# Spectroscopic characterization of singlet-triplet doorway states of aluminum
monofluoride

N. Walter https://www.fhi.mpg.de/207024/mp-department<EMAIL_ADDRESS>J. Seifert  S. Truppe  H.C. Schewe  Fritz-Haber-Institut der Max-Planck-
Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany  B.G. Sartakov  Prokhorov
General Physics Institute, Russian Academy of Sciences, Vavilovstreet 38,
119991 Moscow, Russia  G. Meijer<EMAIL_ADDRESS>Fritz-Haber-
Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany

###### Abstract

Aluminum monofluoride (AlF) possesses highly favorable properties for laser
cooling, both via the A${}^{1}\Pi$ and a${}^{3}\Pi$ states. Determining
efficient pathways between the singlet and the triplet manifold of electronic
states will be advantageous for future experiments at ultralow temperatures.
The lowest rotational levels of the A${}^{1}\Pi,v=6$ and
b${}^{3}\Sigma^{+},v=5$ states of AlF are nearly iso-energetic and interact
via spin-orbit coupling. These levels thus have a strongly mixed spin-
character and provide a singlet-triplet doorway. We here present a hyperfine
resolved spectroscopic study of the A${}^{1}\Pi,v=6$ //
b${}^{3}\Sigma^{+},v=5$ perturbed system in a jet-cooled, pulsed molecular
beam. From a fit to the observed energies of the hyperfine levels, the fine
and hyperfine structure parameters of the coupled states, their relative
energies as well as the spin-orbit interaction parameter are determined. The
standard deviation of the fit is about 15 MHz. We experimentally determine the
radiative lifetimes of selected hyperfine levels by time-delayed ionization,
Lamb dip spectroscopy and accurate measurements of the transition lineshapes.
The measured lifetimes range between 2 ns and 200 ns, determined by the degree
of singlet-triplet mixing for each level.

††preprint: AIP/123-QED

##  I Introduction

In 1939, Rochester recorded the first electronic absorption spectrum of
aluminum monofluoride (AlF).Rochester (1939) He attributed the bands centered
around 227 nm to the A${}^{1}\Pi$ – X${}^{1}\Sigma^{+}$ system and concluded
that the vibrational and rotational constants of these coupled electronic
states are almost the same. By 1974, a total of nine singlet and seven triplet
electronic states had been assigned to AlF and ordered into a common energy
scheme.Barrow, Kopp, and Malmberg (1974) The energy of the triplet relative to
the singlet manifold of states was determined from observed spectral
perturbations, in particular those due to spin-orbit coupling between the
A${}^{1}\Pi$ and b${}^{3}\Sigma^{+}$ states. The direct a${}^{3}\Pi$ –
X${}^{1}\Sigma^{+}$ singlet-triplet transition was only observed two years
later. Kopp, Lindgren, and Malmberg (1976); Rosenwaks, Steele, and Broida
(1976)

A few years ago, our group became interested in AlF in view of its predicted
favorable properties for laser cooling and magneto-optical trapping. Di Rosa,
M. D. (2004); Wells and Lane (2011) We determined precise molecular parameters
for the lowest singlet and triplet states, Truppe  _et al._ (2019);
Doppelbauer  _et al._ (2021); Walter  _et al._ (2021) thereby establishing AlF
as a new benchmark molecule for quantum chemistry calculations. We studied the
spin-orbit interaction between the $v=0$ level in the A${}^{1}\Pi$ state and
the various vibrational levels in the b${}^{3}\Sigma^{+}$ state to determine
the fraction of triplet character of the A${}^{1}\Pi,v=0$ state.Doppelbauer
_et al._ (2021) This fraction is low enough, i.e. this particular interaction
is weak enough, to not cause a significant loss (< 10-6) for optical cycling
on the A${}^{1}\Pi,v=0$ – X${}^{1}\Sigma^{+},v=0$ band.Hofsäss  _et al._
(2021)

Although the perturbation of the A${}^{1}\Pi,v=0$ state is weak, there is a
strong interaction between the nearly iso-energetic lowest rotational levels
of the A${}^{1}\Pi,v=6$ and b${}^{3}\Sigma^{+},v=5$ states. Barrow, Kopp, and
Malmberg (1974) This results in a mixed singlet-triplet character of these
rotational levels that allows them to act as a path from the singlet to the
triplet manifold. Such strongly mixed levels have been proposed as doorways,
for instance, to produce translationally cold CO molecules in their absolute
ground-state, starting from trapped molecules in the a${}^{3}\Pi$
state.Blokland  _et al._ (2011)

Here, we present a detailed spectroscopic investigation of the lowest
rotational levels of the perturbed A${}^{1}\Pi,v=6$ // b${}^{3}\Sigma^{+},v=5$
system of AlF. In Figure 1 the relevant energy levels are shown together with
the excitation and detection schemes that are used. All experiments are
carried out on a jet-cooled supersonic molecular beam containing AlF, as
briefly described in Section II. Using two-color resonance enhanced
multiphoton ionization ((1+1′) REMPI), we record rotationally resolved spectra
of the perturbed region, presented in Section III. Subsequently, hyperfine
resolved laser-induced fluorescence (LIF) spectra are recorded using a narrow-
band continuous wave laser for A${}^{1}\Pi,v=6$ //
b${}^{3}\Sigma^{+},v=5\leftarrow$ a${}^{3}\Pi,v=5$ excitation (see Section
IV). The radiative lifetimes of the perturbed levels range between 2 ns and
200 ns and are determined from spectral line broadening, Lamb dip spectrosopy
and time-delayed ionization, as described in Section V.

##  II Experimental Setup

Figure 1:  Scheme of the energy levels and of the excitation and detection
methods relevant for this study. The top right part shows in detail the
unperturbed rotational levels of the A${}^{1}\Pi,v=6$ and
b${}^{3}\Sigma^{+},v=5$ states.  Figure 2:  Scheme of the molecular beam
setup. The arrangement of the rf interaction region, the Time-of-Flight mass
spectrometer (ToF-MS) and the Laser Induced Fluorescence (LIF) zone can
readily be changed.

The experimental setup is shown schematically in Figure 2, and has been
described in detail elsewhere.Walter  _et al._ (2021) A gas mixture of 2 % SF6
seeded in neon with a backing pressure of 3 bar is released from a pulsed
valve, operated at 10 Hz, and passes through a narrow channel. The radiation
of a pulsed Nd:YAG laser (5 ns pulse duration, 16 mJ pulse energy, 1064 nm) is
mildly focused on a rotating aluminum rod, injecting aluminum atoms in this
channel. AlF is produced in a chemical reaction of the laser ablated aluminum
atoms and SF6. The molecules are cooled while expanding from the narrow
channel into vacuum, reaching a translational temperature of about 3 K and a
rotational temperature of about 5 K. The vibrational cooling is comparatively
inefficient, and as a result about 1 % of the population occupies the
X${}^{1}\Sigma^{+},v=5$ state. This is a sufficient starting point for the
experiments presented here. The molecules pass through a skimmer and then
enter the differentially pumped preparation region. Here, the radiation from a
TiSa-seeded, frequency doubled pulsed dye amplifier (PDA) crosses the
molecular beam perpendicularly and excites the molecules on selected
rotational lines of the a${}^{3}\Pi,v=5$ $\leftarrow$ X${}^{1}\Sigma^{+},v=5$
band around 365 nm.

The lifetime of the molecules in the a${}^{3}\Pi,v=5$ state is several
msWalter  _et al._ (2021) and their forward velocity is about 750 m/s.
Therefore, manipulation and detection can take place in spatially separated
regions. By passing through a parallel plates transmission line, the molecules
can be exposed to rf radiation to populate selected $F$ levels in the
a${}^{3}\Pi,v=5$ state. The metastable molecules can be monitored by
ionization or LIF detection via the b${}^{3}\Sigma^{+},v=5$ state in regions
further downstream. Rotationally resolved electronic spectra are recorded via
(1+1′) REMPI using a pulsed dye laser around 580 nm (0.05 cm-1 bandwidth,
several $\mu$J pulse energy) for resonant excitation, followed by a counter-
propagating pulsed dye laser around 280 nm (several mJ pulse energy) for
ionization. The time delay between the excitation and ionization laser pulse
can be varied with ns precision. Parent AlF+ ions are detected in a compact,
linear Time-of-Flight mass spectrometer (ToF-MS) of the Wiley-McLaren type.
For LIF detection, a narrowband cw ring dye laser is used to resonantly excite
the molecules to the A${}^{1}\Pi,v=6$ // b${}^{3}\Sigma^{+},v=5$ system. The
UV fluorescence to the X${}^{1}\Sigma^{+}$ state, allowed for by the singlet
character of the excited levels, is detected with a photomultiplier tube
(PMT). The absolute frequency of the ring dye laser is recorded with a
wavemeter (HighFinesse WS8) to an absolute accuracy of 10 MHz.

##  III Rotationally Resolved Spectroscopy

Figure 3: Overview of the A${}^{1}\Pi,v=6$ // b${}^{3}\Sigma^{+},v=5$
perturbation. The energy of the levels of the b${}^{3}\Sigma^{+},v=5$ and
A${}^{1}\Pi,v=6$ states, reduced by $\left(48908.0+0.52\,J(J+1)\right)$ cm-1,
are plotted as a function of $J$, on a scale linear in $J(J+1)$. The $e$
levels are plotted in blue, the $f$ levels in green. Solid (dashed) curves
connect the (un)perturbed levels.

The $v=0$ level of the A${}^{1}\Pi$ state of AlF is located about 855 cm-1
below the $v=0$ level of the b${}^{3}\Sigma^{+}$ state.Barrow, Kopp, and
Malmberg (1974) This spacing is only slightly larger than the vibrational
spacing in the A${}^{1}\Pi$ state, locating the $v+1$ level of the
A${}^{1}\Pi$ state close in energy to the $v$ level in the b${}^{3}\Sigma^{+}$
state. As the vibrational frequency in the A${}^{1}\Pi$ state is somewhat
larger than the one in the b${}^{3}\Sigma^{+}$ state, these vibrational levels
come gradually closer together with increasing $v$ and eventually cross. The
energy gap is the smallest for the A${}^{1}\Pi,v=6$ and the
b${}^{3}\Sigma^{+},v=5$ states, for which the lowest rotational levels are
schematically depicted in the top right part of Figure 1.

