[a]J. Juryšek,

# Observation of the galactic PeVatron candidate LHAASO J2108+5157 with the
Large-Sized Telescope for Cherenkov Telescope Array

M. Balbo  D. Eckert  A. Tramacere  G. Pirola  and R. Walter  for the CTA-LST
project

###### Abstract

The Cherenkov Telescope Array (CTA) Observatory will be the next generation
ground-based very-high-energy gamma-ray observatory, sensitive from 20 GeV up
to 300 TeV. The Large-Sized Telescope prototype (LST-1), currently in the
commissioning phase, was inaugurated in October 2018 on La Palma (Spain). It
is the first of four LST telescopes for CTA, to be built in La Palma. In 2021,
LST-1 performed observations of one of the Galactic PeVatron candidates,
LHAASO J2108+5157, recently discovered by the LHAASO collaboration. We present
results of our analysis of the LST-1 data, putting strong constraints on the
emission of the source in the multi-TeV band. We also present results of
multi-wavelength modelling using 12-years Fermi-LAT data and Target of
Opportunity observations with XMM-Newton. We test different scenarios for the
parent particles producing the high energy emission and put constraints on
their spectra.

##  1 Introduction

Galactic PeVatrons are enigmatic cosmic accelerators, believed to produce
cosmic rays (CRs) with energies up to at least $E\approx 1\,\mathrm{PeV}$.
When accelerated CR protons collide with the surrounding ambient gas, they
subsequently emit gamma rays via $\pi^{0}$ decay with the energy of the gamma-
ray photons about 1/10 that of the incident relativistic protons. Therefore,
careful morphological and spectral studies of Ultra-High-Energy (UHE) gamma-
ray sources emitting radiations with energies beyond $100\,\mathrm{TeV}$ are
needed to shed some light on the puzzle of the origin of Very-High-Energy
(VHE) Galactic CRs (for a review, see e.g. [1]). However, the presence of UHE
gamma-ray emission is not evidence of a hadronic accelerator, since gamma rays
can also be produced by electrons and positrons via inverse Compton (IC)
scattering on low-energy photon fields, or via bremsstrahlung on atomic nuclei
of surrounding matter. They can also emit synchrotron radiation when
travelling across a magnetic field. Although the IC emission of gamma rays
with energies greater than $\sim 100\,\mathrm{TeV}$ is suppressed due to the
Klein-Nishina effect, IC can still dominate UHE emission in radiation-
dominated environments [e.g. 2, 3]. The nature of PeVatrons thus remains
unknown despite substantial observational efforts in the last decades, when
several UHE gamma-ray sources have been discovered with ground-based gamma-ray
observatories, like Tibet AS$\gamma$ [4, 5], High Altitude Water Cherenkov
(HAWC) Observatory [6] and Large High Altitude Air Shower Observatory (LHAASO)
[7].

LHAASO J2108+5157 is the first gamma-ray source discovered in the UHE band,
with no VHE counterpart known beforehand [8]. There is also no X-ray
counterpart to the source, but a close High-Energy (HE) soft point source 4FGL
J2108.0+5155 was identified at angular distance of $0.13^{\circ}$ [8]. There
are also two molecular clouds in the direction coincident with LHAASO
J2108+5157, which would support the hypothesis of hadronic origin of the
emission, if CR protons collide with the ambient gas and emit gamma rays via
$\pi^{0}$ decay. Leptonic emission, however, cannot be a priory excluded. The
$95\%$ confidence level upper limit (UL) on the UHE source extension is
$0.26^{\circ}$ [8] .

In this contribution, we present results of a dedicated observation of the
source with the Large-Sized Telescope prototype (LST-1), results of Target of
Opportunity observations with XMM-Newton, and also results of binned analysis
of Fermi-LAT data, which together provide strong constraints on gamma-ray and
X-ray emission of the source. We also present a modeling of multi-wavelength
emission of the source and discuss possible emission scenarios. This
contribution is based on a study of the LST Collaboration [9], where one can
find further details on data analysis, results and modeling of the source
emission.

##  2 Observation and data analysis

###  2.1 LST-1

LST-1 is the first single mirror dish (23 m in diameter) Imaging Air Cherenkov
Telescope prototype built for the future Cherenkov Telescope Array Observatory
(CTAO)111https://www.cta-observatory.org/ in La Palma island, Spain [10]. In
single-telescope regime it is most sensitive in the energy range between tens
of GeV and $\approx 5$ TeV, with the energy and angular resolution of $20\,\%$
and $0.15^{\circ}$, respectively [11], [LST Collaboration, in prep.].

We observed LHAASO J2108+5157 with LST-1 for 91 hours in 2021. After
application of quality cuts based on trigger rate stability, rate of CR events
and atmospheric transmission, we were left with 49.3 hours of good quality
data for the further analysis. For data calibration and event reconstruction
we used a pipeline implemented in lstchain v0.9.6. [12]. We analyzed
$\theta^{2}$ distributions using reflected background method in four
logarithmically spaced energy bins in the energy range between 100 GeV and 100
TeV, which resulted in no significant source detection. At the highest
energies $E>3\,\mathrm{TeV}$, however, we see a point-like excess with a
significance of $3.7\sigma$ [13].

As a next step, we performed 1D spectral analysis with the use of the gammapy
package [14] in full LST-1 energy range. We assumed power law (PL) spectral
shape $\mathrm{d}N/\mathrm{d}E=N_{0}(E/E_{0})^{-\Gamma}$ for the source
spectrum. The reference energy was fixed as $E_{0}=1$ TeV. The best fit
spectral parameters are $N_{0}=(8.0\pm 5.4)\times
10^{-14}\,\mathrm{cm^{-2}s^{-1}TeV^{-1}}$ and $\Gamma=1.6\pm 0.2$, suggesting
hard emission of the source in the TeV band. To estimate source flux in
individual energy bins, we run a joint likelihood fit of the LST-1 data and
LHAASO flux-points, considering spectral shape in a form of power-law with
exponential cutoff. Even though significance of a point-like source in the
full LST-1 energy range is not high enough to claim a detection ($2.2\sigma$
for PL spectral model), LST-1 data provide strong ULs on the source emission
in the TeV band (see Figure 1).

###  2.2 Fermi-LAT

We performed a dedicated binned analysis of the 12-year Fermi-LAT data for the
region. In contrast with the Cao et al. [8], that used 4FGL-DR2 catalogue, we
used more recent 12-years 4FGL-DR3 catalogue [15] to create the source model.
The spectrum of LHAASO J2108+5157 counterpart, 4FGL J2108.0+5155, presents
steep decrease above few GeVs and its link to the VHE-UHE emission is
therefore not straightforward (see Figure 1). We also found that the emission
above $\approx 4$ GeV is dominated by a new hard source ($\Gamma=1.9\pm 0.2$),
located at coordinates $l=92.35^{\circ}$, $b=2.56^{\circ}$, detected with a
significance of $4\sigma$. This new source is not spatially correlated with
LHAASO J2108+5157, but including it in the background model we were able to
improve spectral representation of the HE counterpart 4FGL J2108.0+5155.

###  2.3 XMM-Newton

Target of Opportunity XMM-Newton observation, centered at
$\mathrm{R.A.}=317.0170^{\circ}$, $\mathrm{Dec}=+51.9275^{\circ}$, was carried
out on June 11, 2021, collecting 13.6 ks of data. We reduced the data from all
three (MOS1, MOS2, PN) EPIC cameras using XMMSAS v19.1 and the X-COP data
analysis pipeline [16, 17]. We extracted images in the soft (0.5-2 keV) and
hard (2-7 keV) bands. The brightest X-ray source in the field is the eclipsing
binary V1061 Cyg (RX J2107.3+5202), which exhibits very soft thermal spectrum
and thus cannot be a counterpart of LHAASO J2108+5157. The other sources in
the field are substantialy fainter.

In the case of a leptonic origin of the source emission, we would expect a
synchrotron nebula with an angular size of a few arcminutes, spatially
correlated with the extended UHE emission. We used the public code pyproffit
[18] to derive ULs on the X-ray flux of an extended source centered on the
position of LHAASO J2108+5157 assuming PL source spectrum with a photon index
of 2. Provided the unknown distance to the source, we derived ULs for two
extreme cases - completely unabsorbed and completely absorbed source, for the
source being located nearby, or on the other side of the Galaxy, respectively.
Figure 1 shows $95\%$ absorbed ULs on the X-ray emission in a region with
radius of $6^{\prime}$, which is a maximum reasonable X-ray extension of a
Galactic Pulsar Wind Nebula (PWN) at a distance $d\leq 10$ kpc [19].

##  3 Discussion of possible emission scenarios

Figure 1: Left: Spectral Energy Distribution (SED) of LHAASO J2108+5157
showing leptonic emission scenario. Data from different instruments are shown
- XMM-Newton (green), Fermi-LAT (red), LST-1 (purple), LHAASO-KM2A (yellow).
The solid line represents the best fitting IC emission of LST-1 and LHAASO
data and dashed line represents the synchrotron radiation of the same
electrons in magnetic field of $B=1.2\,\mathrm{\mu G}$. Dotted line represents
a phenomenological model of the tentative pulsar. Right: SED with hadronic
models of emission. The solid line represents the best fitting $\pi^{0}$ decay
emission model. Grey dashed line represents $\pi^{0}$ decay emission model
with proton spectral index $\alpha=2$ shown for reference [8].

###  3.1 Leptonic emission scenario

The fact that most of the currently known VHE sources have been identified as
PWN [20] and most of the potential Galactic PeVatrons can be associated with a
nearby pulsar [7] favours a leptonic emission mechanism. We performed leptonic
modelling of the multi-TeV emission in naima package [21], assuming
exponential cutoff power law distribution of electrons interacting via IC with
Cosmic Microwave Background (CMB) and Far Infrared Radiation of the dust
(FIR). The electron distribution best fitting the observations shown in Figure
1 has a cutoff at $E_{\mathrm{cutoff}}=100^{+70}_{-30}$ TeV and spectral index
$\alpha=1.5\pm 0.4$. The total energy in electrons is
$E(>1\,\mathrm{GeV})\approx 1\times 10^{45}(d/1\,\mathrm{kpc)^{2}\,erg}$.
X-ray ULs on synchrotron nebula emission strongly constrain magnetic field in
the source ($B\leq 1.2\,\mathrm{\mu G}$), which is hard to explain in PWN
scenario, but consistent with TeV halo object [e.g. 22], where one can expect
magnetic field even below Galactic background level [23].

The soft GeV emission of 4FGL J2108.0+5155 has spectral features compatible
with a population of gamma-ray pulsars with spin-down power in the range of
$\dot{E}\approx 10^{35}-10^{37}\,\mathrm{erg\,s^{-1}}$ [24]. Cooling time of
relativistic electrons of the cutoff energy is $t_{\mathrm{cool}}\approx 30$
kyr and thus such a pulsar could release enough energy in $t_{\mathrm{cool}}$
to drive the UHE emission of the TeV halo, provided the distance to the source
$d\leq 10$ kpc. The possible extent of the UHE source is also consistent with
the size of a typical TeV halo object, assuming Geminga-like diffusion
coefficient and source distance $d\geq 2$ kpc.

###  3.2 Hadronic emission scenario

We created a model of $\pi^{0}$ decay dominated multi-TeV emission in naima
package, assuming relativistic protons following exponential cutoff power law
distribution, interacting with one of the two molecular clouds in the
direction coincident with LHAASO J2108+5157. The best fitting proton
distribution shown in Figure 1 has cutoff at
$E_{\mathrm{cutoff}}=250^{+200}_{-90}$ TeV and spectral index $\alpha=0.8\pm
0.5$. While the total energy in relativistic protons is low enough
($E_{\mathrm{total}}=9\times
10^{45}(d/1\,\mathrm{kpc})^{2}(100\,\mathrm{cm^{-3}}/n)\,\mathrm{erg}$, where
$n$ is molecular density of the molecular cloud with distance $d$ in the
direction of the source) compared to the typical energy of a supernova
explosion, required spectral index is too hard compared to the standard
diffusive shock acceleration (DSA) [25]. However, it does not necessarily
exclude hadronic emission scenario as hard spectral indices can be expected
e.g. in the non-linear regime of DSA [26].

