree $d\geq 2$.

######  Theorem 1.3.

Any rational Collet–Eckmann map of degree $d$ at least two with only simple
critical points and with precisely one critical point in the Julia set is a
Lebesgue density point of hyperbolic maps in $\mathcal{R}_{d}$.

This also partially improves our previous result in [4] where we imposed the
additional assumption of slow recurrence of the critical points in the Julia
set.

######  Remark 1.4.

In case that rational maps have critical points of higher multiplicities, one
can also get a corresponding result but in a subspace of $\mathcal{R}_{d}$.
This subspace was defined by G. Levin [19].

#### Ideas of the proof.

We trace the critical orbits for all parameters close to a Collet–Eckmann map
and use the parameter exclusion technique developed by M. Benedicks and L.
Carleson in the seminal works [7, 8]. However, since we start with arbitrary
Collet–Eckmann maps, the dynamical constants, e.g. the recurrence rate, can be
quite wild in general (deep central returns) and traditional methods break
down. This difficulty will be overcome by introducing the idea of _promotion_.
Roughly speaking, promotion means that almost all parameters will reach some
kind of slow recurrence condition.

######  Acknowledgement.

Partially this work was written when all the authors were participating in the
program “Two Dimensional Maps” at Mittag-Leffler Institute during the spring
2023. We thank the organisers for providing a nice working place. We are also
grateful to Michael Benedicks for stimulating discussions on the topic of this
paper. The second author gratefully acknowledges support from the Knut and
Alice Wallenberg Foundation. The third author was gratefully supported by
Vergstiftelsen and the Qilu Young Scholar Program of Shandong University.

##  2 Preliminaries

We collect some preliminary notions and notations in this section. For some
general notions such as the Fatou and Julia sets, and other related notions in
complex dynamics, we refer to [9, 22]. We will always use the Euclidean metric
in this paper and thus distances and diameters are measured with respect to
this metric. The notation $\operatorname{dist}(A,B)$ means the Hausdorff
distance between two sets $A$ and $B$, and $\operatorname{diam}(A)$ denotes
the diameter of $A$. Also, $m(A)$ is the two-dimensional Lebesgue measure
(i.e., area) of the set $A$.

Throughout the paper we fix $d\geq 2$. Let $f_{c_{0}}$ be a Collet–Eckmann
unicritical polynomial of degree $d$. The only critical point is $0$ with the
corresponding critical value being $c_{0}$. Let
${\mathcal{Q}}={\mathcal{Q}}(c_{0},\varepsilon)$ be a square of side length
$\varepsilon$ in the parameter plane centred at the parameter $c_{0}$ with
$\varepsilon>0$ to be sufficiently small (cf. Lemma 2.9 and Lemma 2.8). For a
parameter $c\in{\mathcal{Q}}$ the corresponding map is denoted, as usual, by
$f_{c}$. We study the evolution of the critical orbit for nearby parameters.
Let

$\xi_{n}(c)=f_{c}^{n}(0)\quad\text{for}\quad n\geq 0.$

We will need to fix some neighbourhood for the critical point and study
recurrence of $\xi_{n}(c)$ to this neighbourhood for $c$ close to $c_{0}$ (cf.
Lemma 2.8). Let $0<\Delta^{\prime}<\Delta$ be large integers to be determined
later and put

$U=D(0,e^{-\Delta})\quad\text{and}\quad U^{\prime}=D(0,e^{-\Delta^{\prime}}).$
(2.1)

In the following, we also put $\delta=e^{-\Delta}$ and
$\delta^{\prime}=e^{-\Delta^{\prime}}$.

Starting with a Collet–Eckmann map, we have exponential growth of derivative
along the critical orbit. This expansion will be inherited by nearby
parameters with possibly slightly slower growth at the initial piece of their
critical orbits. This is ensured by the transversality at Collet–Eckmann
parameters (see Lemma 2.9). To gain growth for later orbits, we divide the
orbit into pieces: _bound periods_ and _free periods_. During the bound
periods, the orbits shadow the initial iterates of its critical orbit and
expansion is ensured (see Section 3.1). During free periods, the orbits stay
away from the critical point and hence gain exponential growth of derivatives
(Lemma 2.8) similar as the classical Mañé theorem (Lemma 2.7).

In the rest of this section we present necessary notions which are crucial in
our later analysis and also some other related results concerning recurrence
rates of Collet–Eckmann maps, expansions during bound periods and finally the
transversality result.

###  2.1 Recurrence, bound periods and partitions

With the critical neighbourhoods defined in (2.1), we say that $\xi_{n}(c)$ is
a _return_ if $\xi_{n}(c)\in U^{\prime}$. It is a pseudo-return if
$\xi_{n}(c)\in U^{\prime}\setminus U$. Sometimes we specify and say that, for
instance, $\xi_{n}(c)$ is a return into $U$. If $\xi_{n}(c)$ is a return, the
following orbit will be shadowed by its early piece of the critical orbit of
$f_{c}$ until they separate in the following sense. The number $n$ in this
case is called a return time.

######  Definition 2.1.

Let $\beta>0$. Let $\xi_{n}(c)$ be a return to $U^{\prime}$. The bound period
for this return is defined as the indices $j>0$ such that

$|\xi_{n+j}(c)-\xi_{j}(c)|\leq e^{-\beta j}|\xi_{j}(c)|.$

The largest number $p$ with the above inequality is called the length of the
bound period (for this return).

The number $\beta>0$ should be thought of as a comparatively small number.
There will be some conditions imposed on $\beta$, e.g. in Section 5.1.

After the bound period ends, the two critical orbits will then separate and
stay away from critical points until the next return happens. A return might
be deep in the sense that $\xi_{n}(c)$ is quite close to $0$. To achieve our
goal, we will need to disregard parameters with very deep returns. For later
purpose we will define $\alpha_{n}(c)$ for a parameter $c$ as follows:

$|\xi_{n}(c)|=e^{-\alpha_{n}(c)n}\quad\text{for all}\quad n.$

It is clear that if one is considering a Misiurewicz or semi-hyperbolic
parameter, then $|\xi_{n}(c)|$ has a uniform lower bound. For general
Collet–Eckmann parameters, this need not be true. The map $f_{c}$ is
(critically) slowly recurrent (see e.g. [21]) if

$\lim\limits_{n\rightarrow\infty}\alpha_{n}(c)=0.$

Recall that the lower and upper Lyapunov exponents at the critical value $c$
are defined respectively as

$\underline{\gamma}(c):=\liminf_{n\to\infty}\frac{1}{n}\log|Df_{c}^{n}(c)|\quad\text{and}\quad\overline{\gamma}(c):=\limsup_{n\to\infty}\frac{1}{n}\log|Df_{c}^{n}(c)|$

The Collet–Eckmann condition simply means that $\underline{\gamma}(c_{0})>0$
and that $f$ is not hyperbolic. Let $\gamma_{n}(c)$ be defined such that

$|Df_{c}^{n}(c)|=e^{\gamma_{n}(c)n}\quad\text{for all~{}}~{}n,$

where $\gamma_{n}(c)$ depends on the parameter $c$. Note also that
$|Df_{c}^{n}(z)|$ grows at most exponentially in $n$ for any $c$ and any $z$.
So $\overline{\gamma}(c)$ indeed has an upper bound. Both
$\alpha_{j}=\alpha_{j}(c)$ and $\gamma_{j}=\gamma_{j}(c)$ depend on the
parameter, but we usually suppress it. Suppose that the starting map
$f=f_{c_{0}}$ satisfies, for some $C_{0}>0$ and $\gamma>0$,

$|Df_{c_{0}}^{n}(c_{0})|\geq C_{0}e^{\gamma n}\quad\text{ for all ~{}}\,n>0.$

In other words, $0<\gamma\leq\underline{\gamma}(c_{0})$. We now make the
following definition. Let $\tilde{N}>0$ be a number which satisfies
$\gamma_{k}(c_{0})=\frac{1}{k}\log|Df_{c_{0}}^{k}(c_{0})|\geq(9/10)\underline{\gamma}(c_{0})>0$,
for all $k\geq\tilde{N}$, and that
$\gamma_{\tilde{N}}(c)=\frac{1}{\tilde{N}}\log|Df_{c}^{\tilde{N}}(c)|>(9/10)\underline{\gamma}(c_{0})$
for all $c\in{\mathcal{Q}}$. Moreover, we also require that all bound periods
for return times $n$ considered in the paper are larger than $\tilde{N}$ (so
$n$ is typically much larger than $\tilde{N}$). This can be achieved by
choosing $\delta^{\prime}$ small enough (see the Definition 2.1).

######  Definition 2.2.

Let $\gamma>0$. We say that $c\in{\mathcal{E}}_{n}(\gamma)$ if
$|Df_{c}^{k}(c)|\geq e^{\gamma k}$ for all $\tilde{N}\leq k\leq n$.

Note that we drop the constant $C_{0}$ in front, which appears in the starting
function $f_{c_{0}}$. So in principle,
$c_{0}\in{\mathcal{E}}_{n}(\underline{\gamma}(c_{0})-\varepsilon)$, for very
small $\varepsilon>0$. In our analysis later, we will specify the constant
$\gamma$ that suits our needs.

Using the chain rule

$\displaystyle\left|Df_{c}^{n}(c)\right|=\left|Df_{c}^{n-1}(c)\right|\,\left|Df_{c}(f_{c}^{n-1}(c))\right|$
$\displaystyle=\left|Df_{c}^{n-1}(c)\right|\left|Df_{c}(\xi_{n}(c))\right|$
(2.2) $\displaystyle=\left|Df_{c}^{n-1}(c)\right|d|\xi_{n}(c)|^{d-1},$

we get that

$e^{\gamma_{n}n}=e^{\gamma_{n-1}(n-1)}\,d\,e^{-\alpha_{n}n(d-1)}.$ (2.3)

The following simple inequalities, for $n$ large enough,

$\gamma_{n}\geq\gamma_{n-1}-(d-1)\alpha_{n}-\frac{C}{n}$ (2.4)

and

$\gamma_{n-1}\geq\gamma_{n}+(d-1)\alpha_{n}-\frac{C}{n}$ (2.5)

will be used throughout the paper. Here $C$ can be chosen to be $\log d$.

Our first observation is that one still has some control over the recurrence
of the critical point for a Collet–Eckmann map.

######  Lemma 2.3 (Initial recurrence rate).

Let $f_{c}$ be a Collet–Eckmann map. Then there exist $K>0$ and $\alpha>0$
such that

$\left|\xi_{n}(c)\right|\geq Ke^{-\alpha n}~{}\,~{}\text{for all}~{}\,n\geq
0.$

###### Proof.

From the definition of the upper and lower Lyapunov exponents, given any
$\epsilon>0$ we can find $N\geq 0$ such that, with
$\underline{\gamma}=\underline{\gamma}(c)$ and
$\overline{\gamma}=\overline{\gamma}(c)$,

$e^{(\underline{\gamma}-\epsilon)n}\leq|Df_{c}^{n}(c)|\leq
e^{(\overline{\gamma}+\epsilon)n}\quad(n\geq N).$

By (2.2), we have

$d|\xi_{n}(c)|^{d-1}=\frac{\left|Df_{c}^{n}(c)\right|}{\left|Df_{c}^{n-1}(c)\right|}\geq\frac{e^{(\underline{\gamma}-\epsilon)n}}{e^{(\overline{\gamma}+\epsilon)(n-1)}}=e^{\overline{\gamma}+\epsilon}e^{-(\overline{\gamma}-\underline{\gamma}+2\epsilon)n}\quad(n\geq
N).$

Therefore, there exists a constant $K>0$ such that

$|\xi_{n}(c)|\geq Ke^{-\alpha n}\quad(n\geq 1),$

where $\alpha=(\overline{\gamma}-\underline{\gamma}+2\epsilon)/(d-1)$. ∎

By Lemma 2.3, for a Collet–Eckmann map the numbers $\alpha_{n}$ cannot tend to
$\infty$, so the critical orbits cannot return too close to the critical
point.

######  Remark 2.4.

We notice from the above lemma that if the Lyapunov exponent exists, i.e. if
$\underline{\gamma}=\overline{\gamma}$, then the map satisfies the so-called
slow recurrence condition. In this situation Theorem 1.1 follows from [4].
(Compare also with [15, 16].) In what follows, we therefore assume that
$\underline{\gamma}<\overline{\gamma}$.

Actually, we take the upper exponent $\overline{\gamma}$ to be a trivial upper
bound for all exponents in the cube ${\mathcal{Q}}$ at all times; i.e.

$\overline{\gamma}=\sup\limits_{|z|\leq 2,c\in{\mathcal{Q}}}\log|Df_{c}(z)|,$

so that we have $\gamma_{n}(c)\leq\overline{\gamma}$, for all $n>0$ and all
$c\in{\mathcal{Q}}$. Also, in order to have some space to play with when we
change the parameter, we define

$\underline{\gamma}=\frac{9}{10}\underline{\gamma}(c_{0}).$

The number $\underline{\gamma}$ will stand as a kind of lower Lyapunov
exponent, at least up to the start-up time $N$, described in Lemma 3.9. Up
until this time, we have $\alpha_{n}(c)<\overline{\gamma}-\underline{\gamma}$,
($\tilde{N}\leq n\leq N$), for $c\in{\mathcal{Q}}$.

Although a Collet–Eckmann map has controlled recurrence rates in the sense of
Lemma 2.3, nearby parameters may return very deep due to the expansion. As
mentioned before, we will delete these parameters in our analysis. For this
purpose we need the notion of partition elements.

######  Definition 2.5 (Partition elements).

Let $S>0$ be given. A square $A\subset{\mathcal{Q}}$ is called a partition
element at time $n$ if the following holds for all $k\leq n$:

$\operatorname{diam}\xi_{k}(A)\leq\begin{cases}~{}\,~{}\dfrac{\operatorname{dist}(\xi_{k}(A),0)}{(\log\operatorname{dist}(\xi_{k}(A),0))^{2}}~{}&\text{if}\,~{}\xi_{k}(A)\cap
U\neq\emptyset,\\\ \quad~{}\,~{}S~{}&\text{if}\,~{}\xi_{k}(A)\cap
U=\emptyset.\end{cases}$ (2.6)

For clarification, in the rest of the paper we will write $A_{n}$ for a
partition element at time $n$. More specifically, we use $A_{n}(c)$ for a
partition element at time $n$ that contains the parameter $c$. It is useful to
have in mind that partitions are getting finer and finer as time proceeds; in
other words, $A_{n+1}(c)\subset A_{n}(c)$ for all $n$.

The term $S$ is referred as _large scale_ in this paper. It is always a good
situation if a partition element reaches the large scale. However, this is not
clear for all partition elements. Our main effort is to show that almost all
partition elements will reach the large scale. Clearly, the obstruction to
reach the large scale is caused by the frequent passages near the critical
point.