For an overview of the interaction between the rotational levels in the
A${}^{1}\Pi,v=6$ and b${}^{3}\Sigma^{+},v=5$ states, we first neglect the
hyperfine structure as well as splittings due to $\Lambda$-doubling, spin-spin
and spin-rotation interaction. We assume that $J$, the quantum number
belonging to the total angular momentum $\mathbf{J}$, is a good quantum
number. The energy of the unperturbed rotational levels in the
A${}^{1}\Pi,v=6$ state (Hund’s case (a)) is given by $B_{\text{A}6}J(J+1)$,
where $B_{\text{A}6}$ is the rotational constant of the A${}^{1}\Pi,v=6$
state. Each $J$ level consists of a positive and a negative component of the
$\Lambda$-doublet whose splitting is neglected, i.e. the $e$ levels and $f$
levels are assumed to be degenerate. In the b${}^{3}\Sigma^{+},v=5$ state
(Hund’s case (b)), $\mathbf{J}$ is the vectorial sum of the end-over-end
rotation $\mathbf{N}$ and the total spin of the electrons $\mathbf{S}$. The
expression for the unperturbed rotational energy is given by
$B_{\text{b}5}N(N+1)$, where $B_{\text{b}5}$ is the rotational constant of the
b${}^{3}\Sigma^{+},v=5$ state. In the absence of spin-spin and spin-rotation
interaction, the levels with $J=N+1$, $J=N$ and $J=N-1$, referred to as
$\mathcal{F}_{1}$, $\mathcal{F}_{2}$ and $\mathcal{F}_{3}$ levels,
respectively, are degenerate. The $\mathcal{F}_{1}$ and $\mathcal{F}_{3}$
levels are $f$ levels whereas the $\mathcal{F}_{2}$ levels are $e$ levels.
Using the spectroscopic constants as deduced by Barrow et al. the unperturbed
$J=1$ level of the A${}^{1}\Pi,v=6$ state is only 0.09(3) cm-1 below the
unperturbed $N=2$ level of the b${}^{3}\Sigma^{+},v=5$ state.Barrow, Kopp, and
Malmberg (1974)

Interaction due to spin-orbit coupling occurs between levels that have the
same values of $J$ and the same $e$ or $f$ character. For the $e$ levels, the
energy is found as the Eigenvalues of the $2\times 2$ matrix Kovács (1969)

$\left(\begin{array}[]{cc}E_{6}(\text{A}_{e,J})&-\frac{\xi_{6,5}}{\sqrt{2}}\\\
-\frac{\xi_{6,5}}{\sqrt{2}}&E_{5}({\text{b}_{\mathcal{F}_{2},J}})\end{array}\right)$
(1)

where $E_{6}(\text{A}_{e,J})$ and $E_{5}({\text{b}_{\mathcal{F}_{2},J}})$ are
the energies of the unperturbed $J,e$ levels of the A${}^{1}\Pi,v=6$ state and
b${}^{3}\Sigma^{+},v=5$ state, respectively. The parameter $\xi_{6,5}$ is the
spin-orbit interaction constant for these two states. As $J=N$ for the $e$
levels in the b${}^{3}\Sigma^{+}$ state, and as the rotational constants of
the interacting states only differ by 1 %, the effect of this perturbation is
detectable over a wide range of $J$ levels, as shown with the blue dots in
Figure 3. From this perturbation between the $e$ levels, Barrow et al.Barrow,
Kopp, and Malmberg (1974) determined the relative energies of the singlet and
the triplet levels and found the spin-orbit interaction constant $\xi_{6,5}$
to be about 1.16(1) cm-1.

Figure 4:  Energy of the levels of the b${}^{3}\Sigma^{+},v=5$ and
A${}^{1}\Pi,v=6$ states minus $48900$ cm-1 as a function of $J$, for $J=0-4$.
The $e$ levels are plotted in blue, the $f$ levels in green. Solid (dashed)
curves connect the (un)perturbed levels.

Both the $\mathcal{F}_{1}$ and $\mathcal{F}_{3}$ levels of the
b${}^{3}\Sigma^{+}$ state are $f$ levels, so the energies of the perturbed $f$
levels for a given $J$ are found as the Eigenvalues of the $3\times 3$ matrix
Kovács (1969)

$\left(\begin{array}[]{ccc}E_{6}(\text{A}_{f,J})&\sqrt{\frac{J+1}{4J+2}}\xi_{6,5}&\sqrt{\frac{J}{4J+2}}\xi_{6,5}\\\
\sqrt{\frac{J+1}{4J+2}}\xi_{6,5}&E_{5}({\text{b}_{\mathcal{F}_{1},J}})&0\\\
\sqrt{\frac{J}{4J+2}}\xi_{6,5}&0&E_{5}({\text{b}_{\mathcal{F}_{3},J}})\end{array}\right)$
(2)

where $E_{6}(\text{A}_{f,J})$, $E_{5}({\text{b}_{\mathcal{F}_{1},J}})$ and
$E_{5}({\text{b}_{\mathcal{F}_{3},J}})$ are the energies of the unperturbed
$f$ levels of the A${}^{1}\Pi,v=6$ state and of the $\mathcal{F}_{1}$ and
$\mathcal{F}_{3}$ levels of the b${}^{3}\Sigma^{+},v=5$ state, respectively.
Due to the different $J$-dependence of the rotational energy of the
interacting $f$ levels, their interaction is localized to the lowest values of
$J$, almost like an accidental resonance, as shown by the green dots in Figure
3.

In Figure 4 the effect of the perturbation on the lowest rotational levels in
the A${}^{1}\Pi,v=6$ and b${}^{3}\Sigma^{+},v=5$ states is shown in more
detail. On the left side of this Figure, the calculated energies of the lowest
unperturbed levels are shown, up to $N=3$ in the b${}^{3}\Sigma^{+},v=5$
state. The perturbation lifts the degeneracy of the $J$ levels and on the
right side the resulting $J$ level structure is shown. The shift for the $J=1$
levels is the largest and up to 0.5 cm-1; the magnitude of the shift decreases
for higher values of $J$. The $J=0$ level of the b${}^{3}\Sigma^{+},v=5$ state
is the only one that is not affected, as there is no level in the A${}^{1}\Pi$
state to interact with.

Figure 5: Overview excitation spectra to the lowest rotational levels of the
perturbed A${}^{1}\Pi,v=6$ // b${}^{3}\Sigma^{+},v=5$ system, recorded via
(1+1′) REMPI. The spectra originate from single rotational levels in the
a${}^{3}\Pi,v=5$ state at energy $E_{\text{a}}$ and that are indicated on the
right. The spectra are plotted such that the energy level structure in the
excited state, shown as grey bars near the bottom, can be directly recognized.

Overview excitation spectra from selected rotational levels in the
a${}^{3}\Pi,v=5$ state to the lowest rotational levels of the perturbed
A${}^{1}\Pi,v=6$ // b${}^{3}\Sigma^{+},v=5$ system are shown in Figure 5.
These spectra are recorded via (1+1′) REMPI, using a time delay between the
excitation and ionization laser of less than 10 ns, and the spectral
resolution is limited by the bandwidth of the laser. Using the known energies
of the rotational levels in the a${}^{3}\Pi,v=5$ state, the spectra are
plotted such that the horizontal axis represents the absolute energies of the
perturbed levels. The fourteen energy levels given on the right in Figure 4
are indicated as grey bars above the spectra in Figure 5. The filled fraction
of the grey bars corresponds to the fraction of triplet character of the
corresponding $J$ levels.