The HE gamma-ray emission of the Fermi-LAT counterpart cannot be explained in
a single component model together with UHE emission in hadronic scenario. Cao
et al. [8] suggested that 4FGL J2108.0+5155 show spectral features of an old
SNR. Our dedicated analysis of Fermi-LAT data, however, results in the photon
index of HE emission $\Gamma=-3.2$ above 1 GeV, which seems to be too soft
compared to the spectra of old SNRs interacting with molecular clouds [27].

##  4 Conclusions

We conducted a multi-wavelength study of the unidentified UHE gamma-ray source
LHAASO J2108+5157 combining data from LST-1, XMM-Newton, Fermi-LAT and LHAASO.
Strong ULs on emission of the source derived from the LST-1 data prove that
LST-1 is capable of providing significant scientific results although still in
commissioning phase. We present a leptonic emission scenario suggesting that
the UHE source may be related to a TeV halo object. There is only a handful of
TeV halo objects currently known, and these objects are of a particular
interest because they appear to have particle diffusion coefficients that are
much lower than in the interstellar medium, and thus may provide important
insight into CR particle diffusion within the Galaxy. Soft HE counterpart can
be explained as a pulsar, which could provide the relativistic electrons with
enough energy to power the UHE emission. In the hadronic emission scenario, on
the other hand, the HE and UHE emission cannot be explained as interaction of
protons accelerated in an old SNR with dense molecular cloud, and thus more
complex model would be needed.

## Acknowledgments

We gratefully acknowledge financial support from the following agencies and
organisations listed here https://www.lst1.iac.es/acknowledgements.html

This work was conducted in the context of the CTA-LST Project. This research
has made use of the observations obtained with _XMM-Newton_ , an ESA science
mission with instruments and contributions directly funded by ESA Member
States and NASA. This research has made use of the SIMBAD database, operated
at CDS, Strasbourg, France.

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# Infinite-dimensional Thurston theory
and transcendental dynamics III:
entire functions with escaping singular orbits
in the degenerate case

Konstantin Bogdanov

###### Abstract.

We classify transcendental entire functions that are compositions of a
polynomial and the exponential for which all singular values escape on
disjoint rays. We focus on the case where the escape is degenerate in the
sense that points from different singular orbits are arbitrarily close to each
other. As in the general case we employ an iteration procedure on an
_infinite-dimensional_ Teichmüller space, analogously to the Thurston’s
classical Topological Characterization of Rational Functions, but for an
_infinite_ set of marked points.

###### Contents

  1. 1 Introduction
  2. 2 Prerequisites
    1. 2.1 Escaping set of functions in $\mathcal{N}_{d}$
    2. 2.2 Setup of Thurston iteration and contraction
    3. 2.3 $\operatorname{Id}$-type maps
    4. 2.4 Spiders
  3. 3 Preliminary constructions
    1. 3.1 Clusters
    2. 3.2 No combinatorics near $\infty$ (almost)
    3. 3.3 Separation of preimages in $D_{\rho}$
  4. 4 Compact invariant subset
  5. 5 Appendix A
  6. 6 Acknowledgements

This is the third out of four articles publishing the results of the author’s
doctoral thesis [B0]. The machinery needed for the Thurston iteration on an
infinite-dimensional Teichmüller space is developed in [B1]. In this article
and in [B2] we classify transcendental entire functions that are compositions
of a polynomial and the exponential for which their singular values escape on
disjoint dynamic rays. [B2] covers the general case of such escape when the
distance between points on different singular orbits is bounded below, whereas
in this paper we get rid of this restriction. In the fourth article we
investigate continuity of such families of functions with respect to
potentials and external addresses and use the continuity argument to extend
our classification to the case of escape on the (pre-)periodic rays.

##  1\. Introduction

In transcendental dynamics, as for polynomials, an important dynamically
natural subset of the complex plane is the _escaping set_. For a
transcendental entire function $f$, its _escaping set_ is defined as

$I(f)=\\{z\in\mathbb{C}:f^{n}(z)\to\infty\text{ as }n\to\infty\\}.$

Since the Julia set is the boundary of $I(f)$, the Julia and Fatou sets can be
reconstructed from $I(f)$, which is thus one of the most fundamental invariant
sets for the dynamics of entire functions. It is proved in [RRRS] for
transcendental entire functions of bounded type and finite order that the
escaping set of every function in the family is organized as the union
_dynamic rays_ and their endpoints. About the escaping points that belong to a
dynamic ray one says that they _escape on rays_. This is the standard mode of
escape for many important families of transcendental entire functions.

In [B2] we classified entire functions of the form $p\circ\exp$ where $p$ is a
monic polynomial such that all their singular points escape on disjoint rays
(i.e. every dynamic ray contains no more than one post-singular point) and
distance between points on different singular orbits is bounded below. Denote

$\mathcal{N}_{d}:=\\{p\circ\exp:p\text{ is a monic polynomial of degree
}d\\}.$

As for the exponential family [SZ], for functions of the form $p\circ\exp$ the
points escaping on rays can be described by their potential (or “speed of
escape”) and external address (or “combinatorics of escape”, i.e. the sequence
of dynamic rays containing the escaping orbit). So the classification was
accomplished in terms of potentials and external addresses. To be more
precise, for every function $f=p\circ\exp$ the classification was obtained
within the _parameter space_ of $f$. The parameter space of an entire function
$g$ is the set of entire functions $\varphi\circ f\circ\psi$ where
$\varphi,\psi$ are quasiconformal self-homeomorphisms of $\mathbb{C}$. For the
exponential family $\mathcal{N}_{1}$ this definition of the parameter space
coincides with the parametrization of $e^{z}+\kappa$ (up to affine
conjugation). Earlier classification results for $\mathcal{N}_{1}$ were
obtained in [FRS, FS, F].

In this article we want to extend this classification to the case when points
from different singular orbits come arbitrarily close to each other, that is,
to prove the following more general version of [B2, Classification Theorem
1.4].

######  Theorem 1.1 (Classification Theorem).

Let $f_{0}\in\mathcal{N}_{d}$ be a transcendental entire function with
singular values $\\{v_{i}\\}_{i=1}^{m}$. Let also
$\\{\underline{s}_{i}\\}_{i=1}^{m}$ be exponentially bounded external
addresses that are non-(pre-)periodic and non-overlapping, and
$\\{T_{i}\\}_{i=1}^{m}$ be real numbers such that
$T_{i}>t_{\underline{s}_{i}}$. Then in the parameter space of $f_{0}$ there
exists a unique entire function $g=\varphi\circ f_{0}\circ\psi$ such that each
of its singular values $\varphi(v_{i})$ escapes on rays, has potential $T_{i}$
and external address $\underline{s}_{i}$.

Conversely, every function in the parameter space of $f_{0}$ such that its
singular values escape on disjoint rays is one of these.

Precise definitions of terms used the Classification Theorem are given in
Subsection 2.1. In particular, the condition that external addresses are
non-(pre-)periodic and non-overlapping means that the singular values of $g$
escape on disjoint rays.

The second paragraph of the Classification Theorem 1.1 follows immediately
from Theorem 2.3, and we added it to the statement in order to show that we
indeed have a _classification_ of function in $\mathcal{N}_{d}$ with singular
values escaping on rays (subject to some restrictions).

We use the same general scheme of the proof developed in [B0, B1, B2]
involving a special generalization of the _Thurston’s Topological
Characterization of Rational Functions_ [DH, H] for infinitely many marked
points.

  1. (1)

Construct a (non-holomorphic) “model map” $f$ for which the singular values
escape on rays with the desired potential and external address. It will define
the Thurston $\sigma$-map acting on the Teichmüller space $\mathcal{T}_{f}$ of
$\mathbb{C}$ with $P_{f}$ removed, where $P_{f}$ is the postsingular set of
$f$ (Definition 2.4).

  2. (2)

Construct a compact subset $\mathcal{C}_{f}$ of $\mathcal{T}_{f}$ which is
invariant under $\sigma$.

  3. (3)

Prove that $\sigma$ is strictly contracting in the Teichmüller metric on
$\mathcal{C}_{f}$.

  4. (4)

Prove that $\sigma$ has a unique fixed point in $\mathcal{C}_{f}$, and this
point corresponds to the entire function with the desired conditions on its
singular orbits.

Items (1),(3) and (4) and a big part of (2) are almost completely proved in
[B1, B2], so in this article we will mainly be focused on the upgrade of (2).

The compact invariant subset $\mathcal{C}_{f}$ for the non-degenerate case is
constructed in [B2] approximately as follows. For some initially chosen
constant $\rho>0$, it is the set of points represented by quasiconformal maps
$\varphi$ that can be obtained from the identity via an isotopy $\varphi_{u}$
such that:

  1. (1)

for very point $z\in P_{f}$ outside of $\mathbb{D}_{\rho}(0)$,
$\varphi_{u}(z)$ is contained in a small disk $U_{z}$ around $z$ so that the
disks $U_{z}$ are mutually disjoint;

  2. (2)

inside of $\mathbb{D}_{\rho}(0)$ the distances between marked points
$\varphi_{u}(z)$ are bounded from below;

  3. (3)

the isotopy type of $\varphi_{u}$ relative points of $P_{f}$ outside of
$\mathbb{D}_{\rho}(0)$ is “almost” the isotopy type of identity;

  4. (4)

the isotopy type of $\varphi_{u}$ relative points of $P_{f}$ inside of
$\mathbb{D}_{\rho}(0)$ is not “too complicated”, i.e. there are quantitative
bounds on how many times the marked points “twist” around each other.

Conditions (2),(3) create a separation of the plane into the two subsets:
$\mathbb{D}_{\rho}(0)$ without much control on the behavior of $\varphi_{u}$
but containing only finitely many marked points, and the complement of
$\mathbb{D}_{\rho}(0)$ where the homotopy information is trivial but there are
infinitely many marked points.

However, it is not enough to consider such $\mathcal{C}_{f}$ for the case when
points belonging to different singular orbits come arbitrarily close to each
other: it might be simply impossible to construct such disjoint disks $U_{z}$
so that they stay invariant under the $\sigma$-map (see [B2, Remark 5.10]).
Instead, outside of $\mathbb{D}_{\rho}(0)$ we consider _clusters_ of points
(Definition 3.2) — finite subsets of $P_{f}$ formed by points that are “very
close” to each other. Therefore, in condition (1) we require that the disjoint
disks $U_{z}$ contain clusters of points rather than a single point and add
the condition

  1. (5)

If $U_{z}$ contains more than one point, then these points “almost” do not
move relative to each other.

What exactly this “almost” means is described in Proposition 3.7.

Another rather technical addition comparing to the older $\mathcal{C}_{f}$
from [B2] is that we also need to partially control the clusters inside of
$\mathbb{D}_{\rho}(0)$, that is, we need an additional estimate like in (2) to
prevent the marked points come too close to the asymptotic value — otherwise
after one iteration of the $\sigma$ map they might be “thrown” towards
$-\infty$, i.e. outside of $\mathbb{D}_{\rho}(0)$. The construction of
$\mathcal{C}_{f}$ in full generality is the content of Theorem 4.1.

##  2\. Prerequisites

In this section we assemble some statements and definitions mainly from [B1,
B2] that will be needed in our constructions.

###  2.1. Escaping set of functions in $\mathcal{N}_{d}$

It follows from [RRRS] that for every function $f\in\mathcal{N}_{d}$ its
escaping set is modeled on dynamic rays. In particular, that every escaping
point is mapped on a dynamic ray after finitely many iterations. A related
notion is a ray tail.