For a partition element, we can also give a notion of returns. Let $A$ be a
partition element at time $k-1$. Then $\xi_{k}(A)$ is called a _return_ to $U$
if $\xi_{k}(A)\cap U\neq\emptyset$ and a return to $U^{\prime}$ if
$\xi_{k}(A)\cap U^{\prime}\neq\emptyset$. It also may happen that
$\xi_{k}(A)\cap U^{\prime}\neq\emptyset$ but $\xi_{k}(A)\cap U=\emptyset$. In
this case we say that $\xi_{k}(A)$ is a _pseudo-return_. Now bound periods can
also be defined similarly if a return occurs for a partition element.

######  Definition 2.6.

Suppose that $A$ is a partition element at time $n$. The bound period for $A$
is the indices $j>0$ such that for any $c_{1},c_{2}\in A$ and for any
$z\in\xi_{n}(A)$ the following holds:

$|f_{c_{1}}^{j}(z)-\xi_{j}(c_{2})|\leq e^{-\beta j}|\xi_{j}(c_{2})|.$

The length of this bound period, denoted by $p$, is the largest number such
that the above relation holds.

After bound periods end, the next return will be called a _free return_. At
this point, there are two possibilities: essential and inessential returns.
Let $A_{n-1}$ be a partition element. A return $\xi_{n}(A_{n-1})$ to $U$ or
$U^{\prime}$ is _essential_ if

$\operatorname{diam}(\xi_{n}(A_{n-1}))\geq\frac{1}{2}\dfrac{\operatorname{dist}(\xi_{n}(A_{n-1}),0)}{(\log\operatorname{dist}(\xi_{n}(A_{n-1}),0))^{2}}.$

Otherwise the return is said to be _inessential_.

Suppose that $\xi_{n}(A_{n-1})$ is an essential return to $U$. Then partition
at time $n$ is made from $A_{n-1}$ as follows. If $A_{n-1}$ is a partition
element according to Definition 2.5 then we do not partition further, i.e. we
put $A_{n}=A_{n-1}$. If not, we partition $A_{n-1}$ into four squares of equal
size. If all of them are partition elements, then we are done. If not, we
continue with each square not being a partition element, and partition it into
four new squares of equal size. Continuing in this way we obtain a partition
of $A_{n-1}$ into sub squares $A_{n}^{j}$ satisfying

$\frac{1}{2}\dfrac{\operatorname{dist}(\xi_{n}(A_{n}^{j}),0)}{(\log\operatorname{dist}(\xi_{n}(A_{n}^{j}),0))^{2}}\leq\operatorname{diam}(\xi_{n}(A_{n}^{j}))\leq\dfrac{\operatorname{dist}(\xi_{n}(A_{n}^{j}),0)}{(\log\operatorname{dist}(\xi_{n}(A_{n}^{j}),0))^{2}}.$
(2.7)

Thus each $A_{n}^{j}$ is a partition element at time $n$.

Now we make another definition. After the partition is made as above, for each
set $\xi_{n}(A_{n}^{j})$ put
$\tilde{r}=-\log(\operatorname{dist}(\xi_{n}(A_{n}^{j}),0))$. Define
$r=\lceil\tilde{r}-\frac{1}{2}\rceil$. Then we have

$e^{-r-1/2}\leq\operatorname{dist}(\xi_{n}(A_{n}^{j}),0)<e^{-r+1/2}.$

If this is the case we also write
$\operatorname{dist}(\xi_{n}(A_{n}^{j}),0)\sim e^{-r}$. In general we write
$A\sim B$ if there is a non-dynamical constant $C$ such that

$\frac{1}{C}A\leq B\leq CA.$

###  2.2 Free periods

After the end of a bound period, we enter into a _free period_ before the next
return happens. During this time window, the orbit is away from critical
points which gives growth of derivatives similarly as the classical Mañé
theorem. The following is a Mañé type result which says that a Collet–Eckmann
map is expanding away from critical points. A similar lemma was proved in [4].

In the following, we state and prove the result for rational maps satisfying
the so-called _exponential shrinking property_. A rational map $f$ is said to
satisfy this property if there exist $\lambda_{\operatorname{Exp}}>1$ and
$r>0$ such that for every $z\in{\mathcal{J}}(f)$, each $n>0$ and every
component $W$ of $f^{-n}(D(z,r))$ one has
$\operatorname{diam}W\leq\lambda_{\operatorname{Exp}}^{-n}$. This property is
equivalent to the _topological Collet–Eckmann condition_ and is, in general,
weaker than the Collet–Eckmann condition; see [23]. For the setting considered
in this paper, i.e. maps with exactly one critical point in the Julia set,
these conditions are all equivalent. Let $\operatorname{Crit}(f)$ be the set
of critical points of $f$.

######  Lemma 2.7.

Let $f$ be rational satisfying the exponential shrinking property. Then there
exist a neighbourhood $U^{\prime}$ of
$\operatorname{Crit}(f)\cap{\mathcal{J}}(f)$, $C>0$ and $\gamma>0$ such that
the following holds: Let $z\in{\mathcal{J}}(f)$ and if $f^{j}(z)\not\in
U^{\prime}$ for $0\leq j\leq n-1$, then

$|Df^{n}(z)|\geq Ce^{\gamma n}.$

###### Proof.

Put $z_{n}=f^{n}(z)$ and $z_{0}=z$. Since $f$ satisfies the exponential
shrinking property, per definition one can find
$\lambda_{\operatorname{Exp}}>1$ and $r>0$ such that preimage components
shrink exponentially. Take any $\delta^{\prime}\in(0,r]$ and put

$W^{\prime}_{k}:=f^{-k}(D(z_{n},\delta^{\prime})),$

the component which contains $z_{n-k}$. Thus one has

$\operatorname{diam}W^{\prime}_{k}\leq\lambda_{\operatorname{Exp}}^{-k}$

for all $k\leq n-1.$ So there exists $\ell\in\mathbb{N}$ such that

$\operatorname{diam}W^{\prime}_{k}\leq\lambda_{\operatorname{Exp}}^{-k}<2\delta^{\prime}$

for all $k\geq\ell$. Since $f^{j}(z)\not\in U^{\prime}$ for $j<n$ which means
that $|f^{j}(z)-c|>\delta^{\prime}$ for all $j<n$ and all
$c\in\operatorname{Crit}(f)\cap{\mathcal{J}}(f)$, we see that

$W^{\prime}_{k}\cap(\operatorname{Crit}(f)\cap{\mathcal{J}}(f))=\emptyset$

for all $k\geq\ell$. This implies that there exists
$0<\delta^{\prime\prime}:=\delta^{\prime\prime}(\delta^{\prime})<\delta^{\prime}$
such that if $W^{\prime\prime}_{k}\subset W^{\prime}_{k}$ is the component of
$f^{-k}(D(z_{n},\delta^{\prime\prime}))$ we have

$f^{k}:W^{\prime\prime}_{k}\to D(z_{n},\delta^{\prime\prime})$

is conformal and onto for all $k\leq n$. By the above choice of
$\delta^{\prime\prime}$ and the Koebe one-quarter theorem,

$\lambda_{\operatorname{Exp}}^{-n}>\operatorname{diam}W^{\prime\prime}_{n}\geq\frac{1}{2}\frac{\delta^{\prime\prime}}{|Df^{n}(z_{0})|}.$

This gives

$|Df^{n}(z)|\geq\frac{\delta^{\prime\prime}}{2}\lambda_{\operatorname{Exp}}^{n}=:Ce^{\gamma
n}.$

Here $C=\delta^{\prime\prime}/2$ and
$\gamma=\log\lambda_{\operatorname{Exp}}$.

So the lemma holds with
$U^{\prime}=\bigcup_{c\in\operatorname{Crit}(f)\cap{\mathcal{J}}(f)}D(c,\delta^{\prime})$.
∎

The following lemma follows from the above and is proved in [4, Lemma 3.10].
To state the result, let $\mathcal{N}_{\varepsilon_{0}}$ be a neighbourhood of
the Julia set ${\mathcal{J}}(f_{c_{0}})$, where $\varepsilon_{0}$ is chosen in
a way such that $U^{\prime}\subset\mathcal{N}_{\varepsilon_{0}}$. That is,

$\mathcal{N}_{\varepsilon_{0}}=\left\\{z\in{\mathbb{C}}:\,\operatorname{dist}(z,{\mathcal{J}}(f_{c_{0}}))<\varepsilon_{0}\right\\}.$
(2.8)

Now we return to Collet–Eckmann maps in the unicritical family. Recall that
$U,U^{\prime},\Delta$ and $\Delta^{\prime}$ are defined in (2.1).

######  Lemma 2.8.

Let $f_{c_{0}}$ be a Collet–Eckmann map. Then there exist $\Delta^{\prime}>0$
sufficiently large, $C>0$ and $\gamma_{H}>0$ such that the following holds.
For any $\Delta>\Delta^{\prime}$ there exist $\varepsilon>0$ sufficiently
small and $C_{U}>0$ (dependent on $U$) such that for all $n\geq 1$ and for all
$c\in\mathcal{Q}$, if $z,f_{c}(z),\dots
f_{c}^{n-1}(z)\in\mathcal{N}_{\varepsilon_{0}}\setminus U$, then

$|Df_{c}^{n}(z)|\geq C_{U}e^{\gamma_{H}n}.$ (2.9)

If, in addition, $f_{c}^{n}(z)\in U$, then

$|Df_{c}^{n}(z)|\geq Ce^{\gamma_{H}n}.$ (2.10)

###  2.3 Transversality

One of the ingredients used in the proof of the theorem is a transversality
result at Collet–Eckmann parameters. For the unicritical family, this was
proved by A. Avila [5] and G. Levin [18]. (For general rational Collet–Eckmann
maps, this was due to G. Levin [20].) This result says that for a
Collet–Eckmann unicritical polynomial $f_{c_{0}}$, one has

$\lim_{n\to\infty}\frac{\xi^{\prime}_{n}(c_{0})}{Df^{n-1}_{c_{0}}(c_{0})}=L\,~{}(\neq
0,\,\infty).$

A direct consequence of this is the following lemma proved in [3, Proposition
4.1].

######  Lemma 2.9.

Let $f_{c_{0}}$ be a unicritical polynomial satisfying the Collet–Eckmann
condition with $\gamma_{0}=\underline{\gamma}(c_{0})$. Then for any
$q\in(0,1)$ and any $\gamma\in(0,\gamma_{0})$ there exists $N_{L}>0$ and
$\varepsilon>0$ such that

$\left|\frac{\xi^{\prime}_{n}(c)}{Df^{n-1}_{c}(c)}-L\right|\leq q|L|$ (2.11)

provided that $f_{c}$ satisfies the Collet–Eckmann condition up to time $n\geq
N_{L}$ with exponent $\gamma$ for all $c\in{\mathcal{Q}}$.

It follows from this lemma that, with some a priori specified
$\gamma<\gamma_{0}$, all parameters from ${\mathcal{Q}}$ will have exponential
growth along the critical orbit up to time $N_{L}$ with this exponent.
Moreover, the function $\xi_{n}$ has bounded distortion for all $n\leq N_{L}$.

###  2.4 The weak parameter dependence property

We state here the basic and crucial fact, that, given some lower exponent
$\gamma_{Q}\in(0,\underline{\gamma})$, and $Q>1$ (typically so that $\log Q$
is much smaller than both $\underline{\gamma}$ and $\gamma_{Q}$), there is
$\varepsilon>0$ (the size of the square ${\mathcal{Q}}$) such that, if the
derivative $|Df_{c}^{k}(c)|\geq e^{\gamma_{Q}k}$ for all $N_{L}\leq k\leq n$,
then

$Q^{-(n-k)}|Df_{c}^{n-k}(\xi_{k}(c))||\xi_{k}(a)-\xi_{k}(b)|\\\
\leq|\xi_{n}(a)-\xi_{n}(b)|\\\ \leq
Q^{n-k}|Df_{c}^{n-k}(\xi_{k}(c))||\xi_{k}(a)-\xi_{k}(b)|,$ (2.12)

for all $a,b,c$ belonging to the same partition element
$A_{n}\subset{\mathcal{Q}}$. For a detailed proof see Lemma 3.2 in [4]. The
point is that this result in connection to the Main Distortion Lemma 3.8 gives
a strong version of (2.12) which we present after Lemma 3.8.

##  3 Bound periods, main distortion and start-up

This section is devoted to proving distortion results. This relies on the
expansion during bound periods which in turn depends on estimating the growth
during this period (Lemma 3.3). The expansion during bound periods depends on
the length (Lemma 3.2) and distortion estimates (Lemma 3.1) of bound periods.

###  3.1 Distortion, length and expansion during bound periods

We first prove that we have good distortion during the bound period (see
Definition 2.6) and that, given a return at time $n$, this forces the bound
period to never succeed $n$ in length. With these in hand it will be clear
that one has some definite expansion when the bound periods end.

#### Distortion during bound periods.

The following lemma gives distortion estimate in the bound periods.

######  Lemma 3.1 (Distortion during the bound period).

Let $\varepsilon^{\prime}>0$ be given. Then if
$\delta^{\prime}=e^{-\Delta^{\prime}}$ is sufficiently small the following
holds. Let $n$ be the index of a return, $A$ a partition element at time $n$,
and let $p$ be the associated bound period. Then, for all $c\in A$ and all
$z\in\xi_{n}(A)$,

$\left|\frac{(f_{c}^{j})^{\prime}(f_{c}(z))}{(f_{c}^{j})^{\prime}(c)}-1\right|\leq\varepsilon^{\prime}\qquad(j=1,2,\dots,p).$

###### Proof.

By the chain rule and a standard estimate we have

$\left|\frac{(f_{c}^{j})^{\prime}(f_{c}(z))}{(f_{c}^{j})^{\prime}(c)}-1\right|\leq\exp\left\\{\sum_{\nu=0}^{j-1}\frac{|f_{c}^{\nu}(f_{c}(z))^{d-1}-f_{c}^{\nu}(c)^{d-1}|}{|f_{c}^{\nu}(c)|^{d-1}}\right\\}-1.$

Using the identity

$z^{d-1}-w^{d-1}=(z-w)\sum_{k=0}^{d-2}z^{k}w^{d-2-k},$

we find that

$|f_{c}^{\nu}(f_{c}(z))^{d-1}-f_{c}^{\nu}(c)^{d-1}|\leq|f_{c}^{\nu}(f_{c}(z))-f_{c}^{\nu}(c)|\sum_{k=0}^{d-2}|f_{c}^{\nu}(f_{c}(z))|^{k}|f_{c}^{\nu}(c)|^{d-2-k}.$

During the bound period the following holds,

$\frac{|f_{c}^{\nu}(f_{c}(z))|}{|f_{c}^{\nu}(c)|}\leq\frac{|f_{c}^{\nu}(f_{c}(z))-f_{c}^{\nu}(c)|}{|f_{c}^{\nu}(c)|}+1\leq
e^{-\beta j}+1<2,$

and hence,

$|f_{c}^{\nu}(f_{c}(z))^{d-1}-f_{c}^{\nu}(c)^{d-1}|\leq|f_{c}^{\nu}(f_{c}(z))-f_{c}^{\nu}(c)||f_{c}^{\nu}(c)|^{d-2}(2^{d-1}-1).$