Table 1: Observed A^1Π, v=6 // b^3Σ^+, v=5 ← a^3Π, v=5 transition frequencies, difference between the observed and calculated frequencies (all given in cm^-1), spectroscopic labels of the levels in the upper (primed) and lower (unprimed) state and the fraction of triplet (f_b5) character of the upper level. |  |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  |  |  |
$\tilde{\nu}_{\text{exp}}$ | $\tilde{\nu}_{\text{exp}}-\tilde{\nu}_{\text{calc}}$ | $f_{\text{b}5}$ | $J^{\prime}$ | $F^{\prime}$ | $p^{\prime}$ | $n^{\prime}$ | $\Omega$ | $J$ | $F$ | $p$ | $n$
$17236.4122$ | $0.0002$ | $0.961$ | 1 | 2 | $+$ | 1 | 1 | 1 | 3 | $-$ | 1
$17236.4198$ | $-0.0001$ | $0.961$ | 1 | 2 | $+$ | 1 | 1 | 1 | 2 | $-$ | 1
$17236.4369$ | $0.0000$ | $0.960$ | 1 | 1 | $+$ | 1 | 1 | 1 | 2 | $-$ | 1
$17236.4416$ | $-0.0002$ | $0.960$ | 1 | 1 | $+$ | 1 | 1 | 1 | 1 | $-$ | 7
$17236.5228$ | $-0.0000$ | $0.958$ | 1 | 2 | $+$ | 2 | 1 | 1 | 3 | $-$ | 1
$17236.5232$ | $-0.0001$ | $0.958$ | 1 | 3 | $+$ | 1 | 1 | 1 | 4 | $-$ | 1
$17236.5294$ | $0.0004$ | $0.958$ | 1 | 2 | $+$ | 2 | 1 | 1 | 2 | $-$ | 1
$17236.5339$ | $-0.0001$ | $0.958$ | 1 | 2 | $+$ | 2 | 1 | 1 | 1 | $-$ | 7
$17236.5407$ | $0.0001$ | $0.958$ | 1 | 3 | $+$ | 1 | 1 | 1 | 2 | $-$ | 1
$17236.6592$ | $-0.0002$ | $0.954$ | 1 | 3 | $+$ | 2 | 1 | 1 | 3 | $-$ | 1
$17236.6656$ | $-0.0000$ | $0.954$ | 1 | 3 | $+$ | 2 | 1 | 1 | 2 | $-$ | 1
$17236.6851$ | $0.0002$ | $0.954$ | 1 | 4 | $+$ | 1 | 1 | 1 | 4 | $-$ | 1
$17236.6946$ | $0.0002$ | $0.954$ | 1 | 4 | $+$ | 1 | 1 | 1 | 3 | $-$ | 1
$17237.4064$ | $0.0014$ | $0.774$ | 2 | 2 | $+$ | 5 | 1 | 2 | 3 | $-$ | 1
$17237.4097$ | $0.0004$ | $0.774$ | 2 | 2 | $+$ | 5 | 1 | 2 | 1 | $-$ | 8
$17237.4192$ | $0.0010$ | $0.779$ | 2 | 3 | $+$ | 5 | 1 | 2 | 4 | $-$ | 1
$17237.4257$ | $0.0010$ | $0.779$ | 2 | 3 | $+$ | 5 | 1 | 2 | 2 | $-$ | 1
$17237.4323$ | $0.0013$ | $0.820$ | 2 | 4 | $+$ | 3 | 1 | 2 | 5 | $-$ | 1
$17237.4323$ | $-0.0001$ | $0.833$ | 2 | 3 | $+$ | 6 | 1 | 2 | 4 | $-$ | 1
$17237.4400$ | $-0.0001$ | $0.820$ | 2 | 4 | $+$ | 3 | 1 | 2 | 3 | $-$ | 1
$17237.4400$ | $0.0012$ | $0.833$ | 2 | 3 | $+$ | 6 | 1 | 2 | 2 | $-$ | 1
$17237.4531$ | $-0.0006$ | $0.892$ | 2 | 4 | $+$ | 4 | 1 | 2 | 3 | $-$ | 1
$17237.4579$ | $0.0006$ | $0.894$ | 2 | 5 | $+$ | 1 | 1 | 2 | 4 | $-$ | 1
$17237.5197$ | $-0.0011$ | $0.986$ | 3 | 0 | $+$ | 2 | 1 | 2 | 1 | $-$ | 8
$17237.5635$ | $-0.0009$ | $0.979$ | 3 | 1 | $+$ | 6 | 1 | 2 | 1 | $-$ | 9
$17237.5782$ | $-0.0012$ | $0.969$ | 3 | 2 | $+$ | 7 | 1 | 2 | 2 | $-$ | 1
$17237.6067$ | $0.0001$ | $0.971$ | 3 | 2 | $+$ | 8 | 1 | 2 | 1 | $-$ | 9
$17237.6263$ | $0.0001$ | $0.962$ | 3 | 3 | $+$ | 7 | 1 | 2 | 3 | $-$ | 1
$17237.6566$ | $-0.0004$ | $0.970$ | 3 | 3 | $+$ | 8 | 1 | 2 | 2 | $-$ | 1
$17237.7118$ | $0.0002$ | $0.976$ | 3 | 4 | $+$ | 6 | 1 | 2 | 3 | $-$ | 1
$17237.7371$ | $-0.0001$ | $0.974$ | 3 | 5 | $+$ | 2 | 1 | 2 | 4 | $-$ | 1
$17237.7663$ | $0.0000$ | $0.985$ | 3 | 5 | $+$ | 3 | 1 | 2 | 4 | $-$ | 1
$17237.7899$ | $-0.0001$ | $0.985$ | 3 | 6 | $+$ | 1 | 1 | 2 | 5 | $-$ | 1
$17239.3590$ | $0.0003$ | $0.620$ | 1 | 4 | $+$ | 2 | 1 | 1 | 4 | $-$ | 1
$17239.4221$ | $-0.0003$ | $0.576$ | 1 | 2 | $+$ | 4 | 1 | 1 | 3 | $-$ | 1
$17239.4367$ | $-0.0004$ | $0.526$ | 1 | 1 | $+$ | 2 | 1 | 1 | 1 | $-$ | 7
$17240.2834$ | $0.0002$ | $0.515$ | 1 | 4 | $+$ | 7 | 1 | 1 | 4 | $-$ | 1
$17240.2864$ | $-0.0003$ | $0.519$ | 1 | 3 | $+$ | 9 | 1 | 1 | 3 | $-$ | 1
$17240.2926$ | $-0.0001$ | $0.515$ | 1 | 4 | $+$ | 7 | 1 | 1 | 3 | $-$ | 1
$17240.2926$ | $-0.0003$ | $0.519$ | 1 | 3 | $+$ | 9 | 1 | 1 | 2 | $-$ | 1
$17240.3177$ | $0.0004$ | $0.561$ | 1 | 3 | $+$ | 10 | 1 | 1 | 4 | $-$ | 1
$17240.3177$ | $-0.0006$ | $0.559$ | 1 | 2 | $+$ | 9 | 1 | 1 | 3 | $-$ | 1
$17240.3251$ | $0.0006$ | $0.559$ | 1 | 2 | $+$ | 9 | 1 | 1 | 2 | $-$ | 1
$17240.3293$ | $-0.0002$ | $0.559$ | 1 | 2 | $+$ | 9 | 1 | 1 | 1 | $-$ | 7
$17240.3345$ | $-0.0001$ | $0.561$ | 1 | 3 | $+$ | 10 | 1 | 1 | 2 | $-$ | 1
$17240.3466$ | $0.0005$ | $0.584$ | 1 | 1 | $+$ | 7 | 1 | 1 | 2 | $-$ | 1
$17240.3515$ | $0.0002$ | $0.592$ | 1 | 2 | $+$ | 10 | 1 | 1 | 3 | $-$ | 1
$17240.3515$ | $0.0004$ | $0.584$ | 1 | 1 | $+$ | 7 | 1 | 1 | 1 | $-$ | 7
$17240.3594$ | $0.0002$ | $0.592$ | 1 | 2 | $+$ | 10 | 1 | 1 | 2 | $-$ | 1
$17286.1879$ | $0.0001$ | $0.904$ | 1 | 2 | $-$ | 1 | 0 | 0 | 3 | $+$ | 1
$17286.2099$ | $-0.0000$ | $0.910$ | 1 | 3 | $-$ | 1 | 0 | 0 | 3 | $+$ | 1
$17286.2123$ | $-0.0002$ | $0.899$ | 1 | 1 | $-$ | 1 | 0 | 0 | 2 | $+$ | 1
$17286.2271$ | $0.0001$ | $0.905$ | 1 | 2 | $-$ | 2 | 0 | 0 | 2 | $+$ | 1
$17286.3087$ | $-0.0003$ | $0.896$ | 1 | 3 | $-$ | 2 | 0 | 0 | 2 | $+$ | 1
$17286.3166$ | $0.0001$ | $0.899$ | 1 | 4 | $-$ | 1 | 0 | 0 | 3 | $+$ | 1
$17288.7906$ | $0.0002$ | $0.106$ | 1 | 2 | $-$ | 6 | 0 | 0 | 3 | $+$ | 1
$17288.7945$ | $-0.0000$ | $0.107$ | 1 | 1 | $-$ | 4 | 0 | 0 | 2 | $+$ | 1
$17288.8150$ | $-0.0001$ | $0.115$ | 1 | 3 | $-$ | 7 | 0 | 0 | 2 | $+$ | 1
$17288.8197$ | $0.0000$ | $0.117$ | 1 | 4 | $-$ | 4 | 0 | 0 | 3 | $+$ | 1

Although the overall pattern of the calculated energy levels, shown above the
spectra, agrees with the observations, a detailed inspection shows that the
absolute frequency and splitting of the levels is slightly off. It appears
that both the absolute energies of the unperturbed levels of the
A${}^{1}\Pi,v=6$ and b${}^{3}\Sigma^{+},v=5$ states and the spin-orbit
interaction parameter $\xi_{6,5}$ need to be slightly adjusted. The energy
levels calculated with the improved parameters are shown underneath the
spectra in Figure 5. To find these improved parameters, the three transitions
to the perturbed levels with predominant singlet character (indicated in
black) are used. As the hyperfine structure of these transitions is within the
bandwidth of the laser, their center frequencies can be accurately determined.
The hyperfine structure of the perturbed b${}^{3}\Sigma^{+},v=5,N=0,J=1,+$
level results in three groups of hyperfine levels, spanning over 0.3 cm-1, and
will be similar to that of the corresponding level in the
b${}^{3}\Sigma^{+},v=0$ state.Doppelbauer  _et al._ (2021) From this we know
that the center of gravity of this level, i.e. the frequency of this level in
the absence of fine and hyperfine structure, is about 0.04 cm-1 above the
central group of hyperfine levels (also indicated in black). By defining the
center frequencies of the transitions to the predominantly singlet and triplet
levels in this way, a better agreement with the observations is obtained when
the separation between the unperturbed A${}^{1}\Pi,v=6$ and
b${}^{3}\Sigma^{+},v=5$ states is about 0.07 cm-1 larger than given by Barrow
et al.Barrow, Kopp, and Malmberg (1974), whereas the spin-orbit interaction
parameter is $\xi_{6,5}=1.12$ cm-1, slightly smaller than given by Barrow et
al.Barrow, Kopp, and Malmberg (1974) In addition, the absolute energies of
both the A${}^{1}\Pi,v=6$ and b${}^{3}\Sigma^{+},v=5$ states needed to be
shifted somewhat upwards.

The intensity of the lines in the spectra correlates with the amount of
triplet character of the levels for two reasons: First, the transition
intensity is determined by the amount of triplet character. Second, the
radiative lifetime of the levels in the upper state is longer, i.e. these are
detected more efficiently in the (1+1′) REMPI process, when the amount of
triplet character is larger. Singlet-triplet mixing is seen to be particularly
pronounced for the positive parity, $J=1$ levels belonging to the
A${}^{1}\Pi,v=6,J=1$ // b${}^{3}\Sigma^{+},v=5,N=2$ system. The parity of the
levels is well defined. The apparent violation of the parity selection rule in
some of the spectra is attributed to mixing of closely spaced, opposite parity
hyperfine levels in the a${}^{3}\Pi_{1},v=5$ state due to stray electric
fields in the ionization detection region. The modeling of the perturbation as
given here is seen to be sufficiently accurate for understanding the observed
transitions; additional splittings and energy level shifts are due to
hyperfine, spin-spin and spin-rotation interactions, as discussed in the next
Section.

##  IV Hyperfine Resolved Spectroscopy

Figure 6: Hyperfine resolved LIF spectrum from the a${}^{3}\Pi_{1},v=5,J=1,-$
level to the various positive parity hyperfine components of the perturbed
A${}^{1}\Pi,v=6$ // b${}^{3}\Sigma^{+},v=5$ system. Measured spectra pointing
upwards in black, simulated spectrum pointing downwards in green. The
spectroscopic labeling of the different transitions is given in Table
LABEL:tab:linelist.

The fine structure and hyperfine structure in the $v=0$ levels of the
A${}^{1}\Pi$ and b${}^{3}\Sigma^{+}$ states has recently been precisely
measured and analyzed.Truppe  _et al._ (2019); Doppelbauer  _et al._ (2021)
The model presented in those studies is used to describe the structure in the
excited vibrational levels as well, augmented with the perturbation between
the electronic states as described in the previous Section. Whereas the span
of the hyperfine structure in the A${}^{1}\Pi,v=0$ state is less than 0.02
cm-1, it is about 0.30 cm-1 in the b${}^{3}\Sigma^{+},v=0$ state. The latter
is much larger than the splitting due to spin-rotation interaction, resulting
in $J$ not being a good quantum number; only for the single $N=0$ level in the
b${}^{3}\Sigma^{+},v=0$ state, $J=1$ is well defined. When the hyperfine
interaction is strong compared to the spin-rotation interaction, it is useful
to introduce an intermediate angular momentum $\mathbf{G}$ = $\mathbf{I_{Al}}$
\+ $\mathbf{S}$, where $\mathbf{I_{Al}}$ is the spin of the aluminum nucleus.
This angular momentum $\mathbf{G}$ then couples to $\mathbf{N}$ and
$\mathbf{I_{F}}$, the spin of the fluorine nucleus, to the total angular
momentum $\mathbf{F}$. This approach has been followed and detailed in the
study on the b${}^{3}\Sigma^{+},v=0$ state.Doppelbauer  _et al._ (2021)

In the $v=5$ level of the b${}^{3}\Sigma^{+}$ state, the perturbation with the
A${}^{1}\Pi$ state results in a splitting of the $J$ levels belonging to the
same $N$ that can be considerably larger than the hyperfine splitting.
Interestingly, this makes $J$ a rather good quantum number for all the levels
belonging to $N=0-3$, explaining the overall good agreement between the
measured and calculated energies of the $J$ levels as shown in Figure 5.