######  Definition 2.1 (Ray tails).

Let $f$ be a transcendental entire function. A _ray tail_ of $f$ is a
continuous curve $\gamma:[0,\infty)\to I(f)$ such that for every $n\geq 0$ the
restriction $f^{n}|_{\gamma}$ is injective with
$\lim_{t\to\infty}f^{n}(\gamma(t))=\infty$, and furthermore
$f^{n}(\gamma(t))\to\infty$ uniformly in $t$ as $n\to\infty$.

######  Definition 2.2 (Dynamic rays, escape on rays).

A _dynamic ray_ of a transcendental entire function $f$ is a maximal injective
curve $\gamma:(0,\infty)\to I(f)$ such that $\gamma|_{[t,\infty)}$ is a ray
tail for every $t>0$.

If $z\in I(f)$ belongs to a dynamic ray, we say that $z$ _escapes on rays_.

We showed in [B2] that for functions in $\mathcal{N}_{d}$ its dynamic rays (at
least their parts near $\infty$) are characterized by an exponentially bounded
external address and parametrized by potential.

Fix an integer $d>0$, and denote by $F:\mathbb{R^{+}}\to\mathbb{R^{+}}$ the
function

$F(t):=\exp(dt)-1.$

The _external address_ $\underline{s}=(s_{0}s_{1}s_{2}...)$ is an infinite
sequence of integers $s_{0},s_{1},s_{2},...$ On the set of all external
addresses we can consider the standard shift-operator
$\sigma:(s_{0}s_{1}s_{2}...)\mapsto(s_{1}s_{2}s_{3}...)$. Two external
addresses $\underline{s}_{1}$ and $\underline{s}_{2}$ are called _overlapping_
if there are integers $k,l\geq 0$ so that
$\sigma^{k}\underline{s}_{1}=\sigma^{l}\underline{s}_{2}$.

Further, an external address $\underline{s}=(s_{0}s_{1}s_{2}...)$ is called
_exponentially bounded_ if there exists $t>0$ such that $s_{n}/{F^{n}(t)}\to
0$ as $n\to\infty$. The infimum of such $t$ is denoted by $t_{\underline{s}}$.

As can be seen in the following theorem, exponentially bounded external
addresses correspond to dynamic rays (at least to their part near $\infty$) of
functions in $\mathcal{N}_{d}$. Moreover, from the formula 2.1, dynamic rays
look near $\infty$ almost like straight horizontal lines with the imaginary
part that can be recovered from their external address.

######  Theorem 2.3 (Escape on rays in $\mathcal{N}_{d}$ [B2]).

Let $f\in\mathcal{N}_{d}$. Then for every exponentially bounded external
address there exists a dynamic ray realizing it.

If $\mathcal{R}_{\underline{s}}$ is dynamic ray having an exponentially
bounded external address $\underline{s}=(s_{0}s_{1}s_{2}...)$, and no strict
forward iterate of $\mathcal{R}_{\underline{s}}$ contains a singular value of
$f$, then $\mathcal{R}_{\underline{s}}$ is the unique dynamic ray having
external address $\underline{s}$, and it can be parametrized by
$t\in(t_{\underline{s}},\infty)$ so that

$\mathcal{R}_{\underline{s}}(t)=t+\frac{2\pi is_{0}}{d}+O(e^{-t/2}),$ (2.1)

and
$f^{n}\circ\mathcal{R}_{\underline{s}}=\mathcal{R}_{\sigma^{n}\underline{s}}\circ
F^{n}$. Asymptotic bounds $O(.)$ for
$\mathcal{R}_{\sigma^{n}\underline{s}}(t)$ are uniform in $n$ on every ray
tail contained in $\mathcal{R}_{\underline{s}}$.

If none of the singular values of $f$ escapes, then $I(f)$ is the disjoint
union of dynamic rays and their escaping endpoints.

The parametrization 2.1 allows us to define the potential of a point on a
dynamic ray: if $z\in\mathcal{R}_{\underline{s}}$ corresponds to the parameter
$t$ of formula 2.1, we say that $t$ is the _potential_ of $z$. From Theorem
2.3 follows immediately that the potential is defined in a canonical way, and
if the potential of $z$ is equal to $t$, then the potential of $f(z)$ is equal
to $F(t)$. Note that different points on the same ray have different
potentials.

###  2.2. Setup of Thurston iteration and contraction

In this subsection we overview a few important definitions from [B2] and
sketch the setup of the Thurston iteration.

One starts with the construction of a quasiregular function $f$ modeling
escaping behavior of singular values. Let $f_{0}\in\mathcal{N}_{d}$, and
$v_{1},...,v_{m}$ be the finite singular values of $f_{0}$. Further, let
$\mathcal{O}_{1}=\\{a_{1j}\\}_{j=0}^{\infty},...,\mathcal{O}_{m}=\\{a_{mj}\\}_{j=0}^{\infty}$
be some orbits of $f_{0}$ escaping on disjoint rays $\mathcal{R}_{ij}$, and
let $R_{ij}$ be the part of the ray $\mathcal{R}_{ij}$ from $a_{ij}$ to
$\infty$ (containing also $a_{ij}$).

Now, we introduce the _capture_. Choose some bounded Jordan domain
$U\subset\mathbb{C}\setminus\bigcup_{\begin{subarray}{c}i=\overline{1,m}\\\
j=\overline{1,\infty}\end{subarray}}R_{ij}$ containing all singular values of
$f_{0}$ and the first point $a_{i0}$ on each escaping orbit $\mathcal{O}_{i}$.
Define an isotopy $c_{u}:\mathbb{C}\to\mathbb{C}$ through quasiconformal maps,
such that $c_{0}=\operatorname{id}$, $c_{u}=\operatorname{id}$ on
$\mathbb{C}\setminus U$, and for each $i=\overline{1,m}$ we have
$c_{1}(v_{i})=a_{i0}$. Denote $c=c_{1}$. Thus, $c$ is a quasiconformal
homeomorphism mapping singular values to the first points on the orbits
$\mathcal{O}_{i}$.

Define a function $f:=c\circ f_{0}$. It is a quasiregular function whose
singular orbits coincide with $\\{\mathcal{O}_{i}\\}_{i=1}^{m}$. Its post-
singular set of $f$ is defined in a standard way:

$P_{f}:=\cup_{i=1}^{m}\mathcal{O}_{i}.$

We also call $P_{f}$ the set of _marked points_.

The Thurston $\sigma$-map acts on the Teichmüller space of
$\mathbb{C}\setminus P_{f}$.

######  Definition 2.4 (Teichmüller space of $\mathbb{C}\setminus P_{f}$).

The _Teichmüller space_ $\mathcal{T}_{f}$ of the Riemann surface
$\mathbb{C}\setminus P_{f}$ is the set of quasiconformal homeomorphisms of
$\mathbb{C}\setminus P_{f}$ modulo post-composition with an affine map and
isotopy relative $P_{f}$.

The map

$\sigma:[\varphi]\in\mathcal{T}_{f}\mapsto[\tilde{\varphi}]\in\mathcal{T}_{f}$

is defined in the standard way via the Thurston diagram.

${\mathbb{C},P_{f}}$${\mathbb{C},\tilde{\varphi}(P_{f})}$${\mathbb{C},P_{f}}$${\mathbb{C},\varphi(P_{f})}$$\scriptstyle{\tilde{\varphi}}$$\scriptstyle{f=c\circ
f_{0}}$$\scriptstyle{g}$$\scriptstyle{\varphi}$

More precisely, for every quasiconformal map $\varphi:\mathbb{C}\to\mathbb{C}$
there exists another quasiconformal map
$\tilde{\varphi}:\mathbb{C}\to\mathbb{C}$ such that $g=\varphi\circ
f\circ\tilde{\varphi}^{-1}$ is entire function. We define
$\sigma[\varphi]:=[\tilde{\varphi}]$. For more details and the proof that this
setup is well defined see [B1].

As in [B2], we need to consider the subset of asymptotically conformal points
of $\mathcal{T}_{f}$ and to use the fact that the $\sigma$-map is strictly
contracting on this subset.

######  Definition 2.5 (Asymptotically conformal points [GL]).

A point $[\varphi]\in\mathcal{T}_{f}$ is called _asymptotically conformal_ if
for every $\epsilon>0$ there is a compact set $C\subset\mathbb{C}\setminus
P_{f}$ and a representative $\psi\in[\varphi]$ such that
$\left|\mu_{\psi}\right|<\epsilon$ a.e. on $(\mathbb{C}\setminus
P_{f})\setminus C$.

###  2.3. $\operatorname{Id}$-type maps

One of the key notions in our constructions are $\operatorname{id}$-type maps.

Let $f=c\circ f_{0}$ be the quasiregular function defined in Section 2.2 where
$f_{0}=p\circ\exp$. The union of post-singular ray tails
$S_{0}=\cup_{i,j}R_{ij}$ is called the _standard spider_ of $f$.

######  Definition 2.6 ($\operatorname{Id}$-type maps [B2]).

A quasiconformal map $\varphi:\mathbb{C}\to\mathbb{C}$ is of
$\operatorname{id}$-type if there is an isotopy
$\varphi_{u}:\mathbb{C}\to\mathbb{C},\ u\in[0,1]$ such that
${\varphi_{1}=\varphi},\ \varphi_{0}=\operatorname{id}$ and
$\left|\varphi_{u}(z)-z\right|\to 0$ uniformly in $u$ as $S_{0}\ni
z\to\infty$.

We also say that $[\varphi]\in\mathcal{T}_{f}$ is of $\operatorname{id}$-type
if $[\varphi]$ contains an $\operatorname{id}$-type map. Further,
$\psi_{u},u\in[0,1]$ is an _isotopy of $\operatorname{id}$-type maps_ if
$\psi_{u}$ is an isotopy through maps of $\operatorname{id}$-type such that
$\left|\psi_{u}(z)-z\right|\to 0$ uniformly in $u$ as $S_{0}\ni z\to\infty$.

The following theorem says that the $\sigma$-map is invariant on the subset of
$\operatorname{id}$-type points in $\mathcal{T}_{f}$.

######  Theorem 2.7 (Invariance of $\operatorname{id}$-type points [B2]).

If $[\varphi]$ is of $\operatorname{id}$-type, then $\sigma[\varphi]$ is of
$\operatorname{id}$-type as well.

More precisely, if $\varphi$ is of $\operatorname{id}$-type, then there is a
unique $\operatorname{id}$-type map $\hat{\varphi}$ such that $\varphi\circ
f\circ\hat{\varphi}^{-1}$ is entire.

Moreover, if $\varphi_{u}$ is an isotopy of $\operatorname{id}$-type maps,
then the functions $\varphi_{u}\circ f\circ\hat{\varphi}_{u}^{-1}$ have the
form $p_{u}\circ\exp$ where $p_{u}$ is a monic polynomial with coefficients
depending continuously on $u$.

######  Remark 2.8.

We keep using the hat-notation from Theorem 2.7: $\hat{\varphi}$ denotes the
unique $\operatorname{id}$-type map so that $\varphi\circ
f\circ\hat{\varphi}^{-1}$ is entire.

###  2.4. Spiders

Another important tool in the construction of the invariant compact subset of
$\mathcal{T}_{f}$ are infinite-legged spiders. However, in this article we
only need to prove one rather simple fact about them (Lemma 3.11); all other
applications of spiders are already involved in [B2].

######  Definition 2.9 (Spider).

An image $S_{\varphi}$ of the standard spider $S_{0}$ under an
$\operatorname{id}$-type map $\varphi$ is called a _spider_.

Naturally, the image of a ray tail $R_{ij}$ under a spider map is called a
_leg_.