We conclude that

$\left|\frac{(f_{c}^{j})^{\prime}(f_{c}(z))}{(f_{c}^{j})^{\prime}(c)}-1\right|\leq\exp\left\\{(2^{d-1}-1)\sum_{\nu=0}^{j-1}\frac{|f_{c}^{\nu}(f_{c}(z))-f_{c}^{\nu}(c)|}{|f_{c}^{\nu}(c)|}\right\\}-1.$

Suppose that $\operatorname{dist}(0,\xi_{n}(A))\sim e^{-r}$. Then for all
$z\in\xi_{n}(A)$ we have $|f_{c}(z)-c|\sim e^{-dr}$. We divide the sum into
two parts $[0,j-1]=[0,J]\cup[J+1,j-1]$, where $J=\lceil
r/(\alpha+\overline{\gamma})\rceil$ and where $\alpha$ is the upper bound of
$\alpha_{j}=\alpha_{j}(c)$ for $c\in A$, satisfying
$\alpha<\overline{\gamma}-\underline{\gamma}$. Then the first sum becomes

$\displaystyle\sum_{\nu=0}^{J}$
$\displaystyle\frac{|f_{c}^{\nu}(f_{c}(z))-f_{c}^{\nu}(c)|}{|f_{c}^{\nu}(c)|}\leq
C\sum_{\nu=1}^{J}\frac{|Df_{c}^{\nu}(c)||f_{c}(z)-c|}{e^{-\alpha\nu}}$
$\displaystyle\leq
C\sum_{\nu=0}^{J}e^{\overline{\gamma}\nu}|f_{c}(z)-c|e^{\alpha\nu}\leq
C\sum_{\nu=1}^{J}e^{(\overline{\gamma}+\alpha)\nu-dr}\leq
C^{\prime}e^{r-dr}=C^{\prime}e^{-(d-1)\Delta^{\prime}},$

which can be made arbitrarily small if $\Delta^{\prime}$ is large enough. For
the estimate of the second sum we use the binding condition directly:

$\sum_{\nu=J+1}^{j-1}\frac{|f_{c}^{\nu}(f_{c}(z))-f_{c}^{\nu}(c)|}{|f_{c}^{\nu}(c)|}\leq\sum_{\nu=J+1}^{j-1}e^{-\beta\nu}\leq
Ce^{-\beta(J+1)}\leq
C\exp\left\\{-\frac{\beta}{\alpha+\overline{\gamma}}\Delta^{\prime}\right\\}.$

Adding these two estimates, making $\delta^{\prime}$ sufficiently small,
completes the proof. ∎

#### Length of bound periods.

The bounded distortion above ensures that the bound period will never exceed
its return time, as shown in the following result. Recall that $N_{L}$ was
defined in Section 2.3 and ${\mathcal{E}}_{n}(\gamma)$ was defined in
Definition 2.2.

######  Lemma 3.2 (Length of the bound period).

Let $n\geq N_{L}$ and $\xi_{n}(c)$ be a return where
$c\in{\mathcal{E}}_{n}(\gamma)$ for some $\gamma>0$. Suppose that $p$ is the
associated bound period for this return. Then $p<n$.

###### Proof.

Assume, on the contrary, that $p\geq n$. This means that $2n$ belongs to the
bound period of $\xi_{n}(c)$. By Definition 2.1, we have

$|\xi_{2n}(c)-\xi_{n}(c)|\leq e^{-\beta n}|\xi_{n}(c)|.$

With $z=\xi_{n}(c)$ in Lemma 3.1, we have the following lower bound estimate
for the left hand side of the above inequality:

$\displaystyle|\xi_{2n}(c)-\xi_{n}(c)|$
$\displaystyle=|f_{c}^{n}(\xi_{n}(c))-f_{c}^{n}(0)|$
$\displaystyle=|f_{c}^{n-1}(\xi_{n+1}(c))-f_{c}^{n-1}(c)|$ $\displaystyle\geq
C^{\prime}|(f_{c}^{n-1})^{\prime}(c)||\xi_{n+1}(c)-c|$
$\displaystyle=C^{\prime}|(f_{c}^{n-1})^{\prime}(c)||\xi_{n}(c)|^{d},$

where $C^{\prime}>0$ is some constant which is arbitrarily close to $1$ if
$\varepsilon$ is sufficiently small. So we have that

$C^{\prime}e^{\gamma_{n-1}(n-1)-\alpha_{n}nd}\leq e^{-\beta
n}e^{-\alpha_{n}n},$

which means that

$C^{\prime}e^{\gamma_{n-1}(n-1)-\alpha_{n}n(d-1)}\leq e^{-\beta n}.$

By (2.3), one can see that

$\frac{1}{d}C^{\prime}e^{\gamma_{n}n}\leq e^{-\beta n},$

which is impossible since $\gamma_{n}\geq\gamma>0$ and $n\geq N_{L}$ and
$N_{L}$ can be taken sufficiently large by making $\varepsilon$ small. This
completes the proof. ∎

We will assume that $\tilde{N}\leq N_{L}$ (in fact the difference should be
very large).

#### Expansion during bound periods.

We have the following lemma and its corollary ensuring the derivative growth
of the critical orbit during bound periods.

######  Lemma 3.3.

Let $\xi_{n}(c)\in U^{\prime}$ and $|\xi_{n}(c)|\sim e^{-r}$ and let $p$ be
the associated bound period. Then for any fixed $\eta>0$ the following hold if
$\delta^{\prime}>0$ is sufficiently small:

$\frac{(1-\eta)dr}{\gamma_{p}+\alpha_{p+1}+\beta}\leq
p\leq\frac{(1+\eta)dr}{\gamma_{p-1}+\alpha_{p}+\beta},$ (3.1)

and

$\left|Df_{c}^{p+1}(\xi_{n}(c))\right|\geq
Ce^{(1-\eta)\frac{\gamma_{p}-(d-1)(\alpha_{p+1}+\beta)}{\gamma_{p}+\alpha_{p+1}+\beta}r}.$
(3.2)

###### Proof.

Since $p$ is the bound period for the return $\xi_{n}(c)$, we have, by
Definition 2.1, that

$\left|f_{c}^{p+1}(\xi_{n}(c))-\xi_{p+1}(c)\right|\geq
e^{-\beta(p+1)}|\xi_{p+1}(c)|.$ (3.3)

Therefore, by Lemma 3.1 there exists some constant $C^{\prime}>0$ such that

$|Df_{c}^{p}(c)||\xi_{n+1}(c)-c|\geq C^{\prime}e^{-\beta(p+1)}|\xi_{p+1}(c)|,$

which gives, since $|Df_{c}^{p}(c)|=e^{\gamma_{p}p}$ and
$|\xi_{p+1}(c)|=e^{-\alpha_{p+1}(p+1)}$,

$e^{\gamma_{p}p}e^{-dr}\geq C^{\prime\prime}e^{-(\alpha_{p+1}+\beta)(p+1)},$
(3.4)

where $C^{\prime\prime}=C^{\prime}e^{-1/2}$. By taking $\delta>0$ sufficiently
small, so that $r$ is large enough, (and thus $p$ sufficiently large) we have
the first inequality of (3.1). To see that the second inequality in (3.1) also
holds, we use Definition 2.1 and Lemma 3.1 again at the time $p$. This gives
similar estimate as above but with reversed inequalities. We leave details for
interested readers.

To obtain (3.2), first we note that, by (3.4),

$e^{-r}\geq C^{\prime\prime
1/d}e^{-\frac{1}{d}(\alpha_{p+1}+\beta)(p+1)}e^{-\frac{1}{d}\gamma_{p}p},$

which implies

$e^{-(d-1)r}\geq
C^{\prime\prime(d-1)/d}e^{-\frac{d-1}{d}(\alpha_{p+1}+\beta)(p+1)-\frac{d-1}{d}\gamma_{p}p}.$
(3.5)

Moreover,

$\displaystyle\left|Df_{c}^{p+1}(\xi_{n}(c))\right|$
$\displaystyle=|Df_{c}^{p}(\xi_{n+1}(c))||Df_{c}(\xi_{n}(c))|$
$\displaystyle\geq C^{\prime\prime}|Df_{c}^{p}(c)|e^{-(d-1)r}$
$\displaystyle\geq
C^{\prime\prime\frac{2d-1}{d}}e^{\gamma_{p}p-\frac{d-1}{d}(\alpha_{p+1}+\beta)(p+1)-\frac{d-1}{d}\gamma_{p}p}$
$\displaystyle\geq C^{\prime\prime
2}e^{-2\overline{\gamma}}e^{\frac{1}{d}(\gamma_{p}-(d-1)(\alpha_{p+1}+\beta))p}.$
(3.6)

Combining (3.6) with the first inequality of (3.1), we get

$|Df_{c}^{p+1}(\xi_{n}(c))|\geq
Ce^{(1-\eta)\frac{\gamma_{p}-(d-1)(\alpha_{p+1}+\beta)}{\gamma_{p}+\alpha_{p+1}+\beta}r},$

where $C>0$ is a (non-dynamical) constant. ∎

######  Remark 3.4.

By the definition of the bound period, one can see that $p$ cannot be a return
time. This means, for example, that $\alpha_{p}\approx 0$.

Notice that (2.5) gives us

$\gamma_{k}\geq\gamma_{k+1}+(d-1)\alpha_{k+1}-\frac{C}{k}.$

If we assume that $c\in{\mathcal{E}}_{n}(\gamma)$ then, since $p<n$ by Lemma
3.2,

$\gamma_{p}\geq\gamma+(d-1)\alpha_{p+1}-\frac{C}{p+1}.$

Therefore, assuming $\beta<\gamma/(4d)$,

$\gamma_{p}-(d-1)(\alpha_{p+1}+\beta)\geq\gamma-\beta(d-1)-\frac{C}{p+1}\geq\frac{3\gamma}{4},$
(3.7)

provided $p$ is large enough (which is the case if $\delta^{\prime}$ is small
enough). Now we assume that $\gamma\geq\underline{\gamma}$, and
$\alpha_{p+1}<\overline{\gamma}-\underline{\gamma}\leq\overline{\gamma}$ (see
Remark 2.4). Choosing $\eta$ sufficiently small in Lemma 3.3 we get

$(1-\eta)\frac{\gamma_{p}-(d-1)(\alpha_{p+1}+\beta)}{\gamma_{p}+\alpha_{p+1}+\beta}\geq(1-\eta)\frac{3\gamma}{\left(8+\frac{1}{d}\right)\overline{\gamma}}\geq\frac{\underline{\gamma}}{3\overline{\gamma}}.$

We can get rid of the constant in front of (3.2), such that

$|Df_{c}^{p+1}(\xi_{n}(c))|\geq
e^{\frac{\underline{\gamma}}{4\overline{\gamma}}r},$

given that $\delta^{\prime}$ is sufficiently small.

Now, put

$\kappa=1-\frac{\underline{\gamma}}{4\overline{\gamma}}.$

We arrive at the following corollary.

######  Corollary 3.5.

Suppose that $\xi_{n}(c)\in U^{\prime}$, $|\xi_{n}(c)|\sim e^{-r}$ and
$c\in{\mathcal{E}}_{n}(\gamma)$ for some $\gamma\geq\underline{\gamma}$.
Moreover, suppose that $\beta<\gamma/(4d)$. Then

$|Df_{c}^{p+1}(\xi_{n}(c))|\geq e^{(1-\kappa)r}.$

We finish this section with the following two useful lemmas.

######  Lemma 3.6.

Suppose that $\xi_{n_{1}}(c)$ and $\xi_{n_{2}}(c)$ are two returns,
$c\in{\mathcal{Q}}$, and that $|\xi_{n_{1}}(c)|\sim e^{-r_{1}}$ and
$|\xi_{n_{2}}(c)|\sim e^{-r_{2}}$, where $r_{1}\leq r_{2}-1$. Then the
corresponding bound periods $p_{1}$ and $p_{2}$ satisfy $p_{1}\leq p_{2}$.

###### Proof.

By bounded distortion, Lemma 3.1 (note that we can choose
$\varepsilon^{\prime}$ as small as we like, therefore $C$ below is very close
to $1$), we have, for $j\leq\min\\{p_{1},p_{2}\\}$,

$|\xi_{n_{1}+j}(c)-\xi_{j}(c)|\geq Ce^{-2r_{1}}|Df_{c}^{j}(c)|\geq
Ce^{2}e^{-2r_{2}}|Df_{c}^{j}(c)|\geq|\xi_{n_{2}+j}(c)-\xi_{j}(c)|.$

So the bound period for the second return will be longer than (or at least as
long as) the first one. ∎

######  Lemma 3.7.

Suppose that $\xi_{n_{1}}(A)$ and $\xi_{n_{2}}(A)$ are two consecutive
returns, $n_{2}>n_{1}$, where $A$ is a partition element at time $n_{2}$ and
that $A\subset{\mathcal{E}}_{n_{1}}(\gamma)$ for some $\gamma>0$, and
$\beta<\gamma/(4d)$. Then $|Df_{c}^{n_{2}-n_{1}}(\xi_{n_{1}}(c))|\geq
e^{\gamma_{2}(n_{2}-n_{1})},$ for some $\gamma_{2}\geq\gamma/(3d)$ and for
$c\in A$. In particular,

$\operatorname{diam}(\xi_{n_{2}}(A))\geq
2\operatorname{diam}(\xi_{n_{1}}(A)).$

Moreover, $|Df_{c}^{n_{2}}(c)|\geq e^{(\gamma/3)n_{2}}$ if
$\gamma\leq\gamma_{H}$.

###### Proof.

We use Lemma 3.3 and Lemma 2.8 and equations (3.6) and (3.7) to conclude, with
$p_{1}$ the bound period after $n_{1}$, and $\ell_{1}$ the following free
period,

$|Df_{c}^{p_{1}+1}(\xi_{n_{1}}(c))|\geq e^{\frac{3\gamma_{p_{1}}}{4d}p_{1}}.$

and therefore, since $\gamma_{p_{1}}\geq\gamma$,

$|Df_{c}^{n_{2}-n_{1}}(\xi_{n_{1}}(c))|\geq
e^{\frac{3\gamma}{4d}p_{1}}Ce^{\gamma_{H}\ell_{1}}\geq
e^{\gamma_{2}(n_{2}-n_{1})},$

for some $\gamma_{2}\geq\gamma/(3d)$. Moreover, by the weak parameter
dependence property,

$\operatorname{diam}(\xi_{n_{1}+p_{1}+\ell_{1}}(A))\geq
CQ^{-(p_{1}+\ell_{1})}e^{\gamma_{H}\ell_{1}}e^{(1-\kappa)r}\operatorname{diam}(\xi_{n_{1}}(A))\geq
2\operatorname{diam}(\xi_{n_{1}}(A)),$

since $r$ is large, and $Q>1$ is comparatively small ($\log
Q<\min\\{\gamma_{H},\gamma/(3d)\\}$). Since $p_{1}<n_{1}$ by Lemma 3.2, using
that $n_{1}$ is large,

$|Df_{c}^{n_{2}}(c)|\geq Ce^{\gamma
n_{1}+\frac{3\gamma}{4d}p_{1}+\gamma_{H}\ell_{1}}\geq e^{(\gamma/3)n_{2}}.$

∎

###  3.2 Main Distortion Lemma

Let us first state the main distortion lemma, see Lemma 3.14, [4]. It can be
used in our situation as well, even if the map is not slowly recurrent. The
constant $\tilde{\gamma}$ in the following lemma can be chosen to be
arbitrarily small, but we may have to adjust ${\mathcal{Q}}$, i.e.
$\varepsilon$ to it. We want to be able to use the lemma related to a small
“critical” $\gamma_{C}>0$ defined later.