The resolution in the spectra shown in Figure 5 is sufficient to partly
resolve the fine and hyperfine structure, as is best seen in the transitions
to the $J=1$, $\mathcal{F}_{1}$ level of $N=0$ in the b${}^{3}\Sigma^{+},v=5$
state, for which three (groups of) lines can be recognized. Higher resolution
spectra are recorded using a narrowband cw ring dye laser (< 1 MHz bandwidth,
$1-20$ mW) for excitation on selected transitions of the A${}^{1}\Pi,v=6$ //
b${}^{3}\Sigma^{+},v=5$ $\leftarrow$ a${}^{3}\Pi,v=5$ band, followed by far
off-resonant detection of the laser induced fluorescence back to the
X${}^{1}\Sigma^{+}$ state. On the low frequency side of the spectrum shown in
Figure 6, the hyperfine resolved transitions from the a${}^{3}\Pi_{1},v=5,J=1$
level to the b${}^{3}\Sigma^{+},v=5,N=0$ level are shown. The width of the
spectral lines is about 70 MHz, mainly determined by residual Doppler
broadening in the molecular beam. The hyperfine structure in the
a${}^{3}\Pi_{1},v=5,J=1$ level is known to about 20 kHz accuracy from earlier
measurements.Walter  _et al._ (2021) The perturbation with the
A${}^{1}\Pi,v=6$ state results mainly in an overall shift to lower energy of
the b${}^{3}\Sigma^{+},v=5,N=0$ hyperfine levels. This high resolution
spectrum is therefore well suited to constrain the fine structure and
hyperfine structure parameters in the b${}^{3}\Sigma^{+},v=5$ state.

Figure 7:  Hyperfine resolved LIF spectrum from the a${}^{3}\Pi_{0},v=5,J=0,+$
level to the various negative parity hyperfine components of the perturbed
A${}^{1}\Pi,v=6$ // b${}^{3}\Sigma^{+},v=5$ system. Measured spectra pointing
upwards in black, simulated spectrum pointing downwards in blue. The
spectroscopic labeling of the different transitions is given in Table
LABEL:tab:linelist.

The spectra shown in Figure 7 are recorded from the positive parity, $J=0$
level in the a${}^{3}\Pi_{0}$ state. This $J=0$ level has two hyperfine
components with a separation of only a few MHz (3 MHz for $v=0$, expected to
be similar for $v=5$)Truppe  _et al._ (2019) such that the spectra basically
directly reflect the level structure in the excited state. At low frequency,
the transition to the $N=1$ level of the b${}^{3}\Sigma^{+},v=5$ state is
shown, and about 2.5 cm-1 higher the transition to the negative parity, $J=1$
level of the A${}^{1}\Pi,v=6$ state is shown. The latter level has
predominantly singlet character, and the spectral lines are broadened due to
its short lifetime. The overall hyperfine structure of this $J=1$ level is
similar to what has been observed for the $J=1$ level in the A${}^{1}\Pi,v=0$
state, but it is expanded on the energy scale due to the perturbation by about
a factor two. The observed hyperfine splitting of this level is therefore well
suited to constrain the hyperfine structure parameters in the A${}^{1}\Pi,v=6$
state.

The two spectra shown on the high frequency side of Figure 6 are recorded from
the a${}^{3}\Pi_{1},v=5,J=1$ level, reaching the strongly mixed $N=2,J=1$,
positive parity levels of the A${}^{1}\Pi,v=6$ // b${}^{3}\Sigma^{+},v=5$
system. Two different sets of measurements are shown: the black spectrum is
recorded with the same residual Doppler width as the other spectra in this
Figure, whereas for the recording of the grey spectrum a 4 mm wide vertical
slit is inserted in the molecular beam in front of the LIF region, reducing
the Doppler width by about a factor three. This spectrum is particularly
sensitive to the spin-orbit interaction parameter $\xi_{6,5}$.

From the high resolution LIF spectra, a total of 59 resolved hyperfine
transitions are identified. The observed spectral lines are fitted with a
Voigt profile, and the center frequencies $\tilde{\nu}_{\text{exp}}$ are
determined to an accuracy of better than 10 MHz. The observed frequencies and
assignments are given in Table LABEL:tab:linelist together with the
differences from the calculated values $\tilde{\nu}_{\text{calc}}$. The $F$
quantum number, the parity and a counting integer $n$ (starting from $n=1$ for
the lowest energy level with a given $F$ and a given parity) suffice to
uniquely identify each level, but the quantum number $J$ is given as well.
Primed labels are for levels in the A${}^{1}\Pi,v=6$ //
b${}^{3}\Sigma^{+},v=5$ perturbed system, unprimed ones are for levels in the
a${}^{3}\Pi_{\Omega},v=5$ state. The fraction of triplet character of the
upper levels is given as $f_{\text{b}5}$. The standard deviation of the fit is
15 MHz.

Table 2: Spectroscopic parameters obtained from the fit for the A${}^{1}\Pi$ state (upper part) and for the b${}^{3}\Sigma^{+}$ state (lower part). All values are in MHz. The parameters determined in this study for the $v=6$ and $v=5$ levels of these states are compared to those determined earlier for the $v=0$ levels.Truppe  _et al._ (2019); Doppelbauer  _et al._ (2021) The spin-orbit interaction parameter $\xi_{6,5}$ is 1.1220 cm-1 (SD$\,\sqrt{Q}=0.0012$ cm-1). | A${}^{1}\Pi,v=0$ | A${}^{1}\Pi,v=6$
---|---|---
Parameter | Value111Truppe  _et al._ (2019) (2019) | $\textrm{SD}\sqrt{\textrm{Q}}$ | Value | $\textrm{SD}\sqrt{\textrm{Q}}$
$B_{\text{v}}$ | $16601.9$ | 0.3 | $15583.2120$ | fixed
$D$ | $0.0318$ | fixed | $0.0318$ | fixed
$q$ | $-2.94$ | 0.06 | $-2.94$ | fixed
$a$(Al) | $113$ | 5 | $105.3$ | 5.1
$eq_{0}Q$(Al) | $0.00$ | fixed | $0.00$ | fixed
$eq_{2}Q$(Al) | $40.00$ | fixed | $40.00$ | fixed
$a$(F) | $181$ | 5 | $152$ | 26
| b${}^{3}\Sigma^{+},v=0$ | b${}^{3}\Sigma^{+},v=5$
Parameter | Value222Doppelbauer  _et al._ (2021) (2021) | $\textrm{SD}\sqrt{\textrm{Q}}$ | Value | $\textrm{SD}\sqrt{\textrm{Q}}$
$B_{\text{v}}$ | $16772$ | 5 | $15782.8737$ | fixed
$\lambda$ | $-919$ | 18 | $-652$ | 38
$\gamma$ | $-9$ | 13 | $-2.5$ | 4.3
$b_{F}$(Al) | $1311$ | 3 | $1340.8$ | 1.6
$c$(Al) | $73$ | 18 | $86$ | 9
$eq_{0}Q$(Al) | $-62$ | 99 | $-49.6$ | fixed
$b_{F}$(F) | $870$ | 11 | $834.6$ | 6.9
$c$(F) | $305$ | 53 | $401$ | 35

In all cases, the simulated spectra plotted upside down underneath the
experimental spectra in Figure 6 and 7 reproduce the observed line positions
very well. The relative line intensities are seen to differ. In the simulation
it is assumed that all the hyperfine levels in the a${}^{3}\Pi,v=5$ state are
equally populated. However, this is not the case for the spectra recorded from
the a${}^{3}\Pi_{1},v=5,J=1$ level, as the bandwidth of the a${}^{3}\Pi$
$\leftarrow$ X${}^{1}\Sigma^{+}$ preparation laser is narrower than the span
of the hyperfine structure in this level. The relative intensities of the
lines in the recorded spectra are also influenced by the strongly varying
ratio of the homogeneous (Lorentzian) to the inhomogeneous (Gaussian)
contribution to the linewidth for the various spectral lines. Considering the
near-saturation conditions that these measurements have been made under, this
leads to varying widths of the (off-axis) velocity group that can contribute
to the LIF signal, and thereby to varying signal intensities.

The spectroscopic parameters of the A${}^{1}\Pi,v=6$ and the
b${}^{3}\Sigma^{+},v=5$ states obtained from the fit, are given in Table II
and are compared to the corresponding values for the $v=0$ levels determined
earlier.Truppe  _et al._ (2019); Doppelbauer  _et al._ (2021) The value found
for the spin-orbit interaction constant $\xi_{6,5}$ = 1.1220(12) cm-1, is
slightly smaller than the value of $\sqrt{2}H_{0}$ $\equiv$ $\xi_{6,5}$ =
1.16(1) cm-1 found by Barrow et al.Barrow, Kopp, and Malmberg (1974) Using
these values of the spectroscopic parameters, the unperturbed $J=1$ level of
the A${}^{1}\Pi,v=6$ state is seen to be 0.178(4) cm-1 below the $N=2$ level
of the b${}^{3}\Sigma^{+},v=5$ state. For the a${}^{3}\Pi,v=5$ state, the fine
structure and hyperfine structure parameters as given elsewhere are
taken.Walter  _et al._ (2021) Just as done there, the spin-orbit constant
$A_{5}$ and the spin-spin interaction constant $\lambda_{5}$ are kept fixed.
Given the limited number of rotational levels of the a${}^{3}\Pi,v=5$ state
that are included in the fit, the parameters $A_{5}$, $\lambda_{5}$ and the
$\Lambda$-doubling parameter $o_{5}$ are highly correlated. Only the parameter
$\tilde{A}_{5}=A_{5}-2\lambda_{5}+(o_{5}+p_{5}+q_{5})$ can be determined and
by varying $o_{5}$ in the fit, this is found as $\tilde{A}_{5}$ = 47.730(1)
cm-1. The rotational constant in the a${}^{3}\Pi,v=5$ state is also fitted and
found as $B_{5}=15954.1$(9.7) MHz. Both values are in good agreement with the
values of $\tilde{A}_{5}$ = 47.733(3) cm-1 and $B_{5}=15942.1$(6.3) MHz found
independently earlier.Walter  _et al._ (2021)

##  V Lifetime Measurements

Figure 8:  Measurement of the AlF+ ion signal as a function of the time delay
$\Delta T$ between the excitation and the ionization laser pulses. The
excitation schemes and the levels for which the lifetimes are measured are
indicated on the right.