Let $V=\\{w_{n}\\}\subset\mathbb{C}$ be a finite set. Further, let
$\gamma:[0,\infty]\to\hat{\mathbb{C}}$ be a curve such that $\gamma(0)\in V$,
$\gamma(\infty)=\infty$, $\gamma|_{\mathbb{R}^{+}}\subset\mathbb{C}\setminus
V$ and $\operatorname{Re\,}\gamma(t)\to+\infty$ as $t\to\infty$. On the set of
all such pairs $(V,\gamma)$ one considers a map

$W:(V,\gamma)\mapsto W(V,\gamma)\in F(V),$

where $F(V)$ is the free group on $V$. This map uniquely encodes the homotopy
type of $\gamma$ (with fixed endpoints) in $\mathbb{H}_{r}\setminus V$ where
$\mathbb{H}_{r}=\\{z\in\mathbb{C}:\operatorname{Re\,}z>r\\}$ contains $V$ and
$\gamma$ (a particular value of $r$ is irrelevant). Roughly speaking, we
homotope $\gamma$ into a concatenation of a horizontal straight ray from
$\gamma(0)$ to $+\infty$, and a loop with the base point at $\infty$, and by
$W(V,\gamma)$ denote the representation of the loop via “straight horizontal”
generators of the fundamental group of $(\mathbb{H}_{r}\setminus
V)\cup\infty$. For the details of the construction we refer the reader to [B1,
Subsection 6.1]. We use this type of information for every leg of a spider to
obtain a description of the $\operatorname{id}$-type points.

For every pair of $i\in\\{1,2,...,m\\},j\geq 0$ denote

$\mathcal{O}_{ij}:=\\{a_{kl}\in P_{f}:l<j\text{ or }l=j,k\leq i\\}.$

######  Definition 2.10 (Leg homotopy word).

Let $S_{\varphi}$ be a spider. Then the _leg homotopy word_ of a leg
$\varphi(R_{ij})$ is

$W_{ij}^{\varphi}:=W(\varphi(\mathcal{O}_{ij}),\varphi(R_{ij})).$

Since $W_{ij}^{\varphi}$ is an element of a free group, by
$\left|W_{ij}^{\varphi}\right|$ we naturally denote the length of the word
representing this element.

##  3\. Preliminary constructions

We consider the quasiconformal function $f=c\circ f_{0}$ with $m$ singular
orbits $\mathcal{O}_{i}=\\{a_{ij}\\}$ constructed in Subsection 2.2 with its
associated $\sigma$-map. We denote by $t_{ij}$ the potential of $a_{ij}$ and
by $s_{ij}$ the first number in the external address of $a_{ij}$ (recall that
the imaginary part of $R_{ij}$ tends to $2\pi s_{ij}/d$). Denote by
$\operatorname{SV}(f)$ the set of finite singular values of $f$, i.e. the
image of (finite) singular values of $f_{0}$ under the capture $c$. In this
subsection we prove a few preliminary lemmas needed for the construction of
the compact invariant subset $\mathcal{C}_{f}$. Note that we provide only the
statements that are needed to upgrade the construction in [B2] to the
degenerate case and mainly avoid duplicating results from [B2].

Let $\mathcal{P}=\\{t_{i}\\}_{i=1}^{\infty}$ be the set of all potentials of
points in $P_{f}$ ordered so that $t_{i}<t_{i+1}$. Further, we define a set

$\mathcal{P}^{\prime}:=\\{\rho_{i}:\rho_{i}=\frac{t_{i}+t_{i+1}}{2}\\}.$

The next lemma separates $P_{f}$ and $S_{0}$ into two parts: one near $\infty$
with good estimates and “straight” spider legs, and one near the origin where
the points are located somewhat more chaotically. This is a statement _solely_
about $f_{0}$ and its standard spider.

######  Lemma 3.1 (Good estimates for $S_{0}$ near $\infty$ [B2]).

There exists $t^{\prime}>0$ such that:

  1. (1)

if $t_{i}>t_{j}>t^{\prime}$, then $t_{i}-t_{j}>2$,

  2. (2)

if $t_{kl}>t_{ij}>t^{\prime}$, then
$\left|a_{kl}\right|>\left|a_{ij}\right|+2$,

  3. (3)

if $\rho\in\mathcal{P}^{\prime}$ is bigger than $t^{\prime}$, then all
$a_{ij}$ with potential less that $\rho$ are contained in
$\mathbb{D}_{\rho-1}(0)$, while all $a_{ij}\in P_{f}$ with potential bigger
than $\rho$ are contained in the complement of $\mathbb{D}_{\rho+1}(0)$.

For all $\rho\in\mathcal{P}^{\prime}$ we use the following notation. Denote
$D_{\rho}:=\mathbb{D}_{\rho}(0)$, and for $i\in\\{1,...,m\\}$ let
$N_{i}=N_{i}(\rho)$ be the maximal $j$-index of the points $a_{ij}$ contained
in $D_{\rho}$. For $\rho>t^{\prime}$ the disk $D_{\rho}$ contains first
$N_{i}+1$ points $\\{a_{i0},...,a_{iN_{i}}\\}$ of $\mathcal{O}_{i}$, whereas
the other points of $\mathcal{O}_{i}$ are in $\mathbb{C}\setminus{D}_{\rho}$.

###  3.1. Clusters

First of all, we define the central object of this article.

######  Definition 3.2 (Cluster [B2]).

We say that $a_{ij}$ and $a_{kl}$ are in the same _cluster $Cl(t,s)$_ if they
have the same potential $t=t_{ij}=t_{kl}$ and the same first entry
$s=s_{ij}=s_{kl}$ in the external address.

A cluster is called non-trivial if it contains more than on point of $P_{f}$.

Note that the set $P_{f}$ is a disjoint union of clusters and each cluster
contains at most $m$ points. As can be seen from the asymptotic formula 2.1,
for points $a_{ij}$ and $a_{kl}$ with big potentials being in the same cluster
implies that distance between them is very small, whereas the distance between
any pair of clusters is bounded from below.

The main reason, why we have to take clusters into account, is that if we have
infinitely many non-trivial clusters, we cannot use the same model for
$\mathcal{C}_{f}$ as in the non-degenerate case [B2]: we cannot separate
points in the same clusters by disjoint discs that stable under the Thurston
iteration (see discussion after [B2, Remark 5.10]). On the other hand, we can
control the behavior of marked points in clusters in a sense that under the
Thurston iteration this position is basically getting scaled by a real
constant (see Proposition 3.7).

Now, we are going to define two types of sets that will help us to describe
the behavior of points inside of a cluster under the Thurston iteration. Their
definitions might seem redundantly technical for the moment but this will pay
out by simpler computations later.

######  Definition 3.3 ($A_{x,n}$).

For $x>0$ and integer $n\geq 0$ let $A_{x,n}$ be the set of all complex
numbers $\alpha$ such that

  1. (1)

$\left|\log\left|\alpha\right|\right|<e^{-x/3}+...+e^{-F^{n}(x)/3}$,

  2. (2)

$\left|\arg\alpha\right|<e^{-x/3}+...+e^{-F^{n}(x)/3}$.

Visually $A_{x,n}$ is an annular sector containing $1$, see Figure 3.1. If $x$
increases, due to the fast convergence of the series
$\sum_{n}e^{-F^{n}(x)/3}$, the sets $A_{x,n}$ shrink to arbitrarily small
neighborhoods of $1$ uniformly in $n$.

The next lemma is the direct corollary of Definition 3.3. It presents a
property that allows us to reduce the amount of computations later on.

######  Lemma 3.4 (Pseudo-Multiplicativity of $A_{x,n}$).

If $\alpha_{i}\in A_{F^{i-1}(x),0}$ for $i\in\\{1,...,k\\}$, then
$\alpha_{1}\cdot...\cdot\alpha_{k}\in A_{x,k-1}$.

Figure 3.1. Schematic example of $A_{x,0}$. Figure 3.2. Schematic construction
of $D_{ij}^{kl}$.

The next notion helps to characterize the difference between marked point that
have the same potential but are in different clusters.

######  Definition 3.5 ($D_{ij}^{kl}$).

For $a_{ij}$ and $a_{kl}$ having the same potential but belonging to different
clusters define

$D_{ij}^{kl}:=\left\\{\frac{d(w-z)}{2\pi
i(s_{kl}-s_{ij})}:w\in\mathbb{D}_{1/l}(a_{kl}),z\in\mathbb{D}_{1/j}(a_{ij})\right\\}.$

In other words, if we surround $a_{kl}$ and $a_{ij}$ by the disks of radii
$1/l$ and $1/j$, then $D_{ij}^{kl}$ is just the difference of these two balls
divided over the normalizing coefficient $2\pi i(s_{kl}-s_{ij})/d$. This is
illustrated on Figure 3.2.

It is easy to see from Definition 3.5 that $D_{ij}^{kl}$ is just a disk.

######  Lemma 3.6.

$D_{ij}^{kl}=\mathbb{D}_{r_{ij}^{kl}}\left(\frac{d(a_{kl}-a_{ij})}{2\pi
i(s_{kl}-s_{ij})}\right)$ where
$r_{ij}^{kl}=\frac{d(1/l+1/j)}{2\pi\left|s_{kl}-s_{ij}\right|}$.

If $j$ and $l$ are big enough, then

  1. (1)

$1\in D_{ij}^{kl}$,

  2. (2)

$D_{ij}^{kl}$ is contained in arbitrarily small neighborhoods of $1$.

###### Proof.

That $D_{ij}^{kl}=\mathbb{D}_{r_{ij}^{kl}}\left(\frac{d(a_{kl}-a_{ij})}{2\pi
i(s_{kl}-s_{ij})}\right)$ is a direct consequence of Definition 3.5.

(1) It is implied by Definition 3.5 that the potentials of $a_{ij}$ and
$a_{kl}$ are equal: $t_{ij}=t_{kl}$. Hence from the asymptotic formula follows
that

$\frac{d(a_{kl}-a_{ij})}{2\pi
i(s_{kl}-s_{ij})}=\frac{(s_{kl}-s_{ij})+O(e^{-t_{ij}/2})}{(s_{kl}-s_{ij})}=1+\frac{O(e^{-t_{ij}/2})}{\left|s_{kl}-s_{ij}\right|}.$

The second summand is smaller than $r_{ij}^{kl}$ for big $j,l$. The claim
follows.

(2) Since $a_{ij}$ and $a_{kl}$ belong to different clusters, we have
$\left|s_{kl}-s_{ij}\right|\geq 1$. Hence $r_{ij}^{kl}\to 0$ as
$j,l\to\infty$. ∎

Next lemma provides an estimate on how the Thurston iteration changes the
relative position of marked points inside of the clusters with sufficiently
high potentials: for points $a_{ij},a_{kl}$ belonging to the same cluster with
potential $t$, the number $\hat{\varphi}(a_{kl})-\hat{\varphi}(a_{ij})$ is
equal to $\varphi_{u}(a_{k(l+1)})-\varphi_{u}(a_{i(j+1)})$ scaled by
$F^{\prime}(t)$ and multiplied by a correcting coefficient $\alpha\in A_{t,0}$
(which is getting arbitrarily close to $1$ as $t$ grows).

######  Proposition 3.7 (Negligible rotation within clusters).

For all $\rho\in\mathcal{P}^{\prime}$ big enough holds the following
statement.

If $a_{ij},a_{kl}\notin{D}_{\rho}$ belong to the same cluster $Cl(t,s)$, and
$\varphi_{u},u\in[0,1]$ is an isotopy of $\operatorname{id}$-type maps such
that $\varphi_{0}=c^{-1}$ and for all $u\in[0,1]$:

  1. (1)

$\varphi_{u}(\operatorname{SV}(f))\subset D_{\rho}$,

  2. (2)

$\left|\varphi_{u}(a_{i(j+1)})-a_{i(j+1)}\right|<1/(j+1)$,

  3. (3)

$\left|\varphi_{u}(a_{k(l+1)})-a_{k(l+1)}\right|<1/(l+1)$,

then for every $u\in[0,1]$ holds

$\hat{\varphi}_{u}(a_{kl})-\hat{\varphi}_{u}(a_{ij})=\frac{\varphi_{u}(a_{k(l+1)})-\varphi_{u}(a_{i(j+1)})}{F^{\prime}(t)}\alpha_{u},$

where $\alpha_{u}\in A_{t,0}$.