######  Lemma 3.8 (Main distortion lemma).

Let $\varepsilon^{\prime}>0$ and $\tilde{\gamma}>0$. Then there exists $N$
large enough such that the following holds: If
$A\subset{\mathcal{E}}_{\nu}(\tilde{\gamma})$ is a partition element and
$\nu\geq N$ is a return time or does not belong to the bound period, and
$\nu^{\prime}$ is the next free return, then we have

$\left|\frac{Df_{c_{1}}^{n}(c_{1})}{Df_{c_{2}}^{n}(c_{2})}-1\right|\leq\varepsilon^{\prime}$
(3.8)

for $c_{1},c_{2}\in A$ and for $\nu\leq n\leq\nu^{\prime}$, provided that $A$
is still a partition element at the time $n$.

For consistency we include the proof of this important lemma, although its
proof is almost the same, verbatim, as the proof of Lemma 3.14 in [4]
(alternatively see [3, Lemma 7.3]).

###### Proof.

The exponent cannot drop too much from $\nu$ until $\nu^{\prime}$ due to Lemma
3.7. In fact, the exponent $\gamma_{n}=\gamma_{n}(c)$ at $n$, satisfies
$\gamma_{n}\geq\tilde{\gamma}/3$. Note that the weak parameter dependence
property holds as soon as $c\in{\mathcal{E}}_{n}(\tilde{\gamma}/3)$ (if we
choose $\gamma=\gamma_{Q}>0$ small enough in Lemma 2.9), so we can use Lemmas
3.2 and 3.4 in [4]. The lemma then reduces to check whether the following sum
can be made arbitrarily small:

$\Upsilon:=\sum_{j=1}^{n-1}\frac{|\xi_{j}(a)-\xi_{j}(b)|}{|\xi_{j}(b)|}.$
(3.9)

We divide the sum into three parts as follows. Let $(\nu_{k})$ be the free
returns before time $n$, where $k\leq s$. In other words, $\nu=\nu_{s}$ and
$\nu^{\prime}=\nu_{s+1}$. Let also $p_{k}$ be the length of the associated
bound period of the return $\nu_{k}$. The estimate of $\Upsilon$ is then
divided into the following parts:

$\displaystyle\Upsilon=\sum_{k=1}^{s}\sum_{j=\nu_{k-1}}^{\nu_{k-1}+p_{k-1}}\frac{|\xi_{j}(a)-\xi_{j}(b)|}{|\xi_{j,l}(b)|}$
$\displaystyle+\sum_{k=1}^{s}\sum_{j=\nu_{k-1}+p_{k-1}+1}^{\nu_{k}-1}\frac{|\xi_{j}(a)-\xi_{j}(b)|}{|\xi_{j}(b)|}$
$\displaystyle+\sum_{j=\nu_{s}}^{n-1}\frac{|\xi_{j}(a)-\xi_{j}(b)|}{|\xi_{j}(b)|}$
$\displaystyle=:\Upsilon_{B}+\Upsilon_{F}+\Upsilon_{T}.$

Here $\Upsilon_{B}$ denotes the contribution from bound periods, while
$\Upsilon_{F}$ the contribution from free periods, and $\Upsilon_{T}$ the
contribution from the last return $\nu_{s}$ up until time $n$.

For the bound periods, we use the definition of bound period, distortion, to
estimate each sum in the sums in $\Upsilon_{B}$. First we note that, by the
weak parameter dependence property, if
$|\xi_{\nu_{k-1}}(a)-\xi_{\nu_{k-1}}(b)|\sim e^{-r_{k-1}}/r_{k-1}^{2}$,

$\displaystyle|\xi_{\nu_{k-1}+1}(a)-\xi_{\nu_{k-1}+1}(b)|$ $\displaystyle\sim
e^{-r_{k-1}(d-1)}|\xi_{\nu_{k-1}}(a)-\xi_{\nu_{k-1}}(b)|$ (3.10)
$\displaystyle\sim\frac{e^{-r_{k-1}d}}{r_{k-1}^{2}}\sim\frac{|\xi_{\nu_{k-1}+1}(b)-\xi_{1}(b)|}{r_{k-1}^{2}}.$
(3.11)

Therefore,

$\displaystyle\Upsilon_{B}^{k}=\sum_{j=\nu_{k-1}}^{\nu_{k-1}+p_{k-1}}$
$\displaystyle\frac{|\xi_{j}(a)-\xi_{j}(b)|}{|\xi_{j}(b)|}\lesssim\frac{1}{r_{k-1}^{2}}+\sum_{j=\nu_{k-1}+1}^{\nu_{k-1}+p_{k-1}}\frac{|\xi_{j}(a)-\xi_{j}(b)|}{|\xi_{j}(b)|}$
(3.12)
$\displaystyle\lesssim\frac{1}{r_{k-1}^{2}}+\sum_{j=\nu_{k-1}+1}^{\nu_{k-1}+p_{k-1}}\frac{|Df^{j-\nu_{k-1}-1}(\xi_{\nu_{k-1}+1}(b))||\xi_{\nu_{k-1}+1}(a)-\xi_{\nu_{k-1}+1}(b)|}{|\xi_{j}(b)|}$
(3.13)
$\displaystyle\lesssim\frac{1}{r_{k-1}^{2}}+\sum_{j=\nu_{k-1}+1}^{\nu_{k-1}+p_{k-1}}\frac{|Df^{j-\nu_{k-1}-1}(\xi_{\nu_{k-1}+1}(b))|e^{-r_{k-1}d}}{r_{k-1}^{2}|\xi_{j}(b)|}$
(3.14)
$\displaystyle\lesssim\frac{1}{r_{k-1}^{2}}+\sum_{j=\nu_{k-1}+1}^{\nu_{k-1}+p_{k-1}}\frac{|Df^{j-\nu_{k-1}-1}(\xi_{\nu_{k-1}+1}(b))||\xi_{\nu_{k-1}+1}(b)-\xi_{1}(b)|}{r_{k-1}^{2}|\xi_{j}(b)|}$
(3.15)
$\displaystyle\lesssim\frac{1}{r_{k-1}^{2}}+\sum_{j=\nu_{k-1}+1}^{\nu_{k-1}+p_{k-1}}\frac{|\xi_{j}(b)-\xi_{j-\nu_{k-1}}(b)|}{|\xi_{j}(b)|}$
(3.16)
$\displaystyle\lesssim\frac{1}{r_{k-1}^{2}}+\sum_{j=\nu_{k-1}+1}^{\nu_{k-1}+p_{k-1}}\frac{e^{-\beta(j-\nu_{k-1})}}{r_{k-1}^{2}}\leq\frac{C}{r_{k-1}^{2}}.$
(3.17)

Given $r\geq\Delta$, let $K(r)$ be the set of indices $k$ such that
$|\xi_{\nu_{k}}(A)|\sim e^{-r}$, and let $\hat{k}(r)$ be the largest index
contained in $K(r)$. Then it follows from Lemma 3.7 that

$\Upsilon_{B}=\sum_{k=1}^{s-1}\Upsilon_{B}^{k}\leq\sum_{r\geq\Delta}\sum_{k\in
K(r)}\Upsilon_{B}^{k}\lesssim\sum_{r\geq\Delta}\Upsilon_{B}^{\hat{k}(r)}\lesssim\sum_{r\geq\Delta}\frac{1}{r^{2}}\lesssim\frac{1}{\Delta},$
(3.18)

To estimate the contribution from the corresponding free period, we use the
weak parameter dependence and Lemma 2.8 to obtain

$\displaystyle|\xi_{\nu_{k}}(a)-\xi_{\nu_{k}}(b)|$
$\displaystyle\geq\frac{1}{Q^{\nu_{k}-j}}|Df_{a}^{\nu_{k}-j}(\xi_{j}(a))|\xi_{j}(a)-\xi_{j}(b)|$
$\displaystyle\gtrsim\left(\frac{e^{\gamma_{H}}}{Q}\right)^{\nu_{k}-j}|\xi_{j}(a)-\xi_{j}(b)|$

for $\nu_{k-1}+p_{k-1}+1\leq j\leq\nu_{k}-1$. Since $Q$ is small compared to
$e^{\gamma_{H}}$ we get the each sum $\Upsilon_{F}^{k}$ in $\Upsilon_{F}$ can
be estimated by

$\Upsilon_{F}^{k}\lesssim\frac{|\xi_{\nu_{k}}(a)-\xi_{\nu_{k}}(b)|}{e^{-r}}\sum_{j=\nu_{k-1}+p_{k-1}+1}^{\nu_{k}-1}\left(\frac{Q}{e^{\gamma_{H}}}\right)^{\nu_{k}-j}\lesssim\frac{|\xi_{\nu_{k}}(a)-\xi_{\nu_{k}}(b)|}{e^{-r}}.$

Using the same argument as in the estimate of the contribution from the bound
periods, we find that

$\Upsilon_{F}=\sum_{k=1}^{s-1}\Upsilon_{F}^{k}\leq\sum_{r\geq\Delta}\sum_{k\in
K(r)}\Upsilon_{F}^{k}\lesssim\sum_{r\geq\Delta}\Upsilon_{F}^{\hat{k}(r)}\lesssim\sum_{r\geq\Delta}\frac{1}{r^{2}}\lesssim\frac{1}{\Delta}.$

We now consider the tail $\Upsilon_{T}$. It is easy to see that we may assume
that $\nu_{s}+p_{s}+1\leq n<\nu_{s+1}$, otherwise $n$ belongs to a bound
period or is simply a free return which we already have considered. For this
purpose, we consider pseudo-returns, and we let $\nu_{s}+p_{s}+1\leq
q_{1}\leq\cdots\leq q_{t}\leq n$ be the indices of these returns. By
definition, $\xi_{q_{k}}(a)\cap U^{\prime}\neq\emptyset$ and
$\xi_{q_{k}}(a)\cap U=\emptyset$. For pseudo-returns, bound periods and free
periods are defined in a similar way. As in the previous estimates, the
contribution to the distortion between any two pseudo-returns of index $q_{k}$
and $q_{k+1}$ is a constant times

$\frac{|\xi_{q_{k}}(a)-\xi_{q_{k}}(b)|}{e^{-r_{k}}}+\frac{|\xi_{q_{k+1}}(a)-\xi_{q_{k+1}}(b)|}{e^{-r_{k+1}}},$

where $r_{k}$ and $r_{k+1}$ such that $|\xi_{q_{k}}(b)|\sim e^{-r_{k}}$ and
$|\xi_{q_{k+1}}(b)|\sim e^{-r_{k+1}}$. The difference here is that, at a
pseudo-return, the only thing we know about the length of our interval is that
$|\xi_{q_{k}}(a)-\xi_{q_{k}}(b)|\leq S$, where $S=\varepsilon_{1}\delta$ is
the large scale. With similar methods and notation used for estimating the
bound and free contributions, we have

$\displaystyle\Upsilon_{T}$
$\displaystyle=\left(\sum_{j=\nu_{s}}^{q_{1}}+\sum_{k=1}^{t-1}\sum_{j=q_{k}}^{q_{k+1}-1}+\sum_{j=q_{t}}^{n-1}\right)\frac{|\xi_{j}(a)-\xi_{j}(b)|}{|\xi_{j}(b)|}$
$\displaystyle\lesssim\frac{1}{r_{s}^{2}}+\sum_{k=1}^{t}\frac{|\xi_{q_{k}}(a)-\xi_{q_{k}}(b)|}{|\xi_{q_{k}}(b)|}+\sum_{j=q_{t}}^{n-1}\frac{|\xi_{j}(a)-\xi_{j}(b)|}{|\xi_{j}(b)|}$
$\displaystyle\lesssim\frac{1}{\Delta^{2}}+\sum_{r=\Delta^{\prime}}^{\Delta}\frac{|\xi_{q_{\hat{k}(r)}}(a)-\xi_{q_{\hat{k}(r)}}(b)|}{|\xi_{q_{\hat{k}(r)}}(b)|}+\frac{S}{\delta^{\prime}}$
$\displaystyle\lesssim\frac{1}{\Delta^{2}}+\varepsilon_{1}\sum_{r=\Delta^{\prime}}^{\Delta}e^{r-\Delta}+\varepsilon_{1}\lesssim\frac{1}{\Delta^{2}}+\varepsilon_{1},$

where we in the sum from $q_{t}$ to $n-1$ used Lemma 2.8 (the first
inequality, now with respect to $U^{\prime}$) and that
$|\xi_{j}(b)|>\delta^{\prime}>\delta$ during this time.

Combining all these estimate above we arrive at

$\Upsilon=\sum_{j=1}^{n-1}\frac{|\xi_{j}(a)-\xi_{j}(b)|}{|\xi_{j}(b)|}\lesssim\frac{1}{\Delta}+\varepsilon_{1},$

and if $\delta$ and $\varepsilon_{1}$ are small enough, we reach the desired
conclusion of strong distortion.

∎

From this lemma we can now deduce the strong version of (2.12), namely the
following. For any $Q>1$, there exists $\varepsilon>0$, $N_{L}>0$, and
$\delta^{\prime}>0$ such that, for $N_{L}\leq k\leq n$ we have

$Q^{-1}|Df_{c}^{n-k}(\xi_{k}(c))||\xi_{k}(a)-\xi_{k}(b)|\\\
\leq|\xi_{n}(a)-\xi_{n}(b)|\\\ \leq
Q|Df_{c}^{n-k}(\xi_{k}(c))||\xi_{k}(a)-\xi_{k}(b)|$ (3.19)

for all $a,b,c\in A_{n}$ belonging to the same partition element.

###  3.3 Start-up

Next we prove the following lemma that provides us with a suitable initial
parameter square. (Compare with Lemma 3.15 in [4].)

######  Lemma 3.9.

Let $f_{c_{0}}$ be a Collet–Eckmann unicritical map. Let $N_{L}$ be as in
Lemma 2.9, let $\varepsilon^{\prime},\varepsilon_{1}>0$ be sufficiently small.
Then there is a neighbourhood $U$ of the critical point such that, with
$S=\varepsilon_{1}\delta$ and for each sufficiently small $\varepsilon>0$
there is $N,\tilde{N}^{\prime}$ such that $\tilde{N}^{\prime}\leq N_{L}\leq N$
and such that for all $c\in{\mathcal{Q}}$ we have the following:

  1. $(i)$

For some $\gamma\geq\underline{\gamma}(c_{0})(1-\varepsilon^{\prime})$, one
has for all $c\in{\mathcal{Q}}$

$\left|Df_{c}^{k}(c)\right|\geq e^{\gamma k}\quad(\tilde{N}^{\prime}\leq k\leq
N-1).$

  2. $(ii)$

$\mathcal{Q}$ is a partition element at time $N-1$, and $\xi_{N}(Q)$ either is
an essential return or reaches the large scale $S$.