The fit of the hyperfine resolved transitions yields the fraction of singlet
and triplet character for each of the hyperfine levels belonging to the
perturbed A${}^{1}\Pi,v=6$ // b${}^{3}\Sigma^{+},v=5$ system. To verify the
fraction of triplet character, $f_{\text{b5}}(F,n,\pm)$, of the hyperfine
levels, lifetime measurements are performed. The lifetime $\tau$ of a certain
hyperfine level characterized by the quantum number $F$, the counting integer
$n$ and the parity, is given by

$\frac{1}{\tau(F,n,\pm)}=\frac{1-f_{\text{b}5}(F,n,\pm)}{\tau_{\text{A}6}}+\frac{f_{\text{b}5}(F,n,\pm)}{\tau_{\text{b}5}}$
(3)

and largely differs for the different hyperfine levels.

The lifetime of the A${}^{1}\Pi,v=0$ state has been measured to be
$\tau_{\text{A}0}=1.90$(3) ns.Truppe  _et al._ (2019) The amount of singlet-
triplet mixing in this $v=0$ state is very small and its lifetime is the
radiative lifetimes of the unperturbed vibrational ground state. The found
value agrees very well with the theoretically predicted value of 1.89
nsLanghoff, Bauschlicher, and Taylor (1988). These calculations predict that
the lifetime of the bare A${}^{1}\Pi,v$ states increases with $v$, and that
$\tau_{\text{A}6}=2.05$ ns.

The lifetime of the b${}^{3}\Sigma^{+},v=0$ state is found to be
$\tau_{\text{b}0}=190$(2) ns.Doppelbauer  _et al._ (2021) The lifetimes of the
$v=1$ and $2$ states are measured in the context of this study using time-
delayed ionization to be $\tau_{\text{b}1}=189$(4) ns and
$\tau_{\text{b}2}=191$(8) ns. For these three states, the effect on the
lifetime due to the perturbation with the A${}^{1}\Pi,v+1$ state can be
neglected. In the absence of the perturbation with the A${}^{1}\Pi,v=6$ state,
the b${}^{3}\Sigma^{+},v=5$ state can therefore be assumed to have a lifetime
of $\tau_{\text{b}5}=190$ ns. Using these values for $\tau_{\text{A}6}$ and
$\tau_{\text{b}5}$, the lifetime as a function of $f_{\text{b}5}$ is shown as
the solid curve in Figure 10. The shaded area corresponds to an error bar of
10 % in both of these lifetime values.

Figure 9: LIF intensity as a function of the laser frequency measured in a
Lamb dip setup on the A${}^{1}\Pi,v=6$ //
b${}^{3}\Sigma^{+},v=5,N=1,J=1,F=4,-$ $\leftarrow$
a${}^{3}\Pi_{0},v=5,F=3,J=0,+$ transition at $\nu_{0}=17286.316$ cm-1.

The transitions shown on the high frequency side in Figure 7 reach levels that
only have around 10 % triplet character. The lifetime of these levels is only
slightly larger than $\tau_{\text{A}6}$ and the lifetime broadening of the
spectral lines is in the same range as the Doppler broadening. Voigt fits to
the observed lineshapes of the A${}^{1}\Pi,v=6,J=1,F=3,-$ $\leftarrow$
a${}^{3}\Pi_{0},v=5,J=0,F=2,+$ and A${}^{1}\Pi,v=6,J=1,F=4,-$ $\leftarrow$
a${}^{3}\Pi_{0},v=5,J=0,F=3,+$ transitions, marked in red in Figure 7, yield a
lifetime of 2.1(3) ns for the upper state hyperfine levels. Since the values
of $f_{\text{b}5}$ of both these levels differ by only 0.002, the lifetimes of
these levels can be assumed to be identical and are shown as a single data
point in Figure 10.

The transitions shown on the high frequency side in Figure 6 reach levels that
have around 50 % triplet character and the lifetimes of these levels are
around twice as large as $\tau_{\text{A}6}$. Lorentzian profiles convoluted
with an adapted Doppler profile (70 MHz wide Doppler multiplied by a 20 MHz
wide block function) are used to fit the spectral lineshapes of the two most
isolated lines in the Doppler reduced spectrum. These spectral lines belong to
the A${}^{1}\Pi,v=6$ // b${}^{3}\Sigma^{+},v=5,J=1,F=3,+$ $\leftarrow$
a${}^{3}\Pi_{1},v=5,J=1,F=2,-$ and A${}^{1}\Pi,v=6$ //
b${}^{3}\Sigma^{+},v=5,J=1,F=2,+$ $\leftarrow$ a${}^{3}\Pi_{1},v=5,J=1,F=2,-$
transitions, marked in red in Figure 6. The thus determined lifetimes are
3.9(5) ns and 4.2(5) ns, respectively.

The longest lifetimes are expected for the two hyperfine levels that belong to
the $N=1$, $J=0$ level of the b${}^{3}\Sigma^{+},v=5$ state, as there is no
$J=0$ level in the A${}^{1}\Pi,v=6$ state to interact with. As $J$ is a good,
but not a perfect quantum number, however, these hyperfine levels are
calculated to possess a fraction of about 0.01 of singlet character, already
sufficient to reduce their lifetimes relative to $\tau_{\text{b}5}$ by about a
factor of two, i.e. down to about 100 ns. Such lifetimes can be conveniently
measured via two-color, time-delayed ionization, using lasers with ns duration
pulses. It is difficult to exclusively excite the $J=0$ level with a pulsed
laser, as there are many hyperfine levels belonging to the same $N$ nearby.
Instead, the lifetime of the hyperfine levels belonging to $N=0$ of the
b${}^{3}\Sigma^{+},v=5$ state is measured, shown as the green curve in Figure
8. Using time-delayed ionization, the lifetime for these levels is found as
44(3) ns. Also indicated in Figure 8 is a triple resonance scheme that is used
to selectively excite to the $F=6$ level of $N=2$, $J=3$ in the
b${}^{3}\Sigma^{+},v=5$ state. This is one of the other levels predicted to
have a large triplet character, and the lifetime of that level is found as
80(5) ns.

Lifetimes in the intermediate, $5-40$ ns range can be measured via spectral
line broadening when Doppler-free techniques are employed. We perform Lamb dip
measurements in the molecular beam by retro-reflecting the cw laser beam and
recording the time-integrated LIF intensity generated by both the incoming and
the reflected beam. The spectrally isolated transition from the $F=3$, $J=0$,
$e$ level in the a${}^{3}\Pi_{0},v=5$ state to the $F=4$, $J=1$, $N=1$, $e$
level of the A${}^{1}\Pi,v=6$ // b${}^{3}\Sigma^{+},v=5$ system at around
17286.316 cm-1, the line marked in red in Figure 7, is used for this. The
calculated peak absorption cross section for this transition is equal to
$\sigma_{0}=7.6\cdot 10^{-12}$ cm2. The $F=4$ level in the
b${}^{3}\Sigma^{+},v=5$ state has a calculated fraction of triplet character
of 0.899, and a calculated lifetime of $\tau(F=4)$ = 18.5(3.1) ns. The error
bar of this lifetime is calculated assuming an uncertainty of 10 % of the
lifetimes of both $\tau_{\text{A}6}$ and $\tau_{\text{b}5}$.

Figure 10:  Measured lifetimes for six levels with a different amount of
b${}^{3}\Sigma^{+}$ character ($f_{\text{b}5}$). The solid curve represents
the predicted lifetime for $\tau_{\text{A}6}=2.05$ ns and
$\tau_{\text{b}5}=190$ ns; varying both values by up to 10 % gives the shaded
area.

In the experiment, we use about 3.5 mW of laser power at around 578 nm in a
0.4 cm $\times$ 1.0 cm area, thus we have a photon flux of $\Phi\approx
2.5\cdot 10^{16}$ cm-2 s-1 in both the incoming and the retro-reflected beam.
The interaction time $T$ is well defined by the 4 mm long interaction region
through which the molecules pass to be $T\approx 5.3$ $\mu$s. The product of
the photon flux and the interaction time is thus given as $\Phi T\approx
1.3\cdot 10^{11}$ cm-2. The resulting LIF signal as a function of the laser
frequency is shown in Figure 9. The observed spectral line is well described
by a Voigt profile (dashed red curve), with a Gaussian contribution due to
Doppler broadening of about 70 MHz, and with a Lorentzian Lamb dip at the
center. The inset shows the (inverted) Lamb dip data together with the
Lorentzian fit (blue curve). The fit reveals the relative depth at the center
frequency $\nu_{0}$ to be $D(\nu_{0})=0.26$(3), and the full width at half
maximum to be $\gamma_{\textrm{s}}=12.0$(2.4) MHz. The error bars are the
statistical errors from the fit.

In the Appendix, we derive how the values for the cross section and the
lifetime can be deduced from these two experimental observables in the
limiting case of a fully open two-level system. Using Equations (17) and (18),
we find that $\sigma_{0}=5.3\text{(2.0)}\cdot 10^{-12}$ cm2 and
$\tau=18.7$(3.7) ns. The uncertainty in the experimental value for
$\sigma_{0}$ is an estimate, based on the error bar in the value of
$D(\nu_{0})$ and the inhomogeneity of the laser intensity distribution within
the rectangular area. The uncertainty in the value of $\tau$ is solely
determined by the statistical uncertainty of the fit. Both the experimentally
determined value for the cross section and the value for the lifetime agree
very well with the calculated values, contributing to an overall consistent
picture of the singlet-triplet perturbation.

##  VI Discussion and Outlook

Figure 11:  Hyperfine levels of the perturbed A${}^{1}\Pi,v=6,J=1$ and
b${}^{3}\Sigma^{+},v=5,N=2$ system.

The spectroscopic study of the perturbed A${}^{1}\Pi,v=6$ //
b${}^{3}\Sigma^{+},v=5$ system of AlF presented here has resulted in an
accurate set of fine and hyperfine structure parameters of the coupled states
as well as in an accurate determination of their spin-orbit interaction
parameter. A pictorial presentation of the main result of this study, showing
the most relevant portion of the energy level diagram, is given in Figure 11.
A total of 34 positive parity and 6 negative parity hyperfine levels with a
strongly mixed singlet-triplet character are located in an interval of about 1
cm-1. On the left hand side in this Figure, the simulated hyperfine structure
of the A${}^{1}\Pi,v=6,J=1$ level is shown in the absence of spin-orbit
coupling. On the right hand side, the same is shown for the
b${}^{3}\Sigma^{+},v=5,N=2$ level. The actual energy levels, including the
spin-orbit coupling, are depicted in the center. The $F$ quantum number and
the parity of all the levels are given. The $J$ quantum number is less well
defined and the $J$-fraction is indicated via color coding: green for $J=1$,
blue for $J=2$ and violet for $J=3$. In the unperturbed A${}^{1}\Pi,v=6$
state, $J$ is perfectly defined, whereas in the unperturbed $N=2$ level of the
b${}^{3}\Sigma^{+},v=5$ state $J$ is not a useful quantum number. For the
perturbed levels, the $J$ quantum number is seen to be quite well defined,
with the fraction of a particular $J$ being always more than 50 % and in most
cases more than 80 %.