###### Proof.

Note that the statement of the theorem is vacuous if $d=1$: we do not have
clusters containing more than one point if we only have one singular orbit.

Let $g_{u}=\varphi_{u}\circ f\circ\hat{\varphi}_{u}^{-1}=q_{u}\circ\exp$ with
$q_{u}(z)=z^{d}+b_{d-1}^{u}z^{d-1}+...+b_{1}^{u}z+b_{0}^{u}$ and $g_{0}=f_{0}$

From [B2, Proposition 5.3] we know that for $\rho$ big enough and every
$a_{ij}\notin D_{\rho}$ holds

$\left|\hat{\varphi}_{u}(a_{ij})-a_{ij}\right|<\frac{1}{j}.$

On the other hand, from the formula 2.1 we know that
$\operatorname{Im\,}a_{ij}-2\pi s_{ij}\to 0$ as $j\to\infty$. Hence
$\operatorname{Im\,}\hat{\varphi}_{u}(a_{ij})-2\pi s_{ij}\to 0$ as
$j\to\infty$.

Now let $a_{ij},a_{kl}\notin D_{\rho}$ belong to the same cluster $Cl(t,s)$,
and $\gamma_{u}$ is the straight line segment joining
$\hat{\varphi}_{u}(a_{ij})$ and $\hat{\varphi}_{u}(a_{kl})$. For all $\rho$
big enough we have

$\varphi_{u}(a_{kl})-\varphi_{u}(a_{ij})=g_{u}(\hat{\varphi}_{u}(a_{kl}))-g_{u}(\hat{\varphi}_{u}(a_{ij}))=$
$\int\limits_{\gamma_{u}}g_{u}^{\prime}(z)dz=F^{\prime}(t)\int\limits_{\gamma_{u}}\frac{g_{u}^{\prime}(z)}{F^{\prime}(t)}dz$

From Lemma 5.1 we know that $\left|b_{k}^{u}\right|<L\rho^{\frac{d-k}{d}}$.
Since $t/\rho\to\infty$ as $\rho\to\infty$ and $d>1$, the last expression is
equal to

$F^{\prime}(t)\int\limits_{\gamma_{u}}\left(e^{d\left(z-t\right)}+O\left(e^{-t/2}\right)\right)dz$

In the proof of the item (3) of [B2, Proposition 5.3] we obtained the estimate
$\hat{\varphi}_{u}(a_{ij})-a_{ij}=O\left(e^{-t/2}\right)$, and from the
asymptotic formula 2.1 we know that $a_{ij}=t+2\pi is/d+O(e^{-t/2})$. Hence,
$\hat{\varphi}_{u}(a_{ij})=t+2\pi is/d+O\left(e^{-t/2}\right)$. Analogous
formula holds for $a_{kl}$: $\hat{\varphi}_{u}(a_{kl})=t+2\pi
is/d+O\left(e^{-t/2}\right)$. Consequently, for every $z\in\gamma_{u}$ we also
have $z=t+2\pi is/d+O\left(e^{-t/2}\right)$. Thus,

$F^{\prime}(t)\int\limits_{\gamma}\left(e^{d\left(z-t\right)}+O\left(e^{-\frac{t}{2}}\right)\right)dz=F^{\prime}(t)\int\limits_{\gamma}\left(e^{O\left(\exp\left(-\frac{t}{2}\right)\right)}+O\left(e^{-\frac{t}{2}}\right)\right)dz=$
$F^{\prime}(t)\int\limits_{\gamma}\left(1+O\left(e^{-\frac{t}{2}}\right)\right)dz=F^{\prime}\left(t\right)\left(\hat{\varphi}_{u}\left(a_{kl}\right)-\hat{\varphi}_{u}\left(a_{ij}\right)\right)\left(1+O\left(e^{-\frac{t}{2}}\right)\right)=$
$\frac{F^{\prime}(t)}{1+O\left(e^{-\frac{t}{2}}\right)}\left(\hat{\varphi}_{u}(a_{kl})-\hat{\varphi}_{u}(a_{ij})\right).$

Let $\alpha_{u}=1+O\left(e^{-t/2}\right)$ be the denominator in the last
expression, and let
$\left|O\left(e^{-\frac{t}{2}}\right)\right|<Ce^{-\frac{t}{2}}$ for some
$C>0$. Choose $\rho$ big enough so that

$2Ce^{-\frac{t}{2}}<e^{-\frac{t}{3}}<1/2.$

Then

$1-e^{-\frac{t}{3}}<\left|\alpha_{u}\right|<1+e^{-\frac{t}{3}}.$

After taking logarithm we obtain
$\left|\log\left|\alpha_{u}\right|\right|<e^{-t/3}$. Also we have

$\left|\sin\arg\alpha_{u}\right|<\frac{Ce^{-t/2}}{1-Ce^{-t/2}}<2Ce^{-t/2}.$

Hence,

$-e^{-t/3}<\arg\alpha_{u}<e^{-t/3}.$

Thus $\alpha_{u}\in A_{t,0}$. ∎

We also need another piece of notation.

######  Definition 3.8 ($H,L$).

For every pair $a_{ij},a_{kl}$ belonging to the same cluster define

  * •

$H=H(a_{ij},a_{kl})$ to be the smallest positive integer such that
$a_{i(j+H)}$ and $a_{k(l+H)}$ belong to different clusters,

  * •

$L=L(a_{ij},a_{kl})$ to be the smallest positive integer such that
$a_{i(j-L)}$ and $a_{k(l-L)}$ are defined and belong to different clusters, if
there is no such integer we define $L:=\infty$.

Note that $H(a_{ij},a_{kl})$ is always less than infinity: otherwise this
would mean that the points $a_{ij},a_{kl}$ had the same potential and external
address. From Theorem 2.3 it then follows $a_{ij}=a_{kl}$.

Now, we prove an easy corollary of Proposition 3.7 which provides cluster
estimates for the singular orbits of $f$. Roughly speaking, it says that if
two points are in the same cluster, we can estimate the distance between them
if we know after how many iterations they land in different clusters.
Basically, this is a scaling by the real constant $d(F^{\circ H}(t))^{\prime}$
multiplied by correcting coefficients $\nu$ and $\delta$.

######  Lemma 3.9 (Cluster estimates for $P_{f}$).

For all $j$ big enough holds the following statement.

If $a_{ij}$ and $a_{kl}$ are in the same cluster, then

$a_{kl}-a_{ij}=\frac{2\pi i(s_{k(l+H)}-s_{i(j+H)})}{d(F^{\circ
H}(t))^{\prime}}\nu\delta,$ (3.1)

where $t$ is the potential of $a_{ij}$ and $a_{kl}$, $H=H(a_{ij},a_{kl})$,
$\nu=\nu(i,j,k,l)\in A_{t,H-1}$ and $\delta=\delta(i,j,k,l)\in
D_{i(j+H)}^{k(l+H)}$.

###### Proof.

One simply needs to note that

$a_{k(l+H)}-a_{i(j+H)}=\frac{2\pi i(s_{k(l+H)}-s_{i(j+H)})}{d}\delta$

where $\delta\in D_{i(j+H)}^{k(l+H)}$, and to apply $H$ times the Proposition
3.7 with $\varphi_{u}\equiv c^{-1}$, obtaining at the step $n$ the estimates
for $a_{k(l+H-n)}-a_{i(j+H-n)}$. ∎

We also need to introduce special coefficients $\beta(a_{ij},a_{kl})$, in
order to estimate relative distances between marked points outside of
$D_{\rho}$.

######  Definition 3.10 ($\beta$).

Let $a_{ij},a_{kl}\in P_{f}$. Denote

$\beta(a_{ij},a_{kl}):=\begin{cases}\frac{\pi(s_{k(l+H)}-s_{i(j+H)})}{d(F^{\circ
H}(t))^{\prime}}&\mbox{if\,}a_{ij},a_{kl}\mbox{\,are in the same cluster
$Cl(t,s)$;}\\\ \pi/{2d}&\mbox{if\,}a_{ij},a_{kl}\mbox{\,are in different
clusters;}\end{cases}$

where $H=H(a_{ij},a_{kl})$.

From Lemma 3.9 and the asymptotic formula 2.1 follows immediately that for all
pairs $a_{ij},a_{kl}$ such that $j$ is big enough holds

$\left|a_{ij}-a_{kl}\right|>\beta(a_{ij},a_{kl});$ (3.2)

and for all pairs $a_{ij},a_{kl}$ belonging to the same cluster such that $j$
is big enough holds

$\left|a_{ij}-a_{kl}\right|<4\beta(a_{ij},a_{kl}).$ (3.3)

###  3.2. No combinatorics near $\infty$ (almost)

The next proposition is analogous to [B2, Proposition 5.6] and is supposed to
provide estimates on the homotopy type of spider legs in presence of non-
trivial clusters.

######  Proposition 3.11 (No combinatorics near $\infty$).

There exists a constant $C>0$ such that the following statement holds for all
$\rho\in\mathcal{P}^{\prime}$ big enough.

If $\varphi_{u},u\in[0,1]$ is an isotopy of $\operatorname{id}$-type maps
satisfying:

  1. (1)

$\varphi_{0}|_{\mathbb{C}\setminus\mathbb{D}_{\rho}}=\operatorname{id}$,

  2. (2)

$\varphi_{u}(a_{ij})\in D_{\rho}$ for $j\leq N_{i}$,

  3. (3)

$\left|\varphi_{u}(a_{ij})-a_{ij}\right|<1/j$ for $j>N_{i}$,

  4. (4)

for all $a_{ij},a_{kl}$ with $j>N_{i},l>N_{k}$, belonging to the same cluster
holds

$\varphi_{u}(a_{kl})-\varphi_{u}(a_{ij})=\frac{2\pi
i(s_{k(l+H)}-s_{i(j+H)})}{d(F^{\circ H}(t))^{\prime}}\nu_{u}\delta_{u}$

where $t$ is the potential of $a_{ij}$ and $a_{kl}$, $H=H(a_{ij},a_{kl})$,
$\nu_{u}=\nu_{u}(\varphi_{u},i,j,k,l)\in A_{t,H-1}$ and
$\delta_{u}=\delta_{u}(\varphi_{u},i,j,k,l)\in D_{i(j+H)}^{k(l+H)}$,

then for every $u$ and every $j>N_{i}$ we have

$\left|W_{ij}^{\varphi_{u}}\right|<C.$

###### Proof.

The proof is almost the same as of [B2, Proposition 5.6] except that we need
to take care of possible non-trivial clusters.

Note that if $j$ is big enough, than the points $a_{ij}\notin D_{\rho}$
belonging to different clusters move under $\varphi_{u}$ inside of mutually
disjoint disks $\mathbb{D}_{1/j}(a_{ij})$, and mutual distance between such
disks is bounded from below by $\pi/2d$.

On the other hand, since for big $t$ the sets $A_{t,H-1}$ and
$D_{i(j+H)}^{k(l+H)}$ are contained in arbitrarily small neighborhoods of $1$,
for all essentially big $t$ the marked points $\varphi_{u}(a_{ij})$ inside of
a cluster with potential $t$ have different imaginary parts and this holds for
all $u\in[0,1]$, i.e. the points inside of clusters with high enough
potentials are vertically linearly ordered.

In [B2, Lemma 2.14] we proved that for all big enough $j$ the ray tail
$R_{ij}=\varphi_{0}(R_{ij})$ has the strictly increasing real part. Hence, for
every $u\in[0,1]$ in the homotopy class (relative $P_{f}$) of
$\varphi_{u}(R_{ij})$ there is a representative with the strictly increasing
real part.