  3. $(iii)$

For all $c_{1},c_{2}\in{\mathcal{Q}}$ one has

$\left|\frac{Df_{c_{1}}^{k}(c_{1})}{Df_{c_{2}}^{k}(c_{2})}-1\right|\leq\varepsilon^{\prime}\quad(k=0,1,\dots
N).$

  4. $(iv)$

There exists a constant $K=K(\delta,\varepsilon_{1})>1$ such that for all
$c\in\mathcal{Q}$ and $0\leq k\leq N-1$

$K^{-1}|\xi_{k}(c)|\leq|\xi_{k}(c_{0})|\leq K|\xi_{k}(c)|.$

We will use this lemma where $\varepsilon^{\prime}\leq 1/10$, so that
$\tilde{N}^{\prime}$ can be set equal to $\tilde{N}$ in Definition 2.2.

###### Proof.

For any complex $c_{1},c_{2},z,w$ we have

$\left|\frac{Df_{c_{1}}(z)}{Df_{c_{2}}(w)}-1\right|\leq\left|\frac{z-w}{w^{d-1}}\sum_{k=0}^{d-2}z^{k}w^{d-2-k}\right|\leq\left|\frac{z-w}{w}\right|\sum_{k=0}^{d-2}\left(\left|\frac{z-w}{w}\right|+1\right)^{k}.$

Assuming that $z$ and $w$ belong to a partition element, we have that

$\left|\frac{z-w}{w}\right|\leq\begin{cases}\,\Delta^{-2}\ &\mbox{if}\ z\in U\
\mbox{or}\ w\in U,\\\ \,\varepsilon_{1}\ &\mbox{if}\
z,w\in\mathbb{C}\smallsetminus U.\end{cases}$

With $\eta=\min(\Delta^{-2},\varepsilon_{1})$ we therefore have

$\left|\frac{Df_{c_{1}}(z)}{Df_{c_{2}}(w)}-1\right|\leq(1+\eta)^{d-1}-1.$

In particular,

$\left|\frac{Df_{c_{1}}(z)}{Df_{c_{2}}(w)}\right|\leq C,$

with $C>1$ arbitrarily close to $1$ if $\delta$ and $\varepsilon_{1}$ are
sufficiently small. Assuming that ${\mathcal{Q}}$ is a partition element at
time $n$, and that $c\in{\mathcal{Q}}$, we get from the above distortion that

$|Df^{k}_{c}(c)|\geq C^{-k}|Df_{c_{0}}(c_{0})|\geq
C^{-k}C_{0}e^{\gamma_{0}k}\geq C_{0}e^{\gamma k}.$

This tells us that as long as ${\mathcal{Q}}$ is a partition element $f_{c}$,
with $c\in{\mathcal{Q}}$, will satisfy a Collet–Eckmann condition. We may thus
apply Lemma 2.9 to conclude that our interval will expand:

$\operatorname{diam}\left(\xi_{k}({\mathcal{Q}})\right)\sim|\xi_{k}^{\prime}(c^{\prime})|\operatorname{diam}\left({\mathcal{Q}}\right)\sim
e^{\gamma k}\operatorname{diam}\left({\mathcal{Q}}\right).$

In particular we can make the perturbation small enough to find the time $N$
such that (i) and (ii) are valid.

By the choice of $N_{L}$, we can choose $\varepsilon>0$ sufficiently small
such that ${\mathcal{Q}}\subset{\mathcal{E}}_{N_{L}}(\gamma)$ for some
$\gamma$ arbitrarily close to $\gamma_{0}$.

By the definition of a partition element we can use Lemma 3.8 repeatedly
starting from the time $N_{L}$ up to $N-1$ to get the distortion claimed in
(iii).

To prove (iv) suppose first that $\xi_{k}({\mathcal{Q}})$ is a return to $U$,
and let $r$ be the integer satisfying

$e^{-r}\leq\operatorname{dist}(\xi_{k}(\mathcal{Q}),0)\leq e^{-r+1}.$

Then $\operatorname{diam}\left(\xi_{k}(\mathcal{Q})\right)\leq
e^{-r+1}/r^{2}$, hence for every $c\in\mathcal{Q}$ we have that

$\displaystyle|\xi_{k}(c)|$
$\displaystyle\geq|\xi_{k}(c_{0})|-\operatorname{diam}\xi_{k}(\mathcal{Q})$
$\displaystyle\geq\left(1-\frac{e^{-r+1}}{|\xi_{k}(c_{0})|r^{2}}\right)|\xi_{k}(c_{0})|.$

Since $|\xi_{k}(c_{0})|\geq\operatorname{dist}(\xi_{k}(\mathcal{Q}),0)\geq
e^{-r}$ and $r\geq\Delta$, we conclude that

$|\xi_{k}(c)|\geq\left(1-\frac{e}{\Delta^{2}}\right)|\xi_{k}(c_{0})|.$

On the other hand, if we are outside of $U$, then

$|\xi_{k}(c)|\geq\left(1-\frac{S}{\delta}\right)|\xi_{k}(c_{0})|=(1-\varepsilon_{1})|\xi_{k}(c_{0})|.$

This proves one of the inequalities, and the other one follows from exchanging
$c_{0}$ and $c$ in the above. ∎

##  4 Promotion

In this part we introduce the key idea of _promotion_ which means that most of
the parameters in ${\mathcal{Q}}$ will reach a slow recurrence condition after
several essential returns.

By Lemma 3.9, all critical orbits for parameters in ${\mathcal{Q}}$ will
follow the starting critical orbit up to time $N$ with bounded distortion and
exponential growth of derivatives. If ${\mathcal{Q}}$ has grown to the large
scale under $\xi_{N}$, then $\xi_{N}({\mathcal{Q}})$ covers mostly the Fatou
set of $f_{0}$, and by a compactness argument, we can conclude that most
parameters in ${\mathcal{Q}}$ will land in the Fatou set of the corresponding
map under $\xi_{n}$; see next section for more details. Hence a very large
portion of parameters are hyperbolic in ${\mathcal{Q}}$. Therefore, we may
assume, without loss of generality, that $\xi_{N}({\mathcal{Q}})$ has not
reached the large scale yet and thus is an essential return.

At this time we may have to delete parameters that returns too deep at time
$N$ in comparison to the starting map. Let $C_{1}>1$ be given and consider
those $c\in{\mathcal{Q}}$ such that

$\frac{\alpha_{N}(c)}{\alpha_{N}(c_{0})}\leq C_{1}.$ (4.1)

This means that we delete parameters of measure

$m\left(\\{c\in{\mathcal{Q}}:c\text{ does not satisfy \eqref{appr2-ess}
}\\}\right)\leq
m({\mathcal{Q}})\frac{e^{-2C_{1}\alpha_{N}(c_{0})N}}{e^{-2\alpha_{N}(c_{0})N}}=\colon\delta_{0}m({\mathcal{Q}}),$

which can be made arbitrarily small if $N$ is large enough. We make the
deletion so that the smallest collection of whole partition elements are
deleted (this means that we may delete a little more than necessary). The
remaining parameters are partitioned into partition elements as above. Each of
these partition elements, containing $c$, is denoted by $A_{N}(c)$. Sometimes
we just write $A_{N}$. However, note that $A_{N-1}={\mathcal{Q}}$ (there is no
partition before time $N$).

After this time $N$ we lose this kind of dependence on the starting orbit. So
we need to control recurrence rates so that we do not lose a lot of derivative
growth.

Let $N=\nu_{0}<\nu_{1}<\cdots$ be the free returns after time $N$. Suppose
that $A$ is a partition element at time $\nu_{j}$. We will introduce two
sequences $\\{\widetilde{\alpha}_{\nu_{j}}(A)\\}$ and
$\\{\underline{\gamma}_{\nu_{j}}(A)\\}$ inductively which capture two key
features for later analysis. Firstly, parameters $c\in A$ for which
$\xi_{\nu_{j}}(c)$ are at distance
$e^{-\widetilde{\alpha}_{\nu_{j}}(A)\nu_{j}}$ away from the critical point
will be kept (and thus has controlled approach rate to the critical point),
and secondly, we do not lose a lot of the derivative growth between two free
returns.

We start with the time $\nu_{0}$. By Lemma 3.9 (iv), $\xi_{k}(c)$ stays close
to $\xi_{k}(c_{0})$ and thus $\alpha_{k}(c)$ is comparable to
$\alpha_{k}(c_{0})$ for all $k\leq N$ and all $c\in A_{N}$.

#### Definition of $\widetilde{\alpha}_{\nu_{0}}(A)$.

Put

$\widetilde{\alpha}_{\nu_{0}}(A_{N})=\sup_{c\in A_{N}}\alpha_{\nu_{0}}(c).$

Let $p_{0}$ and $\ell_{0}$ be the length of bound and free periods for the
return time $\nu_{0}$, respectively. The following lemma is a consequence of
Lemma 3.3.

######  Lemma 4.1.

Let $|\xi_{\nu_{0}}(c)|\sim e^{-r}$, for some $c\in A_{N}$, where
$A_{N}\subset{\mathcal{Q}}$ is a partition element satisfying (4.1). Then
$|Df_{c}^{p_{0}+1}(\xi_{\nu_{0}}(c))|\geq e^{(1-\kappa)r}$.

###### Proof.

We can make $C_{1}$ as close to $1$ as we want so that $c\in A_{N}$ implies
$c\in{\mathcal{E}}_{n}(\underline{\gamma})$. The lemma now follows from
Corollary 3.5. ∎

Note that by the definition of $\kappa$, Lemma 4.1 says that
$\operatorname{diam}\left(\xi_{\nu_{0}+p_{0}}(A_{N})\right)\geq e^{-\kappa
r}$, where $\operatorname{diam}\left(\xi_{\nu_{0}}(A_{N})\right)\sim e^{-r}$.
Now we fix some $\kappa^{\prime}$ such that $\kappa<\kappa^{\prime}<1$. In the
following, we shall define $\\{\widetilde{\alpha}_{\nu_{j}}(A)\\}$ and
$\\{\underline{\gamma}_{\nu_{j}}(A)\\}$, where $A$ is a partition element at
the free return time $\nu_{j}$.

Assume that $\operatorname{dist}(0,\xi_{\nu_{j}}(A))\sim e^{-r_{j}}\sim
e^{-\alpha_{\nu_{j}}\nu_{j}}$, and $A=A_{\nu_{j}}$ is a partition element. The
lengths of the bound and free periods for this return are $p_{j}$ and
$\ell_{j}$, respectively. Define

$\widetilde{\alpha}_{\nu_{j+1}}(A)=\kappa^{\prime}\frac{\nu_{j}}{\nu_{j+1}}\sup\limits_{c\in
A}{\alpha_{\nu_{j}}(c)}\quad\left(\leq\kappa^{\prime}\sup\limits_{c\in
A}\alpha_{\nu_{j}}(c)\right).$ (4.2)

We do this at each return (including inessential returns).

Note that since $\alpha_{\nu_{j}}(c)$ is almost constant in $A$, we have, for
some constant $\tilde{\kappa}$ arbitrarily close to $1$ with
$1>\tilde{\kappa}>\kappa^{\prime}$, that for any $c,\,c^{\prime}\in A$,

$\tilde{\kappa}\alpha_{\nu_{j}}(c^{\prime})\leq\alpha_{\nu_{j}}(c)\leq\frac{1}{\tilde{\kappa}}\alpha_{\nu_{j}}(c^{\prime}).$

This, together with (4.2), gives

$\tilde{\kappa}\kappa^{\prime}\frac{\nu_{j}}{\nu_{j+1}}\alpha_{\nu_{j}}(c)\leq\widetilde{\alpha}_{\nu_{j+1}}(A)\leq\frac{\kappa^{\prime}}{\tilde{\kappa}}\frac{\nu_{j}}{\nu_{j+1}}\alpha_{\nu_{j}}(c)$
(4.3)

for all $c\in A$.

At the next return we are going to delete from $A$ (which is a partition
element at time $\nu_{j}$) parameters not satisfying

$|\xi_{\nu_{j+1}}(c)|\geq e^{-\widetilde{\alpha}_{\nu_{j+1}}(A)\nu_{j+1}}.$
(4.4)

In other words, if $c$ is deleted, then

$\alpha_{\nu_{j+1}}(c)>\tilde{\alpha}_{\nu_{j+1}}(A).$

We do this exclusion at each free return (actually it is going to be each
essential return), see Lemma 4.3 below. For a partition element $A$ at the
return $\nu_{j}$, let us also define

$\alpha_{\nu_{j}}(A)=\sup\limits_{c\in A}\alpha_{\nu_{j}}(c).$

So we have a procedure; given $A=A_{\nu_{j}}$ for some essential return
$\xi_{\nu_{j}}(A)$ with $\operatorname{dist}(\xi_{\nu_{j}}(A),0)\sim
e^{-\alpha_{\nu_{j}}(A)\nu_{j}}$, this induces an
$\widetilde{\alpha}_{\nu_{j+1}}(A)$ at the next return, i.e. we have a mapping

$\alpha_{\nu_{j}}(A_{\nu_{j}})\mapsto\widetilde{\alpha}_{\nu_{j+1}}(A_{\nu_{j}}),$

so that

$\alpha_{\nu_{j+1}}(A_{\nu_{j+1}})\leq\widetilde{\alpha}_{\nu_{j+1}}(A_{\nu_{j}})$

for all partition elements $A_{\nu_{j+1}}\subset A_{\nu_{j}}$ that did not get
deleted according to the rule (4.4).

######  Lemma 4.2.

Let $c\in A$ satisfy (4.4). Then $|Df_{c}^{p_{j+1}+1}(\xi_{\nu_{j+1}}(c))|\geq
e^{\frac{\underline{\gamma}}{2d\overline{\gamma}}r_{j}}$.

###### Proof.

For $c$ satisfying (4.4), by (4.3) we have

$|\xi_{\nu_{j+1}}(c)|\geq e^{-\widetilde{\alpha}_{\nu_{j+1}}(A)\nu_{j+1}}\geq
e^{-(\kappa^{\prime}/\tilde{\kappa})\alpha_{\nu_{j}}(c)\nu_{j}}>e^{-\alpha_{\nu_{j}}(c)\nu_{j}}=|\xi_{\nu_{j}}(c)|,$
(4.5)

where we have used the fact that $\kappa^{\prime}<\tilde{\kappa}$ in the third
inequality. Inductively, we see that
$|\xi_{\nu_{j+1}}(c)|>|\xi_{\nu_{0}}(c)|$. By Lemma 3.6, the length of the
bound period for $\nu_{j+1}$ satisfies $p_{j+1}\leq p_{0}$, where $p_{0}$ is
the length of the bound period for the return at the start-up time $\nu_{0}$.

Then the lemma follows in the same way as in the proof of Lemma 4.1. ∎

It turns out that parameter exclusions happen only at essential returns due to
the following simple observation.