The gray wiggly lines crossing the energy levels indicate the fraction of
singlet (left) and triplet (right) character. This fraction correlates
directly with the lifetime of the levels, that range for these levels from 2.3
ns to 100 ns. It is seen in Figure 10 that the measured lifetimes agree very
well with these predicted values, confirming the overall model.

Most of these levels are clearly separated from each other and can spectrally
be selectively addressed. All these levels have a well defined and low value
of $F$ ($0\leq F\leq 6$), a well defined parity and, as a result of the
perturbation, a reasonably well defined value of $J$ (when $0\leq J\leq 3$).
The combination of these properties makes these levels ideally suited as
doorway states between the singlet and triplet manifold of electronic states
in AlF.

###### Acknowledgements.

We thank Marco De Pas, Uwe Hoppe, Sebastian Kray and Klaus-Peter Vogelgesang
for excellent technical support. N. W. acknowledges support by the
International Max Planck Research School for Elementary Processes in Physical
Chemistry. This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (CoMoFun, Grant Agreement No. 949119).

##  VII Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Author Contributions

N. W. carried out the data acquisition, analyzed and visualized the data and
wrote the manuscript. J. S, S. T and H.C.S. maintained the ring dye laser
system, B. S. analyzed the data, G. M. supervised and conceptualized the
project and edited the manuscript.

##  Appendix A Lamb dip experiments

In a Lamb dip experiment, a dip with a Lorentzian profile is observed,
centered on top of a Doppler broadened line. The depth of the dip and the
width of the Lorentzian profile contain information on the peak absorption
cross section of the transition and on the lifetime of the upper level. The
functional dependence of the depth and width of the Lamb dip on the cross
section and lifetime can be found in textbooks for a fully closed two-level
system.Demtröder (2015) Here the equations for a fully open two-level system
are derived.

###  A.1 Rate equation modelling of a general two-level system

We start with a certain number of molecules $N_{a}(t=0)=N^{0}_{a}$ at time
$t=0$ in level $a$. This lower level is connected with a laser to an upper
level $b$, that is originally empty, i.e. $N_{b}(t=0)=0$. The energy
separation between the levels is given by $E_{b}-E_{a}$ = $h\,\nu_{0}$, with
the Planck’s constant $h$. Resonant emission from the upper level to the lower
level occurs with rate $\Gamma_{\text{r}}$ (in s-1). The total loss rate from
the upper level, including $\Gamma_{\text{r}}$, is given by $\Gamma$ (in s-1),
which can be expressed as $\Gamma=1/\tau$, where $\tau$ is the lifetime of the
upper level. In the absence of any other broadening mechanisms, the frequency
dependent absorption cross section $\sigma(\nu)$ (in cm2) for the transition
from level $a$ to level $b$ has a Lorentzian lineshape with a full width of
half maximum given by $\gamma$ and can be written as

$\sigma(\nu)=\sigma_{0}\,\frac{(\gamma/2)^{2}}{(\nu-\nu_{0})^{2}+(\gamma/2)^{2}}\equiv\sigma_{0}\,\mathcal{L}(\nu_{0},\gamma)$
(4)

where $\sigma_{0}$ is the peak absorption cross section and
$\mathcal{L}(\nu_{0},\gamma)$ is a normalized Lorentzian profile (peak
intensity normalized to one) centered around frequency $\nu_{0}$ and with a
full width at half maximum $\gamma=\Gamma/(2\pi)=1/(2\pi\tau)$. When this two-
level system interacts with a laser at frequency $\nu$ and with flux $\Phi$
(photons cm-2 s-1), the rate equations for this system can be written as:

$\left(\begin{array}[]{c}\frac{\textrm{d}N_{a}}{\textrm{d}t}\\\\[2.84544pt]
\frac{\textrm{d}N_{b}}{\textrm{d}t}\\\
\end{array}\right)=\left(\begin{array}[]{rr}-\sigma(\nu)\Phi&\sigma(\nu)\Phi+\Gamma_{\text{r}}\\\\[2.84544pt]
\sigma(\nu)\Phi&-\sigma(\nu)\Phi-\Gamma\\\\[2.84544pt]
\end{array}\right)\left(\begin{array}[]{c}N_{a}\\\\[2.84544pt] N_{b}\\\
\end{array}\right)$ (5)

The Eigenvalues $\lambda_{\pm}$ of this $2\times 2$ matrix are given by:

$\displaystyle\lambda_{\pm}=-(\sigma(\nu)$ $\displaystyle\Phi+\Gamma/2)$ (6)
$\displaystyle\pm$
$\displaystyle\sqrt{(\sigma(\nu)\Phi)^{2}+(\Gamma/2)^{2}+\sigma(\nu)\Phi\Gamma_{\text{r}}}$

The number of molecules in level $b$ at a given time can be expressed as:

$N_{b}(t)=N^{0}_{a}\frac{(\sigma(\nu)\Phi+\lambda_{-})(\sigma(\nu)\Phi+\lambda_{+})}{(\sigma(\nu)\Phi+\Gamma_{\text{r}})(\lambda_{-}-\lambda_{+})}\left(e^{\lambda_{+}t}-e^{\lambda_{-}t}\right)$
(7)

The number of photons, $n_{\textrm{fl}}(\nu,T)$, that are emitted via
spontaneous fluorescence from level $b$ when the laser is interacting with the
molecules for a time duration $T$ is given by

$n_{\textrm{fl}}(\nu,T)\propto\int_{0}^{T}\Gamma N_{b}(t)\,\textrm{d}\,t.$ (8)

###  A.2 Lamb dips in an open two-level system

In an open two-level system, the relation $\Gamma_{\text{r}}\ll\Gamma$ holds.
In the situation that $\sigma_{0}\Phi\ll\Gamma$ we can approximate
$\lambda_{+}\approx-\sigma(\nu)\Phi$ and $\lambda_{-}\approx-\Gamma$. The
number of fluorescence photons, $n_{\textrm{fl}}(\nu,T)$, is given by

$n_{\textrm{fl}}(\nu,T)\propto
N^{0}_{a}\sigma(\nu)\Phi\int_{0}^{T}\left(e^{-\sigma(\nu)\Phi t}-e^{-\Gamma
t}\right)\,\textrm{d}\,t.$ (9)

When the relaxation is fast, the second term under the integral can be
neglected, and we find

$n_{\textrm{fl}}(\nu,T)\propto N^{0}_{a}(1-e^{-\sigma(\nu)\Phi
T})=N^{0}_{a}(1-e^{-\sigma_{0}\Phi T\mathcal{L}(\nu_{0},\gamma)}).$ (10)

Using that we can write

$\frac{1-e^{-\sigma(\nu)\Phi T}}{1-e^{-\sigma_{0}\Phi
T}}=\frac{e^{\sigma_{0}\Phi T/2}(e^{\sigma(\nu)\Phi T/2}-e^{-\sigma(\nu)\Phi
T/2})}{e^{\sigma(\nu)\Phi T/2}(e^{\sigma_{0}\Phi T/2}-e^{-\sigma_{0}\Phi
T/2})}.$ (11)

It is seen from a series expansion, including all terms linear and quadratic
in $\sigma(\nu)\Phi T$ and $\sigma_{0}\Phi T$, that the expression for
$n_{\textrm{fl}}(\nu,T)$ can be approximated by

$n_{\textrm{fl}}(\nu,T)\propto N^{0}_{a}(1-e^{-\sigma_{0}\Phi
T})\mathcal{L}(\nu_{0},\gamma_{\textrm{fl}})$ (12)

with

$\gamma_{\textrm{fl}}=\gamma\sqrt{1+\frac{1}{2}\sigma_{0}\Phi
T+\frac{1}{8}(\sigma_{0}\Phi T)^{2}}.$ (13)

This is a broadened Lorentzian profile that is a very good approximation for
$n_{\textrm{fl}}(\nu,T)$ given by Equation (10) as long as $\sigma_{0}\Phi
T<2$.

In a typical Lamb dip experiment, there is inhomogeneous broadening as well as
saturation broadening of the Lorentzian line-profile, and the absorption line
is described by a convolution of these profiles. When we retroreflect a
mononchromatic laser beam with flux $\Phi$, i.e. when we have two counter-
propagating laser beams, each with flux $\Phi$, the total number of
spontaneous fluorescence photons that is emitted when the laser beams interact
with different velocity-groups of molecules, is given by two times the value
of $n_{\textrm{fl}}(\nu,T)$ given in Equation (12). When both laser beams
interact with the same velocity-group of molecules, then this results in a
total number of spontaneous fluorescence photons given by the value of
$n_{\textrm{fl}}(\nu_{0},T)$ for the doubled laser flux, $2\Phi$, which is
lower. This causes the appearance of the Lamb dip. In an open two-level
system, the relative depth, $D(\nu_{0})$, of the Lamb dip is thus given by

$D(\nu_{0})=1-\frac{(1-e^{-2\sigma_{0}\Phi T})}{2(1-e^{-\sigma_{0}\Phi
T})}=\frac{1}{2}(1-e^{-\sigma_{0}\Phi T})$ (14)

The shape is Lorentzian, saturation broadened due to the excitation rate
$2\sigma_{0}\Phi$, and the full spectral profile of the Lamb dip is given by

$D(\nu)=\frac{(1-e^{-2\sigma_{0}\Phi
T\mathcal{L}(\nu_{0},\gamma)})}{2(1+e^{-\sigma_{0}\Phi T})}\approx
D(\nu_{0})\mathcal{L}(\nu_{0},\gamma_{\textrm{s}})$ (15)

with

$\gamma_{\textrm{s}}=\gamma\sqrt{1+\sigma_{0}\Phi T+\frac{1}{2}(\sigma_{0}\Phi
T)^{2}}.$ (16)

The only experimental observables in a Lamb dip experiment are the relative
depth of the Lamb dip ($0<D(\nu_{0})<1/2$) and its full width at half maximum
$\gamma_{\textrm{s}}$. In an open two-level system, the absorption cross
section $\sigma_{0}$ is given by