Next, due to the asymptotic formula 2.1 there is a universal constant $M>0$
such that for every $a_{ij}\in P_{f}$, every isotopy $\varphi_{u}$ satisfying
conditions of the proposition, and every $u\in[0,1]$ there is at most $M$
points $a_{kl}\in\mathcal{O}_{ij}$ with
$\operatorname{Re\,}\varphi_{u}(a_{kl})>\operatorname{Re\,}\varphi_{u}(a_{ij})-1$.
Hence, from [B1, Lemma 6.6] and [B1, Lemma 6.8] we see that there exists a
universal constant $C$ such that for all $\rho$ big enough and all $j>N_{i}$
we have $\left|W_{ij}^{\varphi_{u}}\right|<C$. ∎

###  3.3. Separation of preimages in $D_{\rho}$

As in [B2], we need a tool to “separate” the marked points inside of
$D_{\rho}$. This tools is a simple “division over the maximum of derivative”.
However, as will be seen in the Theorem 4.1, more control will be needed than
in the non-degenerate case [B2].

######  Definition 3.12 (Maximum of the derivative [B2]).

For $\rho=\frac{t_{n}+t_{n+1}}{2}\in\mathcal{P}^{\prime}$, define

$M_{\rho}:=\sup\limits_{\begin{subarray}{c}g\in\mathcal{N}_{d}\\\ SV(g)\subset
D_{\rho}\end{subarray}}\sup\limits_{\operatorname{Re\,}z<{(d+1)t_{n}}}\left|g^{\prime}(z)\right|.$

For all $\rho$ big enough, $M_{\rho}$ satisfies the following inequalities:

$\rho<M_{\rho}<Ke^{d^{3}t_{n}}$ (3.4)

for some $K>0$ depending only on $d$. The former inequality follows directly
from the definition of $M_{\rho}$, and the latter from Lemma 5.1.

Next proposition helps to estimate mutual distances between marked points
inside of $D_{\rho}$.

######  Proposition 3.13 (Separation of preimages [B2]).

For all $\rho\in\mathcal{P}^{\prime}$ big enough holds the following
statement.

If $\varphi$ is an $\operatorname{id}$-type map such that
$\varphi(\operatorname{SV}(f))\subset D_{\rho}$, and points
$x,y\in\mathbb{C}\setminus\operatorname{SV}(f)$ are satisfy

$\varphi\circ f(x),\varphi\circ
f(y)\in\overline{\mathbb{D}}_{\max\\{|a_{i(N_{i}+1)}|+1\\}}(0),$

then

$\left|\hat{\varphi}(x)-\hat{\varphi}(y)\right|\geq\frac{\left|\varphi\circ
f(x)-\varphi\circ f(y)\right|}{M_{\rho}}.$

Finally, in the last lemma before the construction of the invariant compact
subset $\mathcal{C}_{f}$ we prove certain estimates on mutual distances
between points in $P_{f}$. Roughly speaking, the lemma gives estimates on how
close the marked points may be close to each other based on the combinatorial
information (with a special attention in (2) on the case when the orbit of the
asymptotic value is involved).

Note that it can be in principle formulated without the “$\rho$-notation”, but
we prefer to use it for the sake of the more convenient application in Theorem
4.1.

From now on we agree that $\mathcal{O}_{1}=\\{a_{1j}\\}_{j=0}^{\infty}$ is the
orbit of the _asymptotic_ value of $f$, i.e. $a_{10}=c(\text{as.\ val.\ of
}f_{0})$.

######  Lemma 3.14 (Distance between points in $P_{f}$).

For all $\rho\in\mathcal{P}^{\prime}$ big enough hold the following two
statements.

  1. (1)

If $j\leq N_{i},l\leq N_{k},ij\neq kl$, and $n=\min\\{N_{i}+1-j,N_{k}+1-l\\}$,
then

$\left|a_{kl}-a_{ij}\right|>\frac{\beta\left(a_{i(j+n)},a_{k(l+n)}\right)}{(M_{\rho})^{n}}.$

  2. (2)

If $a_{k(N_{k}+1)}$ and $a_{1(N_{1}+1)}$ are in the same cluster,
$L=L(a_{k(N_{k}+1)},a_{1(N_{1}+1)})$, and $N=\max_{i\leq m}N_{i}$, then:

    1. (a)

if $0\leq n<L$, then

$\left|a_{k(N_{k}+1-n)}-a_{1(N_{1}+1-n)}\right|<4\beta\left(a_{k(N_{k}+1)},a_{1{(N_{1}+1)}}\right)M_{\rho}^{2dNn},$

    2. (b)

if $n\geq L$, then

$\left|a_{k(N_{k}+1-n)}-a_{1(N_{1}+1-n)}\right|>\left(\frac{1}{M_{\rho}}\right)^{2d^{4}N+n-L}.$

###### Proof.

(1) Applying Proposition 3.13 with $\hat{\varphi}=c^{-1}$ to the pair
$a_{i(j+n_{1}+1)},a_{k(l+n_{1}+1)}$ with $0\leq n_{1}\leq n$ we obtain

$\left|a_{i(j+n_{1})}-a_{k(l+n_{1})}\right|>\frac{\left|a_{i(j+n_{1}+1)}-a_{k(l+n_{1}+1)}\right|}{M_{\rho}}.$

At the same time due to formula 3.2

$\left|a_{i(j+n)}-a_{k(l+n)}\right|>\beta\left(a_{i(j+n)},a_{k(l+n)}\right).$

The conclusion follows after multiplication of the inequalities above.

(2) Without loss of generality assume that $L<N_{1}$, otherwise increase
$\rho$. The proof of $(a)$ goes by induction. The case $n=0$ coincides with
formula 3.3.

Assume that $n+1<L$ and

$\left|a_{k(N_{k}+1-n)}-a_{1(N_{1}+1-n)}\right|<4\beta\left(a_{k(N_{k}+1)},a_{1{(N_{1}+1)}}\right)M_{\rho}^{2dNn}.$

Let $\gamma$ be the straight line segment joining $a_{k(N_{k}+1-n)}$ and
$a_{1(N_{1}+1-n)}$. After increasing $\rho$ if needed, we might assume that
the distance from $\gamma$ to the singular values of $f_{0}$ is bigger than
$1/(M_{\rho})^{N+2}$: the number
$4\beta\left(a_{k(N_{k}+1)},a_{1{(N_{1}+1)}}\right)M_{\rho}^{2dNn}$ tends to
$0$ as $\rho\to\infty$ since $a_{k(N_{k}+1)},a_{1(N_{1}+1)}$ are in the same
cluster, which makes $\beta\left(a_{k(N_{k}+1)},a_{1{(N_{1}+1)}}\right)$ very
small even comparing to $1/(M_{\rho})^{2dNn}$.

Further, without loss of generality we might assume that the preimage
$\tilde{\gamma}$ of $\gamma$ under $f_{0}=p\circ\exp$ which starts at
$a_{1(N_{1}-n)}$ ends at $a_{k(N_{k}-n)}$: this is true for all big enough
$\rho$. Hence, it is enough to estimate the length of $\tilde{\gamma}$, in
order to bound the distance between $a_{1(N_{1}-n)}$ and $a_{k(N_{k}+1-n)}$.

If $z\in\tilde{\gamma}$, then the distance from $e^{z}$ to $0$ and the
critical points of $p$ is bigger than the distance between $f_{0}(z)$ and the
singular values of $f_{0}$ divided over $M_{\rho}$, i.e. bigger than
$1/(M_{\rho})^{N+3}$. Hence,

$\left|f_{0}^{\prime}(z)\right|=\left|p^{\prime}(e^{z})e^{z}\right|>d\left(\frac{1}{M_{\rho}^{N+3}}\right)^{d}>\frac{1}{M_{\rho}^{2dN}}.$

Thus,

$\left|a_{k(N_{k}-n)}-a_{1(N_{1}-n)}\right|\leq\frac{\left|a_{k(N_{k}+1-n)}-a_{1(N_{1}+1-n)}\right|}{\min\limits_{z\in\gamma}\left|f_{0}^{\prime}(z)\right|}<$
$\left|a_{k(N_{k}+1-n)}-a_{1(N_{1}+1-n)}\right|M_{\rho}^{2dN}<4\beta\left(a_{k(N_{k}+1)},a_{1{(N_{1}+1)}}\right)M_{\rho}^{2dN(n+1)}.$

To prove $(b)$, first let $n=L$. From Lemma 5.4 we know that the distance
between different preimages of $a_{1(N_{1}+2-L)}$ is bigger than
$(1/(M_{\rho})^{N+2}\rho)^{d^{4}}>1/(M_{\rho})^{d^{4}(N+3)}$. On the other
hand, since $a_{1(N_{1}+1-L)}$ and $a_{k(N_{k}+1-L)}$ are in different
clusters, exactly as in $(a)$ we obtain that

$\left|a_{k(N_{k}+1-L)}-\tilde{a}\right|<4\beta\left(a_{k(N_{k}+1)},a_{1{(N_{1}+1)}}\right)M_{\rho}^{2dNL}$

where $\tilde{a}$ is a preimage of $a_{1(N_{1}+2-L)}$ other than
$a_{1(N_{1}+1-L)}$. But since for big $\rho$ the number
$4\beta\left(a_{k(N_{k}+1)},a_{1{(N_{1}+1)}}\right)M_{\rho}^{2dNL}$ is much
smaller than $1/(M_{\rho})^{d^{4}(N+3)}$, we obtain

$\left|a_{k(N_{k}+1-L)}-a_{1(N_{1}+1-L)}\right|>\frac{1}{2}\frac{1}{M_{\rho}^{d^{4}(N+3)}}>\frac{1}{M_{\rho}^{2d^{4}N}}.$

The inequality $(b)$ for $n>L$ is obtained by straightforward inductive
application of Proposition 3.13 as in the proof of $(a)$. ∎

##  4\. Compact invariant subset

Now, we construct the compact invariant subset $\mathcal{C}_{f}$. First, we
prove invariance. Recall that we agreed that
$\mathcal{O}_{1}=\\{a_{1j}\\}_{j=0}^{\infty}$ is the orbit of the asymptotic
value of $f$.

######  Theorem 4.1 (Invariant subset).

Let $f=c\circ\exp$ be the quasiregular function defined in Subsection 2.2.
Further, let $\rho\in\mathcal{P}^{\prime}$ and
$\mathcal{C}_{f}(\rho)\subset\mathcal{T}_{f}$ be the _closure_ of the set of
points in the Teichmüller space $\mathcal{T}_{f}$ represented by
$\operatorname{id}$-type maps $\varphi$ for which there exists an isotopy
$\varphi_{u},u\in[0,1]$ of $\operatorname{id}$-type maps such that
$\varphi_{0}=\operatorname{id}$, $\varphi_{1}=\varphi$, and the following
conditions are simultaneously satisfied.

  1. (1)

(Marked points stay inside of $D_{\rho}$) If $j\leq N_{i}$,

$\varphi_{u}(a_{ij})\in D_{\rho}.$

  2. (2)

(Precise asymptotics outside of $D_{\rho}$) If $j>N_{i}$, then

$\left|\varphi_{u}(a_{ij})-a_{ij}\right|<1/j.$

  3. (3)

(Separation inside of $D_{\rho}$) If $j\leq N_{i},l\leq N_{k},ij\neq kl$, and
$n=\min\\{N_{i}+1-j,N_{k}+1-l\\}$, then

$\left|\varphi_{u}(a_{kl})-\varphi_{u}(a_{ij})\right|>\frac{\beta\left(a_{i(j+n)},a_{k(l+n)}\right)}{(M_{\rho})^{n}}.$

  4. (4)

(Bounded homotopy) If $j\leq N_{i}$, then

$\left|W_{ij}^{\varphi_{u}}\right|<A^{N_{i}+1-j}\left(\frac{(N_{i}+1)!}{j!}\right)^{4}C$

where $A$ and $C$ are constants from [B2, Theorem 4.11] and Proposition 3.11,
respectively.