######  Lemma 4.3.

Let $A$ be a partition element at time $\nu_{j}$. If the free return
$\xi_{\nu_{j+1}}(A)$ is inessential, then (4.4) is satisfied for all $c\in A$.

###### Proof.

To prove this happens,
$\alpha_{\nu_{j+1}}(c)\leq\widetilde{\alpha}_{\nu_{j+1}}(A)$ for all $c\in A$.
Assume that $|\xi_{\nu_{j+1}}(c)|\sim
e^{-r_{j+1}}=e^{-\alpha_{\nu_{j+1}}(c)\nu_{j+1}}$. By the definition of
inessential returns (cf. Section 2), we have

$\operatorname{diam}\xi_{\nu_{j+1}}(A)\leq\frac{1}{2}\frac{e^{-r_{j+1}}}{r_{j+1}^{2}}.$

By the above Lemma 4.2, we also have

$\operatorname{diam}\xi_{\nu_{j+1}}(A)\geq e^{-\kappa r_{j}}.$

Combining the above we obtain

$\kappa\alpha_{\nu_{j}}(c)\nu_{j}=\kappa r_{j}\geq
r_{j+1}=\alpha_{\nu_{j+1}}(c)\nu_{j+1},$

which gives

$\alpha_{\nu_{j+1}}(c)\leq\kappa\alpha_{\nu_{j}}(c)\frac{\nu_{j}}{\nu_{j+1}}\leq\frac{\kappa}{\kappa^{\prime}}\widetilde{\alpha}_{\nu_{j+1}}(A)<\widetilde{\alpha}_{\nu_{j+1}}(A)$

since $\kappa<\kappa^{\prime}$. ∎

Recall that $A=A_{\nu_{j}}$ is a partition element at time $\nu_{j}$ so we set
$\widetilde{\alpha}_{k}(A)=\widetilde{\alpha}_{\nu_{j+1}}(A)$ for
$\nu_{j}<k\leq\nu_{j+1}$, so that $\widetilde{\alpha}_{k}(A)$ is defined also
between returns.

#### Definition of $\underline{\gamma}_{\nu_{j}}$.

Recall inequality (2.4): for $n$ large enough we have

$\gamma_{n}\geq\gamma_{n-1}-(d-1)\alpha_{n}-\frac{C}{n}.$

Let $A_{n}$ be a partition element at time $n$ and put

$\gamma_{n}(A_{n})=\inf\limits_{c\in A_{n}}\gamma_{n}(c).$

We define

$\underline{\gamma}_{\nu_{0}}(A_{\nu_{0}})=\gamma_{\nu_{0}-1}(A_{\nu_{0}})-(d-1)\alpha_{\nu_{0}}(A_{\nu_{0}})-\frac{C}{\nu_{0}},$

so that
$\underline{\gamma}_{\nu_{0}}(A_{\nu_{0}}(c_{0}))=\underline{\gamma}_{N}$ is
at least $\gamma$ from Lemma 3.9 (i) (for some suitable
$\varepsilon^{\prime}\leq 1/10$). For a partition element $A$ at time
$\nu_{j}$ we inductively define

$\underline{\gamma}_{\nu_{j+1}}(A)=\left(\underline{\gamma}_{\nu_{j}}(A)+\gamma_{H}\frac{\ell_{j}}{\nu_{j}}\right)\frac{\nu_{j}}{\nu_{j+1}-1},$
(4.6)

and put
$\underline{\gamma}_{\nu_{j+1}-1}(A)=\underline{\gamma}_{\nu_{j+1}}(A)$. We
will also define $\underline{\gamma}_{n}$ when $n$ is not a return time below.

######  Remark 4.4.

Note that $\underline{\gamma}_{\nu_{j}}$ assigns a number for each partition
element. So, if $A_{j}$ and $A_{j+1}$ are respectively partition elements at
time $\nu_{j}$ and $\nu_{j+1}$ and moreover $A_{j+1}\subset A_{j}$, we have
$\underline{\gamma}_{\nu_{j}}(A_{j+1})=\underline{\gamma}_{\nu_{j}}(A_{j})$.

Suppose $\underline{\gamma}_{k}(A)$ is defined for $\nu_{0}\leq k\leq\nu_{j}$,
that $\underline{\gamma}_{\nu_{j}}(A)=\underline{\gamma}_{\nu_{j}-1}(A)$, that
(for $j\geq 1$)

$\gamma_{k}(c)\geq\underline{\gamma}_{k}(A)+(d-1)\alpha_{k+1}(c),\text{ for
$\nu_{j-1}<k<\nu_{j}$},$ (4.7)

and that $\gamma_{\nu_{j}}(c)\geq\underline{\gamma}_{\nu_{j}}(A)$ at returns.

If we follow a partition element $A$, and $c\in A$, we get, for those
parameters not deleted, using that $|Df^{p_{j}+1}(\xi_{\nu_{j}}(c))|\geq
e^{(1-\kappa)r_{j}}$ by Corollary 3.5,

$\displaystyle\gamma_{\nu_{j+1}}(c)\geq\gamma_{\nu_{j+1}-1}(c)-(d-1)\alpha_{\nu_{j+1}}(c)-\frac{C}{\nu_{j+1}}$
(4.8)
$\displaystyle\geq\left(\gamma_{\nu_{j}-1}(c)(\nu_{j}-1)+(1-\kappa)\alpha_{\nu_{j}}(c)\nu_{j}+\gamma_{H}\ell_{j}\right)\frac{1}{\nu_{j+1}-1}$
(4.9) $\displaystyle-(d-1)\tilde{\alpha}_{\nu_{j+1}}(A)-\frac{C}{\nu_{j+1}}$
(4.10)
$\displaystyle\geq\left(\underline{\gamma}_{\nu_{j}-1}(A)\left(1-\frac{1}{\nu_{j}}\right)+(d-1)\alpha_{\nu_{j}}(c)\left(1-\frac{1}{\nu_{j}}\right)\right)\frac{\nu_{j}}{\nu_{j+1}-1}$
(4.11)
$\displaystyle+\left((1-\kappa)\alpha_{\nu_{j}}(c)+\gamma_{H}\frac{\ell_{j}}{\nu_{j}}\right)\frac{\nu_{j}}{\nu_{j+1}-1}-(d-1)\frac{\kappa^{\prime}}{\tilde{\kappa}}\alpha_{\nu_{j}}(c)\frac{\nu_{j}}{\nu_{j+1}}-\frac{C}{\nu_{j+1}}$
(4.12)
$\displaystyle\geq\left(\underline{\gamma}_{\nu_{j}-1}(A)\left(1-\frac{1}{\nu_{j}}\right)+\gamma_{H}\frac{\ell_{j}}{\nu_{j}}-\frac{C}{\nu_{j}}\right)\frac{\nu_{j}}{\nu_{j+1}-1}$
(4.13)
$\displaystyle+\left((d-1)\left(1-\frac{1}{\nu_{j}}-\frac{\kappa^{\prime}}{\tilde{\kappa}}\right)+(1-\kappa)\right)\alpha_{\nu_{j}}(c)\frac{\nu_{j}}{\nu_{j+1}-1}.$
(4.14)

Note also that, since $\kappa^{\prime}<\tilde{\kappa}$,

$\frac{\underline{\gamma}_{\nu_{j}-1}+2C}{\nu_{j}}\leq\left((d-1)\left(1-\frac{1}{\nu_{j}}-\frac{\kappa^{\prime}}{\tilde{\kappa}}\right)+(1-\kappa)\right)\alpha_{\nu_{j}}$

for sufficiently large $\nu_{j}\geq N$, since
$\alpha_{\nu_{j}}(c)\nu_{j}=r_{\nu_{j}}\geq\Delta$ and $\Delta$ is large.
Then, using that $p_{j}\leq\nu_{j}$ by Lemma 3.2, we get
$\gamma_{\nu_{j+1}}(c)\geq\underline{\gamma}_{\nu_{j+1}}(A)$. If we define
$\underline{\gamma}_{k}(A)=\underline{\gamma}_{\nu_{j+1}}(A)$ for
$\nu_{j}<k\leq\nu_{j+1}$, then it follows that

$\gamma_{k}(c)\geq\underline{\gamma}_{k}(c)+(d-1)\alpha_{k+1}(c),\text{ for
$\nu_{j}<k<\nu_{j+1}$.}$ (4.15)

So at returns we have the weaker inequality
$\gamma_{\nu_{j+1}}(c)\geq\underline{\gamma}_{\nu_{j+1}}(A)$. With these
definitions, we have that $\alpha_{j}(c)\leq\tilde{\alpha}_{j}(A)$ and
$\gamma_{k}(c)\geq\underline{\gamma}_{k}(c)$, with possible equality only at
returns. We also define $\underline{\gamma}_{j}=\underline{\gamma}$ for $j<N$.
We now make the following definition.

######  Definition 4.5.

Parameters satisfying

$|\xi_{j}(c)|=e^{-\alpha_{j}(c)j},$

where $\alpha_{j}(c)\leq\tilde{\alpha}_{j}(A)$ for all $c\in A=A_{j}(c)$ and
for all $j\leq n$, are called parameters satisfying the basic approach rate
condition until time $n$. We denote such parameters with ${\mathcal{B}}_{n}$.

We will continue updating $\underline{\gamma}_{n}$ and $\tilde{\alpha}_{n}$ as
above until we reach a certain situation as described as follows. Let us now
consider the consecutive free returns $N=\nu_{0},\nu_{1},\nu_{2},\ldots$.

We see that

$\frac{\tilde{\alpha}_{\nu_{j}}(c)}{\underline{\gamma}_{\nu_{j}}(c)}\leq\kappa^{\prime
j}\frac{\tilde{\alpha}_{\nu_{0}}(A)}{\underline{\gamma}_{\nu_{0}}(A)},\qquad\text{
where $c\in A$}.$

Now we choose the smallest $J\geq 0$ so that,
$\tilde{\alpha}_{\nu_{J}}\leq\tilde{C}\underline{\gamma}_{\nu_{J}}$, where
$\tilde{C}$ is sufficiently small (depending on the starting function etc.),
defined later. So $\nu_{J}$ is a time when the $\alpha_{\nu_{J}}(c)$ is
(sufficiently) small compared to $\underline{\gamma}_{\nu_{J}}$. The return
times $\nu_{J}=\nu_{j}(c)$ depend on the parameter and are of course also
constant on each partition element. We call $\nu_{J}=\nu_{J}(c)$ the end of
the promotion, or the time period of the promotion for the corresponding
parameter; the $\alpha_{j}$ becomes promoted in a sense to a small
$\alpha=\alpha_{\nu_{J}}$.

It may seem that the same $\kappa$ cannot be used during the promotion period,
since the Lyapunov exponent after time $N$ may go down. However, since
$1>\kappa^{\prime}>\kappa$ we keep only parameters that return further out
from the critical point, i.e. if $\xi_{\nu_{j}}(A_{j})$ is an essential
return, and $A_{j}\subset{\mathcal{Q}}$ is a partition element, then
$\xi_{\nu_{j+1}}(A_{j})$ either has reached the large scale before it returns
(good situation), is an inessential return, or an essential return. If it is
inessential or essential, the a partition element in $A_{j}$, let us call it
$A_{j+1}\subset A_{j}$, after deletion by the basic approach rate condition,
will satisfy

$|\xi_{\nu_{j+1}}(c)|\geq e^{-\kappa^{\prime}r_{j}}\gg
e^{-r_{j}}\sim|\xi_{\nu_{j}}(c)|,$

for $c\in A_{j+1}$. Hence by Lemma 3.6 the corresponding bound periods satisfy
$p_{j+1}\leq p_{j}\leq\ldots\leq p_{0}<N$ (the last inequality follows from
Lemma 3.2). But this means that the corresponding
$\gamma_{p_{j}}\geq\underline{\gamma}$, hence Corollary 3.5 can be used during
the whole promotion period for all returns $\nu_{j}$ up until $j=J$. For later
use we put

$\gamma_{I}=\min\\{\underline{\gamma},\gamma_{H}\\}\frac{\underline{\gamma}(1-\kappa^{\prime})}{2\alpha_{\nu_{0}}d+\underline{\gamma}(1-\kappa^{\prime})}.$

It will be shown that this is a lower Lyapunov exponent during the promotion
period.

The next observation is that the measure of parameters deleted between two
consecutive essential returns during the promotion period is exponentially
small in terms of the return time of the former return. See Lemma 8.1 in [3].

######  Lemma 4.6.

Let $\xi_{\nu}(A)$ be an essential return and $\nu$ belonging to the promotion
period for $A\subset{\mathcal{E}}_{\nu}(\gamma_{I})\cap{\mathcal{B}}_{\nu}$
and let $\xi_{\nu^{\prime}}(A)$ be the next essential return. Then if
$\hat{A}$ is the set of parameters in $A$ that satisfy the basic approach rate
condition (${\mathcal{B}}_{\nu^{\prime}}$) at time $\nu^{\prime}$, we have

$m(\hat{A})\geq\left(1-e^{-(3/2)(\kappa^{\prime}-\kappa)\alpha_{\nu}\nu}\right)m(A).$

###### Proof.

We know that $\xi_{\nu}(A)$ grows rapidly during the bound period $p$. By
Lemmas 3.1 and 3.8 and the definition of the bound period, we get, for any
$c\in A$,

$\operatorname{diam}(\xi_{\nu+p+1}(A))\sim\frac{e^{-2r}}{r^{2}}|Df_{c}^{p}(\xi_{\nu+1}(c))|\geq
e^{-\kappa r},$ (4.16)

where $r=\alpha_{\nu}\nu$ and $\alpha_{\nu}=\alpha_{\nu}(A)$.

If there are no inessential returns between $\nu$ and $\nu^{\prime}$, then
those parameters that get deleted satisfy
$\alpha_{\nu^{\prime}}=\alpha_{\nu^{\prime}}(c)>\tilde{\alpha}_{\nu^{\prime}}(A)=\kappa^{\prime}(\nu/\nu^{\prime})\alpha_{\nu}(A)$.
However, if we have, say, $s$ inessential returns between $\nu$ and
$\nu^{\prime}$, then we get $\alpha_{\nu^{\prime}}>\kappa^{\prime
s+1}(\nu/\nu^{\prime})\alpha_{\nu}$ (including the last return $\nu^{\prime}$
which is essential). The growth between each pair of inessential returns
$\xi_{m_{j}}(A)$ and $\xi_{m_{j+1}}(A)$ can be estimated by

$|Df_{c}^{p_{j}}(\xi_{m_{j}}(c))|\geq e^{(1-\kappa)r_{j}},$

where, by definition (see proof of Lemma 4.3), $\kappa r_{j}\geq r_{j-1}$. So
the total diameter becomes

$\displaystyle\operatorname{diam}(\xi_{\nu^{\prime}}(A))$ $\displaystyle\geq
e^{-\kappa r+(1-\kappa)r_{1}+(1-\kappa)r_{2}+\ldots+(1-\kappa)r_{s}}$ (4.17)
$\displaystyle\geq e^{-\kappa
r+(1-\kappa)r(\kappa^{-1}+\ldots\kappa^{-s})}=e^{-r(\kappa+1-\kappa^{-s})}.$
(4.18)

Note that by Lemma 4.3, no parameters are deleted during inessential returns.