$\sigma_{0}=\frac{1}{\Phi
T}\textrm{ln}\left(\frac{1}{1-2D(\nu_{0})}\right)\,\,\stackrel{{\scriptstyle
D(\nu_{0})\ll 1}}{{\approx}}\,\,\frac{2D(\nu_{0})}{\Phi T}$ (17)

and the lifetime $\tau$ is given by

$\displaystyle\tau=$
$\displaystyle\frac{1}{2\pi\gamma_{\textrm{s}}}\sqrt{1+\textrm{ln}\left(\frac{1}{1-2D(\nu_{0})}\right)+\frac{1}{2}\left(\textrm{ln}\left(\frac{1}{1-2D(\nu_{0})}\right)\right)^{2}}$
(18) $\displaystyle\stackrel{{\scriptstyle D(\nu_{0})\ll
1}}{{\approx}}\,\,\frac{1+D(\nu_{0})}{2\pi\gamma_{\textrm{s}}}.$

## References

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  * Barrow, Kopp, and Malmberg (1974) R. F. Barrow, I. Kopp, and C. Malmberg, “The electronic spectrum of gaseous AIF,” Physica Scripta 10, 86–102 (1974).
  * Kopp, Lindgren, and Malmberg (1976) I. Kopp, B. Lindgren, and C. Malmberg, “Rotational analysis of the a–X transition of AlF,” Physica Scripta 14, 170–174 (1976).
  * Rosenwaks, Steele, and Broida (1976) S. Rosenwaks, R. Steele, and H. Broida, “Observation of a${}^{3}\Pi$–X${}^{1}\Sigma^{+}$ intercombination emission in AlF,” Chemical Physics Letters 38, 121–124 (1976).
  * Di Rosa, M. D. (2004) Di Rosa, M. D., “Laser-cooling molecules - concept, candidates, and supporting hyperfine-resolved measurements of rotational lines in the A–X(0,0) band of CaH,” Eur. Phys. J. D 31, 395–402 (2004).
  * Wells and Lane (2011) N. Wells and I. C. Lane, “Electronic states and spin-forbidden cooling transitions of AlH and AlF,” Phys. Chem. Chem. Phys. 13, 19018–19025 (2011).
  * Truppe  _et al._ (2019) S. Truppe, S. Marx, S. Kray, M. Doppelbauer, S. Hofsäss, H. C. Schewe, N. Walter, J. Pérez-Ríos, B. G. Sartakov, and G. Meijer, “Spectroscopic characterization of aluminum monofluoride with relevance to laser cooling and trapping,” Phys. Rev. A 100, 052513 (2019).
  * Doppelbauer  _et al._ (2021) M. Doppelbauer, N. Walter, S. Hofsäss, S. Marx, H. C. Schewe, S. Kray, J. Pérez-Ríos, B. G. Sartakov, S. Truppe, and G. Meijer, “Characterisation of the b${}^{3}\Sigma^{+},v=0$ state and its interaction with the A${}^{1}\Pi$ state in aluminium monofluoride,” Molecular Physics 119, e1810351 (2021).
  * Walter  _et al._ (2021) N. Walter, M. Doppelbauer, S. Marx, J. Seifert, X. Liu, J. P. Ríos, B. Sartakov, S. Truppe, and G. Meijer, “Spectroscopic characterization of the $\textrm{a}^{\textrm{3}}{\Pi}$ state of aluminum monofluoride,” (2021), arXiv:2112.09498 .
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  * Blokland  _et al._ (2011) J. H. Blokland, J. Riedel, S. Putzke, B. G. Sartakov, G. C. Groenenboom, and G. Meijer, “Producing translationally cold, ground-state CO molecules,” The Journal of Chemical Physics 135, 114201 (2011).
  * Kovács (1969) I. Kovács,  _Rotational Structure in the Spectra of Diatomic Molecules_ (American Elsevier Publishing Company, 1969).
  * Langhoff, Bauschlicher, and Taylor (1988) S. R. Langhoff, C. W. Bauschlicher, and P. R. Taylor, “Theoretical studies of AlF, AlCl, and AlBr,” The Journal of Chemical Physics 88, 5715–5725 (1988).
  * Demtröder (2015) W. Demtröder,  _Laser Spectroscopy 2_ (Springer, Berlin, Heidelberg, 2015).


# Building Expansion for Generalizations of Viana Maps

V. Horita, N. Muniz, and O. Sester Work partially supporte by FAPESP
(2014/21815-5 and 2019/10269-3).

###### Abstract

In the seminal paper [15], Viana built examples of maps presenting two
positive Lyapunov exponents exploring skew-products of a (uniformly) expanding
map and a quadratic map (order 2 critical point) perturbed by some level of
_noise_. Here we extend that construction replacing the quadratic underlying
dynamics by maps with a more degenerated critical point.

††2020 Mathematics Subject Classification. Primary 37D25, 37C05.††Key words.
Nonuniformly hyperbolic systems, Viana maps, Lyapunov exponents .

##  1 Introduction

In dimension one, much has been studied about the richness of chaotic behavior
(positive Lyapunov exponent) present in families at a parameter, see Jakobson
[10], Benedicks and Carleson [3], Graczyk and Swiatek [7], and Koslovski,
Shen, and van Strien [11]. On the other hand, this phenomenon does not occur
persistently since hyperbolicity is open and dense for one dimensional
$C^{k}$-maps.

In higher dimension, Viana [15] brought us an impressive account of the
existence of multidimensional nonhyperbolic behaviour in a persistent way. His
work introduced what now is referred as Viana maps, i.e., $C^{3}$-perturbation
of a $C^{3}$-skew-product
$\varphi_{\alpha}\colon\mathbb{S}^{1}\times\mathbb{R}\to\mathbb{S}^{1}\times\mathbb{R}$
given by $\varphi_{\alpha}(\theta,x)=(\hat{g}(\theta),a(\theta)-x^{2})$, where
$\hat{g}\colon\mathbb{S}^{1}\to\mathbb{S}^{1}$ is an expanding map of the
circle and $a(\theta)=a_{0}+\alpha\phi(\theta)$. The map $\phi$ is a Morse
function and $a_{0}\in(1,2)$ is fixed such that $x=0$ is a pre-periodic point
for the map $h(x)=a_{0}-x^{2}$. There exists a compact interval
$I_{0}\subset(-2,2)$ such that $\varphi_{\alpha}(\mathbb{S}^{1}\times
I_{0})\subset\operatorname{int}(\mathbb{S}^{1}\times I_{0})$. Particular cases
that satisfy the above condition are $\phi(\theta)=\sin 2\pi\theta$ and and
$\hat{g}(\theta)=d\theta\mod 1$. Thus, $\varphi_{\alpha}$ has the form.

$\varphi_{\alpha}(\theta,x)=(d\theta\mod 1,\alpha\sin 2\pi\theta+h(x)).$

Viana showed that $C^{3}$-perturbation of $\varphi_{\alpha}$ has two positive
Lyapunov exponents, for $d\geq 16$. Later, this result was extended for $d\geq
2$ by Buzzi, Sester, and Tsuji in [4]. The factor $\alpha\sin 2\pi\theta$ can
be though as an $\alpha$-perturbation of the quadratic map $h$.

Skew products of this type, with a curve of neutral fixed points were
exploited by Gouzel [6]. More extensions of the result were obtained by
Schnellmann in [12] and [13] with Misiurewicz-Thurston quadratic maps as the
basis dynamics, Huang-Shen in [9] and Varandas [14] in a generalized context.
Moreover, Gao in [5] consider Benedicks-Carleson maps instead circle uniformly
maps $\hat{g}$ in the first factor to obtain two positive Lyapunov exponents.
All works mentioned deal with a quadratic critical point.

Here, we consider maps $h_{D}\colon\mathcal{M}\to\mathcal{M}$, $D\geq 2$ (to
be defined in a little while), where $\mathcal{M}=\mathbb{S}^{1}$ if $D$ is
odd and $\mathcal{M}=I_{0}\subset\mathbb{R}$ an interval if $D$ is even. The
map $h_{D}$ presents one critical point $\tilde{x}$ of order $D$, that is,
$h_{D}^{(j)}(\tilde{x})=0$, for all $1\leq j<D$, and
$h_{D}^{(D)}(\tilde{x})\neq 0$. Let
$\varphi_{\alpha,D}\colon\mathbb{S}^{1}\times\mathcal{M}\to\mathbb{S}^{1}\times\mathcal{M}$
be defined by

$\varphi_{\alpha,D}(\theta,x)=(d\theta\mod 1,\alpha\sin 2\pi\theta+h_{D}(x)).$

Let us be more specific on $h_{D}$. We consider intervals
$I^{\prime},I^{\prime\prime}\subset\mathcal{M}$ centered in $\tilde{x}$
depending on $D$, such that $I^{\prime\prime}$ is a proper sub-interval of
$I^{\prime}$. We begin defining the map for positive odd number $D\geq 3$. In
[8] are considered one-parameter families maps in $\mathbb{S}^{1}$ with one
cubic critical point and, for a positive measure set of parameter, the maps
are non-uniformly expanding. Here, we consider $C^{2D+2}$-maps
$h_{2D+1}\colon\mathbb{S}^{1}\to\mathbb{S}^{1}$, $D\geq 1$ positive integer,
defined by

$h_{2D+1}(x)=\left\\{\begin{array}[]{lll}2x\mod
1&\text{if}&x\in\mathbb{S}^{1}\setminus I^{\prime}\\\
A(x-1/2)^{2D+1}&\text{if}&x\in I^{\prime\prime}\end{array}\right.,$ (1)

where $A$ is a positive constant chosen such that the derivative of $h_{2D+1}$
at the extreme points of $I^{\prime\prime}$ is equal to $7/4$. In each
component of $I^{\prime}\setminus I^{\prime\prime}$ we suppose that the first
and second derivatives of $h_{2D+1}$ are monotone. Then, $\tilde{x}=1/2$ is
the unique critical point of $h_{2D+1}$.

Figure 1: Map with odd critical point

Now, we take $I^{\prime}=(-1,1)$ and we consider $h_{2D}\colon I_{0}\to
I_{0}$, $D\geq 1$ positive integer, a $C^{2D+1}$-map defined by

$h_{2D}(x)=\left\\{\begin{array}[]{lll}a_{0}-x^{2}&\text{ if }&x\in
I_{0}\setminus I^{\prime}\\\ a_{0}-Ax^{2D}&\text{ if }&x\in
I^{\prime\prime}\end{array}\right.,$ (2)

where, as before, $A$ is such that at the extreme points of $I^{\prime\prime}$
the modulus of derivative of $h_{D}$ is equal $7/4$, and in each component of
$I^{\prime}\setminus I^{\prime\prime}$ the first and second derivatives are
monotone. As defined, $h_{2D}$ has a unique (pre-fixed) critical point
$\tilde{x}=0$ of order $2D$.