  5. (5)

(Rigidity in clusters) For all $a_{ij},a_{kl}$ with $j>N_{i},l>N_{k}$
belonging to the same cluster, holds

$\varphi_{u}(a_{kl})-\varphi_{u}(a_{ij})=\frac{2\pi
i(s_{k(l+H)}-s_{i(j+H)})}{d(F^{\circ H}(t))^{\prime}}\nu_{u}\delta_{u}$

where $t$ is the potential of $a_{ij}$ and $a_{kl}$, $H=H(a_{ij},a_{kl})$,
$\nu_{u}=\nu_{u}(\varphi_{u},i,j,k,l)\in A_{t,H-1}$ and
$\delta_{u}=\delta_{u}(\varphi_{u},i,j,k,l)\in D_{i(j+H)}^{k(l+H)}$.

  6. (6)

(Clusters inside of $D_{\rho}$) If $a_{k(N_{k}+1)}$ and $a_{1(N_{1}+1)}$
belong to the same cluster, $L=L(a_{k(N_{k}+1)},a_{1(N_{1}+1)})$, and
$N=\max_{i\leq m}N_{i}$, then:

    1. (a)

if $0\leq n<L$, then

$\left|\varphi_{u}(a_{k(N_{k}+1-n)})-\varphi_{u}(a_{1(N_{1}+1-n)})\right|<4\beta\left(a_{k(N_{k}+1)},a_{1{(N_{1}+1)}}\right)M_{\rho}^{2dNn},$

    2. (b)

if $n\geq L$, then

$\left|\varphi_{u}(a_{k(N_{k}+1-n)})-\varphi_{u}(a_{1(N_{1}+1-n)})\right|>\left(\frac{1}{M_{\rho}}\right)^{2d^{4}N+n-L}.$

Then if $\rho\in\mathcal{P}^{\prime}$ is big enough, the subset
$\mathcal{C}_{f}(\rho)$ is well-defined, invariant under the $\sigma$-map and
contains $[\operatorname{id}]$.

Before the proof let us explain informally the meaning of conditions (1)-(6).

(1)-(2) say that the maps $\varphi$ are “uniformly of
$\operatorname{id}$-type”, that is, the marked points outside of a disk
$D_{\rho}$ have precise asymptotics, while inside of $D_{\rho}$ we allow some
more freedom.

(3) and (6b) say that the points inside of $D_{\rho}$ cannot come very close
to each other. Moreover, it is required to control the distance to the
asymptotic value — if a marked point is too close to it, then after Thurston
iteration its preimage has its real part close to $-\infty$, and this spoils
condition (1).

(4) describes the homotopy information and provides bounds for leg homotopy
words of legs with endpoints inside of $D_{\rho}$. Analogous bounds for marked
points outside of $D_{\rho}$ are encoded _implicitly_ in conditions (2),(3)
and (5).

(5) helps us to control the behavior of marked points within clusters. Note
that we do not need it if there are no clusters — in this case the marked
points outside of $D_{\rho}$ are contained in small disjoint disks. If there
are clusters, we cannot have such nice separation of orbits, but have a good
control over the relative behavior of points. This condition prevents points
within clusters to approach and rotate around each other.

(6) is needed for the same reason as (3) but in the very special case: when a
marked point $a_{i(N_{i}+1)}$ is in the same cluster with the the marked point
$a_{1(N_{1}+1)}$ on the orbit of the asymptotic value. In this case the
estimate $\beta$ is very small. If we use the same estimates as in (3) to
bound how the distance between their preimages changes under iteration of the
$\sigma$-map, then we obtain a marked point that is too close to the
asymptotic value — after one more iteration its preimage will be “spit”
towards $-\infty$. To avoid this we do the following trick. We make preimages
of this pair of points stay “very close” as long as $a_{i(N_{i}+1-n)}$ and
$a_{1(N_{1}+1-n)}$ are in the same cluster (condition (6a)). The first time
they are in different clusters leads to a better estimate on the distance
(condition (6b)).

###### Proof of Theorem 4.1.

A big part of the proof repeats the proof of [B2, Theorem 5.9]. Whenever this
is the case we avoid unnecessary repetitions and send the reader to [B2,
Theorem 5.9] for details.

Let $\mathcal{C}_{f}^{\circ}(\rho)\subset\mathcal{C}_{f}(\rho)$ be the set of
points in $\mathcal{T}_{f}$ of which we take the closure in the statement of
the theorem, that is, represented by $\operatorname{id}$-type maps $\varphi$
for which there exists an isotopy $\varphi_{u},u\in[0,1]$ of
$\operatorname{id}$-type maps such that $\varphi_{0}=\operatorname{id}$,
$\varphi_{1}=\varphi$ and the conditions (1)-(4) are simultaneously satisfied.
Since the $\sigma$-map is continuous, it is enough to prove invariance of
$\mathcal{C}_{f}^{\circ}(\rho)$.

First, we prove that for big $\rho$ the set $\mathcal{C}_{f}^{\circ}(\rho)$
contains $[c^{-1}]$: $c^{-1}$ can be joined to identity via the isotopy
$c_{u}^{-1}$ where $c_{u}$ was constructed in Subsection 2.2, and this isotopy
satisfies the conditions (1)-(6).

That $c_{u}^{-1}$ satisfies (1)-(4) is discussed in [B2, Theorem 5.9], while
(5) and (6) follow from Lemma 3.9 and Lemma 3.14, respectively.

For all $\varphi\in\mathcal{C}_{f}^{\circ}(\rho)$ after concatenation with
$c_{u}^{-1}$ we can obtain the isotopy $\psi_{u}$ of $\operatorname{id}$-type
maps with $\psi_{0}=c^{-1},\psi_{1}=\varphi$ and satisfying conditions
(1)-(6). Then $\hat{\psi}_{u}$ is an isotopy of $\operatorname{id}$-type maps
with $\hat{\psi}_{1}=\operatorname{id}$. Let $g_{u}(z)=\psi_{u}\circ
f\circ\hat{\psi}_{u}^{-1}(z)=p_{u}\circ\exp(z)$. Now, we prove that
$\hat{\psi}_{u}$ satisfies each of the items (1)-(6): it will follow that
$\hat{\varphi}\in\mathcal{C}_{f}^{\circ}(\rho)$.

We are going to prove that each of the conditions (1)-(6) for $\hat{\psi}$
follows from the conditions (1)-(6) for $\psi$.

(4) Literally repeats the proof of (4) in [B2, Theorem 5.9] using Proposition
3.11 instead of [B2, Proposition 5.6].

(3) Follows directly from Proposition 3.13.

(2) The proof fully repeats the proof of [B2, Theorem 5.9](2).

(1) We show that $\pm\rho/2$ are upper and lower bounds for both the real and
imaginary parts of $\hat{\psi}_{u}(a_{ij})$ for $a_{ij}\in D_{\rho}$: this
will give the desired bound on $\left|\hat{\psi}_{u}(a_{ij})\right|$. We will
only show that $\operatorname{Re\,}\hat{\psi}_{u}(a_{ij})>-\rho/2$, the other
bounds are obtained exactly as in [B2, Theorem 5.9].

Let $a_{ij}$ be such that $j\leq N_{i}$ and $N=\max N_{i}$. If
$j+1+N_{1}=N_{i}$ and $a_{1N_{1}},a_{iN_{i}}$ are in the same cluster, then
from (6b)

$\left|\varphi_{u}(a_{i(j+1)})-\varphi_{u}(a_{10})\right|>\left(\frac{1}{M_{\rho}}\right)^{2d^{4}N+N+1-L}.$

Otherwise for $n=\min\\{N_{i}-j,N_{1}+1\\}$ from (3) we have

$\left|\varphi_{u}(a_{i(j+1)})-\varphi_{u}(a_{10})\right|>\frac{\beta\left(a_{i(j+n)},a_{k(l+n)}\right)}{(M_{\rho})^{N+1}}>\frac{\pi}{2d(M_{\rho})^{N+1}}.$

In any case we might assume that the first inequality holds (this is true for
big enough $\rho$). Then

$\left|p_{u}^{-1}(\psi_{u}(a_{i(j+1)}))-p_{u}^{-1}(\psi_{u}(a_{10}))\right|=\left|\exp(\hat{\psi}_{u}(a_{ij}))-0\right|>\left(\frac{1}{M_{\rho}}\right)^{2d^{4}N+N+1-L+1}.$

Hence,

$\operatorname{Re\,}\hat{\psi}_{u}(a_{ij})>-(N(2d^{4}+1)+2-L)\log M_{\rho}.$

If $\rho$ is big, using the inequality 3.4 we see that the right hand side of
the last expression is bigger than $-\rho/2$.

(5) From Proposition 3.7 we see that for all $j>N_{i},l>N_{k}$,
$a_{ij},a_{kl}$ belonging to the same cluster with potential $t$

$\hat{\psi}_{u}(a_{kl})-\hat{\psi}_{u}(a_{ij})=\frac{\psi_{u}(a_{k(l+1)})-\psi_{u}(a_{i(j+1)})}{F^{\prime}(t)}\alpha_{u}=$
$\frac{2\pi
i\left(s_{k(l+H)}-s_{i(j+H)}\right)}{d\left(F^{\circ(H-1)}\right)^{\prime}\circ
F(t)F^{\prime}(t)}\left(\nu_{u}(\psi_{u},i,j+1,k,l+1)\alpha_{u}\right)\delta_{u}(\psi_{u},i,j+1,k,l+1)=$
$\frac{2\pi i\left(s_{k(l+H)}-s_{i(j+H)}\right)}{d\left(F^{\circ
H}(t)\right)^{\prime}}\nu_{u}(\hat{\psi}_{u},i,j,k,l)\delta_{u}(\hat{\psi}_{u},i,j,k,l)$

where $\alpha_{u}\in A_{t,0}$ and $H=H(a_{ij},a_{kl})$.

(6) The proof of (6) is goes analogously to the proof of Lemma 3.14 (2).

Without loss of generality assume that $L<N_{1}$, otherwise increase $\rho$.
The case $n=0$ follows from (4).

Assume that $n+1<L$ and

$\left|\psi_{u}(a_{k(N_{k}+1-n)})-\psi_{u}(a_{1(N_{1}+1-n)})\right|<4\beta\left(a_{k(N_{k}+1)},a_{1{(N_{1}+1)}}\right)M_{\rho}^{2dNn}.$

Let $\gamma_{u}$ be the straight line segment joining
$\psi_{u}(a_{k(N_{k}+1-n)})$ and $\psi_{u}(a_{1(N_{1}+1-n)})$. After
increasing $\rho$ if needed, assume that the distance from $\gamma_{u}$ to the
singular values of $f_{0}$ is bigger than $1/(M_{\rho})^{N+2}$: the number
$4\beta\left(a_{k(N_{k}+1)},a_{1{(N_{1}+1)}}\right)M_{\rho}^{2dNn}$ tends to
$0$ as $\rho\to\infty$ since $a_{k(N_{k}+1)},a_{1(N_{1}+1)}$ are in the same
cluster. Hence, $\beta\left(a_{k(N_{k}+1)},a_{1{(N_{1}+1)}}\right)$ is very
small even comparing to $1/(M_{\rho})^{2dNn}$.

Without loss of generality assume that the preimage $\tilde{\gamma}_{u}$ of
$\gamma_{u}$ under $g_{u}=p_{u}\circ\exp$ which starts at
$\hat{\psi}_{u}(a_{1(N_{1}-n)})$ ends at $\hat{\psi}_{u}(a_{k(N_{k}-n)})$:
this is true since this holds for $\psi_{0}=c^{-1}$ and under the isotopy
$\psi_{u}$ the singular values of $g_{u}$ do not cross $\gamma_{u}$. Hence, it
is enough to estimate the length of $\tilde{\gamma}_{u}$, in order to bound
the distance between $\hat{\psi}_{u}(a_{1(N_{1}-n)})$ and
$\hat{\psi}_{u}(a_{k(N_{k}+1-n)})$.