Recall that $r=\alpha_{\nu}\nu$. By the main distortion Lemma 3.8 together
with Lemma 2.9, we see that the measure of parameters deleted at time
$\nu^{\prime}$ is

$\displaystyle\frac{m(A)-m(\hat{A})}{m(A)}$ $\displaystyle\leq
C\,\frac{(2e^{-\alpha_{\nu^{\prime}}\nu^{\prime}})^{2}}{\operatorname{diam}(\xi_{\nu^{\prime}}(A))^{2}}$
(4.19) $\displaystyle\leq
4Ce^{-2\alpha_{\nu^{\prime}}\nu^{\prime}+2(\kappa+1-\kappa^{-s})\alpha_{\nu}\nu}\leq
4Ce^{-2(\kappa^{\prime s+1}-1-\kappa+\kappa^{-s})\alpha_{\nu}\nu},$ (4.20)

where $C$ is a distortion constant (close to $1$). Since
$\kappa^{\prime}>\kappa$, we have $\kappa^{\prime
s+1}-1-\kappa+\kappa^{-s}\geq\kappa^{\prime}-\kappa>0$ for all $s\geq 0$. This
finishes the proof since $\alpha_{\nu}\nu=r$ is large (so that the constant
$4C$ in front is eaten up). ∎

We end this section by summarising the results above. We are going to continue
the promotion until $\alpha_{\nu_{j}}$ is small enough, but not too small.
With a prescribed $\alpha=\alpha_{\nu_{J}}>0$ (defined later) we get the
following, if $\alpha_{\nu_{j}}\geq\alpha$.

######  Proposition 4.7.

Let $\nu_{0}=N$ as before. Then during the promotion period, from $\nu_{0}$
until $\nu_{J}$, the set of parameters that satisfy ${\mathcal{B}}_{\nu_{J}}$,
has measure at least

$\left[\prod_{j=0}^{J-1}\left(1-e^{-(3/2)(\kappa^{\prime}-\kappa)\alpha_{\nu_{j}}\nu_{j}}\right)\right](1-\delta_{0})m({\mathcal{Q}})=(1-\delta_{1})(1-\delta_{0})m({\mathcal{Q}}).$

So the remaining parameters constitute a finite union of squares, were the
$\alpha=\alpha_{\nu_{J}}(c)$ are small, sufficiently small in order to use the
large deviation argument. Note that $\delta_{1}$, the portion deleted, can be
made arbitrarily small. Heuristically, the method of promotion indicates that
all CE-maps are density points of slowly recurrent maps, since one can choose
$\alpha>0$ arbitrarily small (possibly diminishing the diameter of
${\mathcal{Q}}$).

##  5 Conclusion

We are going to consider iterates from some starting time $m_{0}$ (not
necessarily $N$) up to $(1+\iota)m_{0}$ for some suitable $\iota>0$. As in
previous papers, [7, 8] and [4] we define escape time as follows.

######  Definition 5.1.

We say that $\xi_{n}(A_{n-1})$ has escaped or is in escape position, if $n$
does not belong to a bound period and
$\operatorname{diam}(\xi_{n}(A_{n-1}))\geq S$. We also speak of escape
situation for $A_{n-1}$ and say that $A_{n-1}$ has escaped if
$\xi_{n}(A_{n-1})$ has escaped.

Escape situations are the ideal situations since it means that a partition
element has grown to the large scale under $\xi_{n}$. If we are lucky, even
${\mathcal{Q}}$ grows to the large scale, and then $\xi_{n}({\mathcal{Q}})$
covers mostly the Fatou set of $f_{0}$, and by a compactness argument, we can
conclude that most parameters in ${\mathcal{Q}}$ will land in the Fatou set of
the corresponding map under $\xi_{n}$. Hence a very large portion of
parameters are hyperbolic in ${\mathcal{Q}}$. If $\xi_{n}({\mathcal{Q}})$ does
not escape, we have to continue with the ordinary partitions. Also since the
starting map is assumed to not be slowly recurrent, we want to delete
parameters so that the remaining parameters are “promoted” in a way that after
a finite number of free returns, these parameters have reached a kind of slow
recurrence condition.

###  5.1 Large deviations

In this section we choose $\tilde{C}$ so that
$\alpha_{\nu_{J}}/\underline{\gamma}_{\nu_{J}}$ is small enough. Let us see
how the Lyapunov exponent changes during the time of promotion. During the
promotion period between $N=\nu_{0}$ and $\nu_{J}=m_{0}$ the Lyapunov exponent
cannot drop too much, since all the bound periods $p_{j}$ are bounded by
$p_{0}$, by Lemma 3.6. But we have something stronger. Recall that
$\underline{\gamma}_{j}$ is also defined for $j<N$, namely
$\underline{\gamma}_{j}=\underline{\gamma}$, and
$\gamma_{p_{j}}\geq(d-1)\alpha_{p_{j}+1}+\underline{\gamma}_{p_{j}}$, where
$\underline{\gamma}_{p_{j}}=\underline{\gamma}$ since $p_{j}\leq p_{0}<N$. By
Lemma 3.3, we have, if $0<\eta<1$,

$p_{j+1}\leq\frac{(1+\eta)dr_{j+1}}{\gamma_{p_{j+1}-1}+\alpha_{p_{j+1}}+\beta}\leq\frac{2d\kappa^{\prime}r_{j}}{\underline{\gamma}}\leq\frac{2d\kappa^{\prime
j}r_{0}}{\underline{\gamma}}.$

So the sum of all bound periods during the promotion time is bounded by

$\sum_{j=0}^{J}p_{j}\leq\sum_{j=0}^{J}\frac{2d\kappa^{\prime
j}r_{0}}{\underline{\gamma}}\leq\frac{2dr_{0}}{\underline{\gamma}}\frac{1}{1-\kappa^{\prime}}.$

If we set the Lyapunov exponent to be zero at the bound periods in the
promotion period, and count only the free periods $\ell_{j}$ after $\nu_{0}$,
we get the following estimate of the Lyapunov exponent after the promotion
period, if $\sum_{j}\ell_{j}$ is all the free periods after $\nu_{0}$:

$\gamma_{\nu_{J}}\geq\frac{\underline{\gamma}\nu_{0}+\gamma_{H}\sum_{j}\ell_{j}}{\frac{2d\alpha_{\nu_{0}}}{\underline{\gamma}}\frac{1}{1-\kappa^{\prime}}\nu_{0}+\nu_{0}+\sum_{j}\ell_{j}}\geq\min\\{\underline{\gamma},\gamma_{H}\\}\frac{\underline{\gamma}(1-\kappa^{\prime})}{2\alpha_{\nu_{0}}d+\underline{\gamma}(1-\kappa^{\prime})}=\gamma_{I}.$
(5.1)

######  Remark 5.2.

According to Remark 3.4, $p_{j}$ cannot be a return time, but $p_{j}+1$ can.
Moreover, if the bound period ends so that $p_{j}+1$ is not a return time,
then $\xi_{\nu_{j}+p_{j}}(A_{\nu_{j}})$ has grown to size $\sim e^{-\beta
p_{j}}e^{-\alpha_{p_{j}}p_{j}}$. But since
$\xi_{\nu_{j}+p_{j}}(A_{\nu_{j}})\cap U=\emptyset$, we have
$\alpha_{p_{j}}\approx 0$ and since $r_{j}=\alpha_{\nu_{j}}\nu_{j}$,

$\displaystyle\operatorname{diam}(\xi_{\nu_{j}+p_{j}}(A_{\nu_{j}}))$
$\displaystyle\sim e^{-\beta p_{j}}\sim
e^{-\frac{2d\beta}{\gamma_{p_{j}-1}+\beta}r_{j}}$
$\displaystyle=e^{-\frac{2d\beta\alpha_{\nu_{j}}}{\gamma_{p_{j}-1}+\beta}\nu_{j}}\geq
e^{-\frac{2d\beta\alpha_{\nu_{j}}}{\gamma_{p_{j}-1}+\beta}(\nu_{j}+p_{j})}\geq
e^{-\frac{2d\beta\alpha_{\nu_{j}}}{\underline{\gamma}}(\nu_{j}+p_{j})}.$ (5.2)

So for a prescribed $\alpha>0$ which we are looking for after promotion has
ended, if $\beta>0$ is chosen sufficiently small, then
$\xi_{\nu_{j}+p_{j}}(A_{\nu_{j}})$ has already reached a large size and the
promotion period has ended. We could choose $\beta$ so that, for instance,

$\frac{\alpha}{2}>\frac{2d\beta\alpha_{\nu_{0}}}{\underline{\gamma}}\geq\frac{2d\beta\alpha_{\nu_{j}}}{\underline{\gamma}}.$

This means that we may assume that during the promotion period, all essential
return occur directly after the bound returns. In particular, we may assume
that $\sum_{j}\ell_{j}=0$ in (5.1).

We may assume without loss of generality that
$\gamma_{H}\geq\underline{\gamma}$. Note also that $\gamma_{I}$ only depends
on the starting map and $\kappa^{\prime}$, since
$\alpha_{\nu_{0}}<\overline{\gamma}-\underline{\gamma}$. Set

$\hat{\kappa}=1-\frac{\gamma_{I}}{4\overline{\gamma}}.$

This is an updated $\kappa$, since we cannot guarantee that
$\gamma\geq\underline{\gamma}$ anymore. We want to choose
$\alpha=\alpha_{\nu_{J}}$ so small such that

$0<\frac{\alpha}{\gamma_{I}}\leq\min\left\\{\frac{1}{16d},\,\frac{\gamma_{C}^{3}}{1000d\overline{\gamma}^{2}}\right\\},$
(5.3)

where $\gamma_{C}=\min\\{\gamma_{I}/(12d),\gamma_{H}\\}$.

In the following lemma, we estimate the maximum escape time for essential
returns after promotion in an interval $[m_{0},2m_{0}]$. Note that during this
period, the exponent cannot drop below $\gamma_{I}/4$.

######  Lemma 5.3 (q-Lemma).

Given an essential return $\xi_{\nu}(A)\subset U$, where
$A\subset{\mathcal{E}}_{\nu}(\gamma_{I}/4)\cap{\mathcal{B}}_{\nu}$,
$\operatorname{diam}(\xi_{\nu}(A))\sim e^{-r}$ and $\nu\in[m_{0},2m_{0}]$. Let
$n$ be the smallest number so that either $\xi_{n}(A)$ is an essential return
or has escaped. Then, for $q=n-\nu$, we have

$q\leq\tilde{M}r,$

where $\tilde{M}$ is a constant only depending on the unperturbed map and
$\kappa^{\prime}$.

###### Proof.

If the bound period $p\leq\nu_{J}=m_{0}$, we have, by the definition of the
bound period, replacing $\underline{\gamma}$ with $\gamma_{I}$ in the proof of
Corollary 3.5 (see also Lemma 3.3), for all $c\in A$,

$\left|Df_{c}^{p+1}(\xi_{\nu}(c))\right|\geq e^{(1-\hat{\kappa})r}.$ (5.4)

Let $m_{j}$ be the inessential returns after $\nu$, i.e.
$\nu<m_{1}<m_{2}<\ldots<m_{s}<n$. Let $p_{j}$ and $q_{j}$ be the bound and
free periods respectively following $m_{j}$. Let $p_{0}$ and $q_{0}$ be the
bound and free periods following the return $\nu$. It can happen that escape
takes place before a return takes place, and then $q_{s}$ is not a complete
free period. It can also happen that $n$ is a time during the bound period for
$m_{s}$. But then we have $m_{s}+p_{s}-\nu$ as an upper bound for $q$ and we
can assume that $q>m_{s}+p_{s}$.

Suppose that $\operatorname{dist}(0,\xi_{m_{j}}(A))\sim e^{-r_{j}}$, and let
$r=r_{0}$. Suppose that $n=\nu^{\prime}$ is a return. As long as the bound
period is bounded by $(8d\alpha/\gamma_{I})\nu\leq\nu/2\leq m_{0}$, since
$\alpha=\alpha_{\nu_{J}}$ and $\alpha/\underline{\gamma}_{I}\leq 1/(16d)$, we
can use the same estimate as (5.4), the Main Distortion Lemma 3.8 and Lemma
2.8, to obtain

$\displaystyle\operatorname{diam}(\xi_{n}(A))$
$\displaystyle\sim|Df_{c}^{n-\nu}(\xi_{\nu}(c))|\operatorname{diam}(\xi_{\nu}(A))$
$\displaystyle=\prod_{j=0}^{s}|Df_{c}^{p_{j}}(\xi_{m_{j}}(c))|C^{\prime}e^{\gamma_{H}q_{j}}\operatorname{diam}(\xi_{\nu}(A))$
$\displaystyle\geq e^{\kappa
r}C^{\prime}e^{\gamma_{H}q_{0}}\operatorname{diam}(\xi_{\nu}(A))\prod_{j=1}^{s}e^{r_{j}(1-\hat{\kappa})}\prod_{j=1}^{s}C^{\prime}e^{q_{j}\gamma_{H}}.$
(5.5)

If $n$ was not a return, then let $q_{1}<\ldots<q_{t}$ be the pseudo-returns
after $m_{s}+p_{s}$. Between each pair of pseudo-returns we have uniform
expansion of the derivative according to Lemma 3.7. Between $m_{s}+p_{s}$ and
$q_{1}$ we also have uniform expansion according to Lemma 2.8. So we only need
to consider the last time period, from $q_{t}$ to $n$. Since $\xi_{q_{t}}(A)$
may belong to $U^{\prime}\setminus U$ we have $|Df_{c}(\xi_{q_{t}}(c))|\geq
e^{-(d-1)\Delta}$ for all $c\in A$. After time $q_{t}$ we can use the binding
information, Lemma 3.1 and the first statement of Lemma 2.8 with
$U=U^{\prime}$, depending on whether $n$ belongs to the bound period or not.
In any case we get uniform expansion;
$|Df_{c}^{n-q_{t}-1}(\xi_{q_{t}+1}(c))|\geq Ce^{\gamma_{H}(n-q_{t}-1)}$, where
$C$ does not depend on $\delta$ (but possibly $\delta^{\prime}$). In other
words, with $z=\xi_{m_{s}+p_{s}}(c)$, for $c\in A$, we have

$\displaystyle|Df_{c}^{n-(m_{s}+p_{s})}(z)|$
$\displaystyle=|Df_{c}^{q_{1}-(m_{s}+p_{s})}(z)||Df_{c}^{q_{2}-q_{1}}(f_{c}^{q_{1}-(m_{s}+p_{s})}(z))|$
$\displaystyle\cdot\ldots\cdot|Df_{c}^{q_{t}-q_{t-1}}(f_{c}^{q_{t-1}-(m_{s}+p_{s})}(z))||Df_{c}(f_{c}^{q_{t}-(m_{s}+p_{s})}(z))|$
$\displaystyle\cdot|Df_{c}^{n-q_{t}-1}(f_{c}^{q_{t}-(m_{s}+p_{s})+1}(z))|$
$\displaystyle\geq
C^{\prime}e^{\gamma_{H}(q_{1}-(m_{s}+p_{s}))}e^{\gamma_{2}(q_{t}-q_{1})}e^{-(d-1)\Delta}Ce^{\gamma_{H}(n-q_{t}-1)}$
$\displaystyle\geq e^{\gamma_{C}(n-(m_{s}+p_{s}))}e^{-(d-1)\Delta},$ (5.6)

where $\gamma_{2}\geq\gamma_{I}/(12d)\geq\gamma_{C}$. So we may have to
replace $q_{s}\gamma_{H}$ with $q_{s}\gamma_{C}-(d-1)\Delta$ in (5.5), where
$q_{s}=n-(m_{s}+p_{s})$ in this case. Since $\gamma_{C}\leq\gamma_{H}$, and
$\operatorname{diam}\xi_{n}(A)$ is assumed to be at most
$S=\varepsilon_{1}\delta\leq 1$, we therefore get

$\sum_{j=1}^{s}r_{j}(1-\hat{\kappa})+\sum_{j=0}^{s}q_{j}\gamma_{C}\leq
r\hat{\kappa}+(d-1)\Delta.$

Hence, if $q=\sum_{j=0}^{s}p_{j}+\sum_{j=0}^{s}q_{j}$, we get, by Lemma 3.3
($p\leq 2dr/(\gamma_{p-1}+\alpha_{p}+\beta)$),

$\displaystyle q-p_{0}$
$\displaystyle=\sum_{j=1}^{s}p_{j}+\sum_{j=0}^{s}q_{j}\leq\sum_{j=1}^{s}\frac{(1+\eta)d}{\gamma_{p_{j}-1}+\alpha_{p_{j}}+\beta}r_{j}+\sum_{j=0}^{s}q_{j}$
$\displaystyle\leq\sum_{j=1}^{s}\frac{(1+\eta)d}{\underline{\gamma}_{\nu_{J}}(1-\hat{\kappa})}(1-\hat{\kappa})r_{j}+\frac{1}{\gamma_{C}}\sum_{j=0}^{s}q_{j}\gamma_{C}$
$\displaystyle\leq\max\biggl{\\{}\frac{(1+\eta)}{(1-\hat{\kappa})\underline{\gamma}_{\nu_{J}}},\frac{1}{\gamma_{C}}\biggr{\\}}\biggl{(}\sum_{j=1}^{s}r_{j}(1-\hat{\kappa})+\sum_{j=0}^{s}q_{j}\gamma_{C}\biggr{)}$
$\displaystyle\leq\max\biggl{\\{}\frac{(1+\eta)}{(1-\hat{\kappa})\underline{\gamma}_{\nu_{J}}},\frac{1}{\gamma_{C}}\biggr{\\}}\biggl{(}\hat{\kappa}r+(d-1)\Delta\biggr{)}$
(5.7) $\displaystyle\leq Mr,$ (5.8)

where $M=2d/(\gamma_{C}(1-\hat{\kappa}))$. Moreover, $p_{0}\leq
8dr/\gamma_{I}$ (since the exponent $\gamma\geq\gamma_{I}/4)$, so

$q\leq
Mr+\frac{2d}{\gamma_{I}}r\leq\frac{4d}{(1-\hat{\kappa})\gamma_{C}}r\leq\frac{16d\overline{\gamma}}{\gamma_{C}^{2}}r.$

From (5.1) and since
$(d-1)\alpha_{\nu_{0}}<\overline{\gamma}-\underline{\gamma}$, we see that
$\gamma_{I}\geq C_{I}\underline{\gamma}$, where $C_{I}$ only depends on the
starting map and $\kappa^{\prime}$. Hence $\gamma_{C}$ only depends on the
starting map. We now put

$\tilde{M}=\frac{16d\overline{\gamma}}{\gamma_{C}^{2}}.$

With $\alpha=\alpha_{\nu_{J}}$, and the conditions on $\alpha$ (see (5.3)) we
obtain

$\tilde{M}\alpha\leq\frac{1}{2}.$

Since the total time is bounded from above by
$\nu+\tilde{M}r\leq\nu+\tilde{M}\alpha_{\nu}\nu\leq(3/2)\nu$, we can use the
binding information the whole time. ∎

We will now estimate the set of parameters that do not escape after a long
time.

######  Definition 5.4.

A partition element $A$ at time $n$ that contains the parameter $c$ is denoted
by $A_{n}(c)$. Given an essential return $\xi_{\nu}(A)$, and $c\in
A=A_{\nu}(c)$ we define the escape time $E(\nu,c)$ to be the time $n-\nu$
where $\xi_{n}(A_{n-1}(c))$ is in escape position for the first time after
$\nu$.

By Lemma 3.9, we can assume without loss of generality that
$\xi_{N}({\mathcal{Q}})$ is an essential return. Otherwise, if
$\xi_{N}({\mathcal{Q}})$ is in escape position, we can go directly to the next
section. The point is that most parameters in ${\mathcal{Q}}$ escape before a
certain time after $N$. Suppose $m_{0}=\nu_{J}$ is an essential return time
for a partition element $A=A_{0}$. We now want to choose $\iota$ so that the
Lyapunov exponent cannot drop under $\gamma_{C}$ anytime in the time window
$[m_{0},(1+\iota)m_{0}]$.

Suppose that $\nu_{J}=m_{0}<m_{1}<\ldots<m_{s}<(1+\iota)m_{0}$ are consecutive
essential returns for a parameter $a\in{\mathcal{Q}}$. Let $A_{j}$ be a
partition element for an essential return $\xi_{m_{j}}(A_{j})\subset U$ where
$\operatorname{dist}(0,\xi_{m_{j}}(A_{j}))\sim e^{-r_{j}}$. Then, by Lemma
3.3,

$\frac{m(A_{j+1})}{m(A_{j})}\leq
C\frac{e^{-2r_{j+1}}}{e^{-2\hat{\kappa}r_{j}}}.$

Consequently,

$\frac{m(A_{s})}{m(A_{0})}\leq
C^{s}\frac{e^{2\sum_{j=0}^{s-1}-r_{j+1}}}{e^{-2\hat{\kappa}\sum_{j=0}^{s-1}r_{j}}}\leq
C^{s}e^{2\hat{\kappa}r_{0}-2(1-\hat{\kappa})\sum_{j=1}^{s}r_{j}}.$

Putting $R=r_{1}+\ldots+r_{s}$ we have, by the q-Lemma,

$E(m_{0},c)\leq\sum_{j=0}^{s}\tilde{M}r_{j}\leq\tilde{M}r_{0}+\tilde{M}R.$

Following the old traditions (the ideas go back to the works [7, 8]), we now
compute the number of combinations of returns, i.e. the number of combinations
of $R=r_{1}+\ldots+r_{s}$ where $R$ is fixed. This can be estimated with
Stirling’s formula: If $\Delta$ is sufficiently large, then

$\binom{R+s-1}{s-1}\leq e^{R(1-\hat{\kappa})/3}(1+\eta(\Delta))^{R},$

where $\eta(\Delta)\rightarrow 0$ as $\Delta\rightarrow\infty$. Let $A_{s,R}$
be the set of parameters in $A_{0}$ that have exactly $s$ free essential
returns in the time interval $[m_{0},(1+\iota)m_{0}]$ before escape, where
$R=r_{1}+\ldots+r_{s}$. Let $\hat{A}$ be the largest of all squares in
$A_{s,R}$. Then,

$m(A_{s,R})\leq m(\hat{A})e^{R(1-\hat{\kappa})/3}(1+\eta(\Delta))^{R}.$

So,

$\displaystyle m\left(\left\\{c\in A_{0}:E(m_{0},c)=t\right\\}\right)$ (5.9)
$\displaystyle\leq\sum_{R\geq\frac{t}{\tilde{M}}-r,s\leq R/\Delta}m(A_{s,R})$
(5.10)
$\displaystyle\leq\sum_{R\geq\frac{t}{\tilde{M}}-r}\sum_{s=1}^{R/\Delta}m(\hat{A})e^{R(1-\hat{\kappa})/3}(1+\eta(\Delta))^{R}$
(5.11) $\displaystyle\leq
m(A_{0})\sum_{R\geq\frac{t}{\tilde{M}}-r}C^{\prime}e^{R(1-\hat{\kappa})/3}(1+\eta(\Delta))^{R}C^{R/\Delta}e^{2\hat{\kappa}r-2(1-\hat{\kappa})R}$
(5.12) $\displaystyle\leq
Cm(A_{0})\sum_{R\geq\frac{t}{\tilde{M}}-r}e^{R(1-\hat{\kappa})/3+R\ln(1+\eta(\Delta))+(R/\Delta)\log
C+2\hat{\kappa}r-2(1-\hat{\kappa})R}$ (5.13) $\displaystyle\leq
Cm(A_{0})\sum_{R\geq\frac{t}{\tilde{M}}-r}e^{-\frac{3}{2}(1-\hat{\kappa})R+2\hat{\kappa}r}$
(5.14) $\displaystyle\leq
Cm(A_{0})e^{-\frac{3}{2}(1-\hat{\kappa})(\frac{t}{\tilde{M}}-r)+2\hat{\kappa}r}$
(5.15)
$\displaystyle=Cm(A_{0})e^{-\frac{3}{2}(1-\hat{\kappa})\frac{t}{\tilde{M}}+\frac{r}{2}(3+\hat{\kappa})}.$
(5.16)

So if $t\geq 2\tilde{M}\frac{3+\hat{\kappa}}{1-\hat{\kappa}}r$ then we have an
estimate of the measure for parameters with long escape times (recall
$r=\alpha m_{0}$):

$m\left(\\{c\in A_{0}:E(m_{0},c)=t\\}\right)\leq
Cm(A_{0})e^{-2(3+\hat{\kappa})r}=Cm(A_{0})e^{-2(3+\hat{\kappa})\alpha m_{0}},$

or

$m(\\{c\in A_{0}:E(m_{0},c)=t\\})\leq
Cm(A_{0})e^{-(1-\hat{\kappa})\frac{t}{\tilde{M}}}.$

This means that a big portion of $A_{0}$ will reach the large scale within the
time period $[m_{0},(1+\iota)m_{0}]$, where
$\iota=10\tilde{M}\frac{3+\hat{\kappa}}{1-\hat{\kappa}}\alpha\leq 1$ (see the
condition on $\alpha$ in (5.3)). Let this portion be $1-\delta_{2}$, where
$0<\delta_{2}<1$, is small. Again note that $\delta_{2}$ can be chosen as
close to zero as we want.

Since this is true for every such $A_{0}\subset{\mathcal{Q}}$, we have that
the measure of parameters that reach the large scale within time
$(1+\iota)m_{0}$ is at least

$m({\mathcal{Q}})(1-\delta_{0})(1-\delta_{1})(1-\delta_{2}).$

From $m_{0}$ to $m_{0}(1+\iota)$, we do not loose too much either, since the
Lyapunov exponent during this time cannot drop below
$\gamma_{I}/(2(1+\iota))\geq\gamma_{C}$. Hence we have good distortion control
for all parameters satisfying the basic approach rate assumption until
$(1+\iota)m_{0}$.

###  5.2 Hyperbolic maps

Now to prove our theorems, we can argue similarly as in [4]. Let
${\mathcal{F}}(f_{c_{0}})$ be the Fatou set for the unperturbed map
$f_{c_{0}}$ which is Collet–Eckmann, and let ${\mathcal{N}}_{\varepsilon_{0}}$
be an $\varepsilon_{0}$-neighbourhood of ${\mathcal{J}}(f_{c_{0}})$ as defined
in (2.8). Here, $\varepsilon_{0}$ can be chosen arbitrarily close to zero
since the Hausdorff dimension of the Julia set of $f_{c_{0}}$ is strictly less
than two and therefore has zero area [11, Corollary 2]. Then since the Fatou
set of $f_{c_{0}}$ consists only of the attracting basin of $\infty$, any
compact subset of the Fatou set is stable under perturbation. So
$\widehat{{\mathbb{C}}}\setminus{\mathcal{N}}_{\varepsilon_{0}}$ is still
contained the Fatou set ${\mathcal{F}}(f_{c})$ for all $c\in{\mathcal{Q}}$,
provided that ${\mathcal{Q}}$ is chosen small enough. By bounded distortion

$m\left(\\{c\in{\mathcal{Q}}:c\in{\mathcal{F}}(f_{c})\\}\right)\geq(1-\delta_{0})(1-\delta_{1})(1-\delta_{2})(1-\delta_{3})m({\mathcal{Q}}),$
(5.17)

where $\delta_{3}$ is a constant depending on the distortion of $\xi_{n}(c)$
on ${\mathcal{Q}}$. Since also $0<\delta_{3}<1$ can be arbitrarily small, it
follows that the Lebesgue density of hyperbolic maps at $c=c_{0}$ is equal to
$1$, by making the diameter of ${\mathcal{Q}}$ tending to zero.

To see that Theorem 1.1 actually holds, we note that for all parameter
$c\in{\mathcal{Q}}$ satisfying $c\in{\mathcal{F}}(f_{c})$, the critical point
is attracted by the super-attracting fixed point at $\infty$. Therefore,
(5.17) actually gives an estimate of parameters which lie outside of
${\mathcal{M}}_{d}$.

######  Remark 5.5.

For Collet–Eckmann rational maps with precisely one critical point in the
Julia set, our proof works completely analogously. In this case, the parameter
space is higher dimensional. Instead of using the transversality result by
Avila and also Levin mentioned before, we shall need a more general result by
Levin [20]. To get Lebesgue density of hyperbolic maps, we consider one-
dimensional slices in the parameter space and get the same estimate as above.

######  Remark 5.6.

In the setting of rational maps with precisely one critical point in the Julia
set, the Collet–Eckmann condition is equivalent to several other well known
non-uniform hyperbolicity conditions, e.g., the topological Collet–Eckmann
condition, the second Collet–Eckmann condition, uniform hyperbolicity on
periodic orbits, etc, see [23]. So our result applies to these maps as well.

######  Remark 5.7.

The methods of promotion and the following parameter exclusion are likely to
generalise to all rational CE-maps. Since the recurrence exponent $\alpha>0$
after the promotion period can be chosen arbitrarily small (with possibly
diminishing the diameter of ${\mathcal{Q}}$), taking into consideration that
there may be several critical points in the Julia set (see [3], [4]), one
could prove that every rational CE-map for which $J(f)=\hat{{\mathbb{C}}}$ is
a Lebesgue density point of slowly recurrent CE-maps, even for real-analytic
families of maps containing the starting CE-map (cf. [17] where complex one-
dimensional families were considered).

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Magnus Aspenberg
Centre for Mathematical Sciences
Lund University, Box 118, 22 100 Lund, Sweden

<EMAIL_ADDRESS>

Mats Bylund
Laboratoire de Mathématiques d’Orsay
Université Paris–Saclay, 91400 Orsay, France

<EMAIL_ADDRESS>

Weiwei Cui
Research Center for Mathematics and Interdisciplinary Sciences
Frontiers Science Center for Nonlinear Expectations, Ministry of Education
Shandong University, Qingdao, 266237, China.

<EMAIL_ADDRESS>