If $z\in\tilde{\gamma}_{u}$, then the distance from $e^{z}$ to $0$ and the
critical points of $p_{u}$ is bigger than $1/(M_{\rho})^{N+3}$. Hence,

$\left|g_{u}^{\prime}(z)\right|=\left|p_{u}^{\prime}(e^{z})e^{z}\right|>d\left(\frac{1}{M_{\rho}^{N+3}}\right)^{d}>\frac{1}{M_{\rho}^{2dN}}.$

Thus, as in Lemma 3.14 (2),

$\left|\hat{\psi}_{u}(a_{k(N_{k}-n)})-\hat{\psi}_{u}(a_{1(N_{1}-n)})\right|\leq\frac{\left|\psi_{u}(a_{k(N_{k}+1-n)})-\psi_{u}(a_{1(N_{1}+1-n)})\right|}{\min\limits_{z\in\gamma}\left|g_{u}^{\prime}(z)\right|}<$
$\left|\psi_{u}(a_{k(N_{k}+1-n)})-\psi_{u}(a_{1(N_{1}+1-n)})\right|M_{\rho}^{2dN}<4\beta\left(a_{k(N_{k}+1)},a_{1{(N_{1}+1)}}\right)M_{\rho}^{2dN(n+1)}.$

To prove $(b)$, first consider $n=L$. From Lemma 5.4 as in Lemma 3.14 (2)
follows that the distance between different preimages of
$\psi_{u}(a_{1(N_{1}+2-L)})$ under $g_{u}$ is bigger than
$1/(M_{\rho})^{d^{4}(N+3)}$. On the other hand, since $a_{1(N_{1}+1-L)}$ and
$a_{k(N_{k}+1-L)}$ are in different clusters, as in $(a)$ we have

$\left|\hat{\psi}_{u}(a_{k(N_{k}+1-L)})-\tilde{a}\right|<4\beta\left(a_{k(N_{k}+1)},a_{1{(N_{1}+1)}}\right)M_{\rho}^{2dNL}$

where $\tilde{a}$ is a preimage of $\psi_{u}(a_{1(N_{1}+2-L)})$ under $g_{u}$
other than $\hat{\psi}_{u}(a_{1(N_{1}+1-L)})$. Since for big $\rho$ the number
$4\beta\left(a_{k(N_{k}+1)},a_{1{(N_{1}+1)}}\right)M_{\rho}^{2dNL}$ is much
smaller than $1/(M_{\rho})^{d^{4}(N+3)}$, we have

$\left|\hat{\psi}_{u}(a_{k(N_{k}+1-L)})-\hat{\psi}_{u}(a_{1(N_{1}+1-L)})\right|>\frac{1}{2}\frac{1}{M_{\rho}^{d^{4}(N+3)}}>\frac{1}{M_{\rho}^{2d^{4}N}}.$

The inequality $(6b)$ for $n>L$ can be obtained by application of Proposition
3.13 as in the proof of $(3)$. ∎

######  Remark 4.2.

Every point of $\mathcal{C}_{f}$ is evidently asymptotically conformal.
Indeed, near infinity the marked points are separated into clusters that move
inside of disjoint small disks, whereas there is only a negligible relative
movement of marked points under isotopies inside of every cluster.

######  Remark 4.3.

Note that Theorem 4.1 defines an invariant set also in the case when
$\underline{s}_{i}$ is (pre-)periodic. Indeed, absence of (pre-)periodicity is
not used in the proof. But we do not have enough tools to prove that in the
(pre-)periodic case the set $\mathcal{C}_{f}(\rho)$ would be compact. For this
one needs to upgrade spider construction to allow each leg contain infinitely
many marked points. An alternative way to prove Theorem 1.1 in the
(pre-)periodic case is to use the continuity argument which we do in the
subsequent article.

Next, we prove that $\mathcal{C}_{f}(\rho)$ is compact.

######  Theorem 4.4 (Compactness).

$\mathcal{C}_{f}(\rho)$ is a compact subset of $\mathcal{T}_{f}$.

###### Proof.

The proof of the theorem is identical to the proof of [B2, Theorem 5.11],
except of a small detail that in our case we are allowed to have non-trivial
clusters. This obviously does not violate the convergence argument [B2,
Theorem 5.11] since there is almost no relative movement of point inside of
clusters outside of $D_{\rho}$. ∎

Now, we have all ingredients for the proof Classification Theorem 1.1.

###### Proof of Classification Theorem 1.1.

The proof literally repeats the proof of [B2, Theorem 1.4] after obvious
replacement of invariance and compactness theorems by our upgraded versions:
Theorem 4.1 and Theorem 4.4. ∎

##  5\. Appendix A

In this appendix we provide some properties of polynomials and of their
compositions with the exponential.

The following lemma gives estimates on the coefficients of polynomials $p$ if
we know an estimate on the magnitude of singular values of $p\circ\exp$.

######  Lemma 5.1 (Singular values bound coefficients [B2]).

Fix some integer $d>0$. There exists a universal constant $L>0$ such that if
$g=p\circ\exp$ where $p(z)=z^{d}+b_{d-1}z^{d-1}+...+b_{1}z+b_{0}$, and
singular values of $g$ are contained in the disk $\mathbb{D}_{\rho}(0)$ with
$\rho$ big enough, then $\left|b_{k}\right|<L\rho^{\frac{d-k}{d}}$.

Next statement describes the behavior of polynomials $p$ outside of a disk
containing singular values of $p\circ\exp$ subject to the condition that this
disk is big enough.

######  Lemma 5.2 (Preimages of outer disks [B2]).

Fix some integer $d>1$. If $\rho>1$ is big enough, then for every
$g=p\circ\exp$ with monic polynomial $p$ of degree $d$ and singular values
contained in $\mathbb{D}_{\rho}(0)$ for all $r\geq\rho$ we have

$p^{-1}(\mathbb{D}_{r}(0))\subset\mathbb{D}_{r}(0).$

For a polynomial $p$ we denote by $\operatorname{Crit}(p)$ the set of its
critical points. By $\operatorname{dist}(A,B)$ we denote the Euclidean
distance between sets $A,B\subset\mathbb{C}$.

######  Lemma 5.3 (Distance between preimages under polynomials).

Let $d>1$ be an integer. Then there exists a real constant $K>0$ such that for
every real $\epsilon\in(0,1)$ and $r>1$ holds the following statement.

If $p$ is a monic polynomial of degree $d$ and $\alpha\in\mathbb{C}$ is such
that all roots $z_{1},z_{2},...,z_{d}$ of the equation $p(z)=\alpha$ are
contained in $\mathbb{D}_{r}(0)$ and
$\operatorname{dist}\left(\\{z_{1},...,z_{d}\\},\operatorname{Crit}(p)\right)>\epsilon$,
then for every pair of indices $i,j$ such that $1\leq i<j\leq d$ we have

$\left|z_{i}-z_{j}\right|>K\left(\frac{\epsilon}{r}\right)^{d^{2}}.$

###### Proof.

It is enough to proof the lemma for $i=1,j=2$.

Let $q(z):=p(z)-\alpha=(z-z_{1})\cdot...\cdot(z-z_{d})$, and let $\Delta_{q}$
be the discriminant of $q$. Then

$\Delta_{q}=\prod_{i<j}(z_{i}-z_{j})^{2}=(-1)^{\frac{d(d-1)}{2}}p^{\prime}(z_{1})p^{\prime}(z_{2})\cdot...\cdot
p^{\prime}(z_{d}).$

If $p^{\prime}(z)=d(z-c_{1})\cdot...\cdot(z-c_{d-1})$, then for
$n\in\\{1,...,d\\}$ we obtain $\left|p^{\prime}(z_{n})\right|\geq
d\epsilon^{d-1}$. Hence

$\left|\Delta_{q}\right|\geq d^{d}\epsilon^{d(d-1)}.$

Finally,

$\left|z_{1}-z_{2}\right|=\frac{\left|\Delta_{q}\right|}{\left|\Delta_{q}/(z_{1}-z_{2})\right|}>\frac{\left|\Delta_{q}\right|}{(2r)^{d(d-1)-1}}\geq\frac{d^{d}\epsilon^{d(d-1)}}{(2r)^{d(d-1)-1}}>K\left(\frac{\epsilon}{r}\right)^{d^{2}}$

where $K=d^{d}/2^{d(d-1)-1}.$ ∎

Recall that for an entire function $g\in\mathcal{N}_{d}$ we denote by
$\operatorname{SV}(g)$ the set of its finite singular values.

######  Lemma 5.4 (Distance between preimages under $f\in\mathcal{N}_{d}$).

Fix an integer $d>1$. For all $\epsilon\in(0,1)$ and all big enough $\rho>0$
holds the following statement.

If $p$ is a monic polynomial of degree $d$ and $g=p\circ\exp$ satisfy
$\operatorname{SV}(g)\subset\mathbb{D}_{\rho}(0)$, and
$\alpha\in\mathbb{D}_{2\rho}(0)$ is such that
$\operatorname{dist}(\alpha,\operatorname{SV}(g))>\epsilon$, then for every
pair of distinct complex numbers $w_{1},w_{2}\in g^{-1}(\alpha)$ holds

$\left|w_{1}-w_{2}\right|>\left(\frac{\epsilon}{\rho}\right)^{d^{4}}.$

###### Proof.

Pick $\zeta\in p^{-1}(\alpha)$, and let $c$ be a critical point of $p$. Due to
Lemma 5.2, for all $\rho$ big enough we have $\left|\zeta\right|<2\rho$ and
$\left|c\right|<2\rho$. But then

$\left|\zeta-c\right|>\frac{\operatorname{dist}(\alpha,\operatorname{SV}(g))}{\max\limits_{\left|z\right|<2\rho}\left|p^{\prime}(z)\right|}>\frac{\epsilon}{\max\limits_{\left|z\right|<2\rho}\left|p^{\prime}(z)\right|}.$

From Lemma 5.1 follows that for all $\rho$ big enough

$\max\limits_{\left|z\right|<2\rho}\left|p^{\prime}(z)\right|<\rho^{d}.$

Hence, if $\zeta_{1},\zeta_{2}\in p^{-1}(\alpha)$, from Lemma 5.3 follows that

$\left|\zeta_{1}-\zeta_{2}\right|>K\left(\frac{\operatorname{dist}(p^{-1}(\alpha),\operatorname{Crit}(p))}{2\rho}\right)^{d^{2}}>\frac{K}{2^{d^{2}}}\left(\frac{\epsilon}{\rho^{d+1}}\right)^{d^{2}}.$

Further, if $e^{w_{1}}=\zeta_{1}$ and $e^{w_{2}}=\zeta_{2}$, then for big
$\rho$

$\left|w_{1}-w_{2}\right|>\frac{\left|\zeta_{1}-\zeta_{2}\right|}{\max\limits_{\operatorname{Re\,}z<\log
2\rho}\left|\exp^{\prime}z\right|}>\frac{K}{2^{d^{2}}}\left(\frac{\epsilon}{\rho^{d+1}}\right)^{d^{2}}\frac{1}{2\rho}>\left(\frac{\epsilon}{\rho}\right)^{d^{4}}.$

∎

##  6\. Acknowledgements

We would like to express our gratitude to our research team in Aix-Marseille
Université, especially to Dierk Schleicher who supported this project from the
very beginning, Sergey Shemyakov who carefully proofread all drafts, as well
as to Kostiantyn Drach, Mikhail Hlushchanka, Bernhard Reinke and Roman Chernov
for uncountably many enjoyable and enlightening discussions of this project at
different stages. We also want to thank Dzmitry Dudko for his multiple
suggestions that helped to advance the project, Lasse Rempe for his long list
of comments and relevant questions, and Adam Epstein for important discussions
especially in the early stages of this project.

Finally, we are grateful to funding by the Deutsche Forschungsgemeinschaft
DFG, and the ERC with the Advanced Grant “Hologram” (695621), whose support
provided excellent conditions for the development of this research project.
This research received support from the European Research Council (ERC) under
the European Union Horizon 2020 research and innovation programme, grant
647133 (ICHAOS).

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