######  Theorem A.

For every sufficiently small $\alpha>0$, and for all integer number $d\geq
16$, the map $\varphi_{\alpha,D}$, has two positive Lyapunov exponents at
Lebesgue every point $(\theta,x)\in\mathbb{S}^{1}\times\mathcal{M}$. Moreover,
the same holds for every map $\varphi$ sufficiently close to
$\varphi_{\alpha,D}$ in $C^{D}(\mathbb{S}^{1}\times\mathcal{M})$.

The proof of Theorem A follows closely the proof in [15] where it is assumed
$d\geq 16$. The general case $d\geq 2$ can be adapted from [4]. In this paper,
we deal only with the case where $d\geq 16$.

Let us mention that the presence of two positive Lyapunov exponents is an
indication of the existence of absolutely continuous invariant probability
measure. For Viana maps and extensions in the case of a quadratic critical
point, see Alves [1], Alves-Viana [2], and Gao [5].

##  2 Preliminary statements and results

We can assume that
$\varphi\colon\mathbb{S}^{1}\times\mathcal{M}\to\mathbb{S}^{1}\times\mathcal{M}$
has the form

$\varphi(\theta,x)=(g(\theta),f(\theta,x)),\quad\text{with
}\partial_{x}f(\theta,x)=0\text{ if and only if }x=\tilde{x},$ (3)

and we prove that the theorem holds as long as

$|\varphi-\varphi_{\alpha,D}|_{C^{D}}\leq\alpha\quad\text{ on
}\mathbb{S}^{1}\times\mathcal{M}.$

To overcome the simplifying hypothesis (3) and so conclude the proof of
Theorem A it is enough to follow exactly the same approach as in [15, Section
2.5].

###  2.1 Admissible curves

We say that a $C^{2}$-curve $\hat{X}$ is an _admissible curve_ if
$\hat{X}=\operatorname{graph}{X}$, with $X\colon\mathbb{S}^{1}\to\mathcal{M}$,
satisfies

$|X^{\prime}(\theta)|\leq\alpha\quad\text{ and
}\quad|X^{\prime\prime}(\theta)|\leq\alpha,\qquad\text{for all
}\theta\in\mathbb{S}^{1}.$

We suppose that $\alpha$ is small such that $\alpha<A^{-1}$.

Given $\omega\subset\mathbb{S}^{1}$, we write
$\hat{X}|_{\omega}=\operatorname{graph}(X|_{\omega})$.

Let $\tilde{\theta}_{0}\in\mathbb{S}^{1}$ be the fixed point of $g$ and denote
by $\tilde{\theta}_{1},\cdots,\tilde{\theta}_{d}=\tilde{\theta}_{0}$ its $d$
pre-images under $g$ ordered according to the orientation of $\mathbb{S}^{1}$.
We consider _Markov partitions_ $\mathcal{P}_{n}$ of $\mathbb{S}^{1}$ defined
by,

$\displaystyle\mathcal{P}_{1}$
$\displaystyle=\\{[\tilde{\theta}_{j-1},\tilde{\theta}_{j})\colon 1\leq j\leq
d\\},\text{ and}$ $\displaystyle\mathcal{P}_{n+1}$
$\displaystyle=\\{\text{connected components of
}g^{-1}(\omega)\colon\omega\in\mathcal{P}_{n}\\}.$

######  Lemma 2.1.

Let $X$ be an admissible curve. If $\omega\in\mathcal{P}_{n}$ then
$\varphi^{n}(\hat{X}|_{\omega})$ is also an admissible curve.

###### Proof.

The proof follows closely [15, Lemma 2.1]. Let $Y\colon\mathbb{S}^{1}\to I$ be
defined by $Y(g(\theta))=f(\theta,X(\theta))$,
$\theta\in\omega\in\mathcal{P}_{1}$. The estimates on the derivatives of $g$
are taken from [15], i.e., $|g^{\prime}|\geq 15$ and
$|g^{\prime\prime}|\leq\alpha$.

The derivatives of $f$ are $\alpha$ perturbations of those derivatives of
$\tilde{f}(\theta,x)=\alpha\sin 2\pi\theta+h_{D}(x)$. So,

$\displaystyle|\partial_{\theta}f|\leq\alpha|2\pi\cos 2\pi\theta|+\alpha\leq
8\alpha,$ $\displaystyle|\partial_{\theta\theta}f|\leq\alpha|4\pi^{2}\sin
2\pi\theta|+\alpha\leq 50\alpha,\text{ and }$ $\displaystyle|\partial_{\theta
x}f|\leq\alpha.$

Note that we have $|h_{D}^{\prime}(x)|\leq 7/4$ in $I^{\prime\prime}$ and
$7/4\leq|h_{D}^{\prime}(x)|\leq 2$ in $I^{\prime}\setminus I^{\prime\prime}$.
In $\mathcal{M}\setminus I^{\prime}$, $|h_{D}^{\prime}(x)|=2$ if $D$ is odd,
and $|h_{D}^{\prime}(x)|\leq 4$ if $D$ is even. Anyway,
$|h_{D}^{\prime}(x)|\leq 4$ for every $x\in\mathcal{M}$.

For the second derivative, in $\mathcal{M}\setminus I^{\prime}$, we have
$|h_{D}^{\prime\prime}(x)|=0$ if $D$ is odd, and
$|h_{D}^{\prime\prime}(x)|\leq 2$ if $D$ is even. We are assuming that in the
extreme points $x_{i}$, $i=1,2$, of $I^{\prime\prime}$ the derivative has
modulus $7/4$, that is to say

$7/4=|h_{D}^{\prime}(x_{i})|=DA|x_{i}-\tilde{x}|^{D-1}=DA\left(\frac{|I^{\prime\prime}|}{2}\right)^{D-1}.$
(4)

Then,

$|h_{D}^{\prime\prime}(x)|\leq D(D-1)A|x-\tilde{x}|^{D-2}\leq
D(D-1)A\left(\frac{|I^{\prime\prime}|}{2}\right)^{D-2}\leq\frac{7(D-1)}{4|I^{\prime\prime}|}.$

From the fact that $|h_{2D+1}^{\prime\prime}(x)|=0$ for every
$x\in\mathcal{M}\setminus I^{\prime}$, $|h_{2D}^{\prime\prime}(x)|=2$ for
every $x\in\mathcal{M}\setminus I^{\prime}=(-1,1)$, and $h_{D}^{\prime\prime}$
is monotone in each connected component of $I^{\prime}\setminus
I^{\prime\prime}$, we have
$|h_{D}^{\prime\prime}(x)|<7(D-1)/(4|I^{\prime\prime}|)$, for every $x$ in
$\mathcal{M}$.

The fact that $\partial_{x}\tilde{f}(\theta,x)=h_{D}^{\prime}(x)$ and
$\partial_{xx}\tilde{f}(\theta,x)=h_{D}^{\prime\prime}(x)$ implies, for all
$x\in\mathcal{M}$,

$|\partial_{x}f|\leq 4+\alpha\qquad\text{ and
}\quad|\partial_{xx}f|\leq\frac{7(D-1)}{4|I^{\prime\prime}|}+\alpha\leq
2\alpha^{-1},$

where the last inequality uses that $\alpha$ is small, and so we may assume
$\alpha<\left(7(D-1)/(4|I^{\prime\prime}|)\right)^{-1}$. Thus,

$\displaystyle|Y^{\prime}|$
$\displaystyle=\left|\frac{1}{g^{\prime}}(\partial_{\theta}f+\partial_{x}fX^{\prime})\right|\leq\frac{1}{15}(8\alpha+5\alpha)\leq\alpha\quad\text{
and}$ $\displaystyle|Y^{\prime\prime}|$
$\displaystyle=\left|\left(\frac{1}{g^{\prime}}\right)^{2}(\partial_{\theta\theta}f+2\partial_{\theta
x}fX^{\prime}+\partial_{xx}f(X^{\prime})^{2}+\partial_{x}fX^{\prime\prime}-Y^{\prime}g^{\prime\prime})\right|\leq$
$\displaystyle\leq\frac{1}{15^{2}}(50\alpha+2\alpha+\frac{7(D-1)}{4|I^{\prime\prime}|}\alpha^{2}+5\alpha+\alpha^{2})\leq\alpha.$

The lemma is proved. ∎

Next, we state a property of admissible curves.

######  Lemma 2.2.

Let $\hat{X}=\operatorname{graph}{X}$ be an admissible curve and denote
$\hat{X}(\theta)=(\theta,X(\theta))$,
$\hat{Z}=\varphi(\hat{X}(\theta))=(g(\theta),Z(\theta))$. Then, given any
interval $I\subset\mathcal{M}$, we have
$m(\\{\theta\in\mathbb{S}^{1}\colon\hat{Z}(\theta)\in\mathbb{S}^{1}\times
I\\})\leq 4|I|/\alpha+2\sqrt{|I|/\alpha}$.

###### Proof.

Analogous to [15, Lemma 2.2]. ∎

######  Corollary 2.3.

there is $C_{1}>0$ such that, given $\hat{X}_{0}=\operatorname{graph}{X_{0}}$
an admissible curve and $I\subset\mathcal{M}$ an interval with $|I|\leq\alpha$
we have

$m(\\{\theta\in\mathbb{S}^{1}\colon\hat{X_{j}}(\theta)\in\mathbb{S}^{1}\times
I\\})\leq C_{1}\sqrt{\frac{|I|}{\alpha}},\text{ for every }j\geq 1.$

###### Proof.

Analogous to [15, Corollary 2.3]. ∎

For $(\theta,x)\in\mathbb{S}^{1}\times\mathcal{M}$ and $j\geq 0$ we write
$(\theta_{j},x_{j})=\varphi^{j}(\theta,x)$. We consider constants $0<\eta\leq
1/3$ and $0<\kappa<1$, that will be made precise in a little while, both
depending only on $h_{D}$.

######  Lemma 2.4.

There are $\delta_{1}>0$ and $\sigma_{1}>1$ such that

  * a)

for every $\alpha>0$ small, there exists $N=N(\alpha)\geq 1$ such that

$\prod_{j=0}^{N-1}\left\lvert\partial_{x}f(\theta_{j},x_{j})\right\rvert\geq\left\lvert
x-\tilde{x}\right\rvert^{D-1}\alpha^{-1+\frac{\eta}{D-1}},$

whenever $\left\lvert x-\tilde{x}\right\rvert<2\sqrt[D]{\alpha}$;

  * b)

for each $(\theta,x)\in\mathbb{S}^{1}\times\mathcal{M}$ with
$\sqrt[D]{\alpha}\leq\left\lvert x-\tilde{x}\right\rvert<\delta_{1}$, there
exists $p(x)\leq N$ such that

$\prod_{j=0}^{p(x)-1}\left\lvert\partial_{x}f(\theta_{j},x_{j})\right\rvert\geq\frac{1}{\kappa}\sigma_{1}^{p(x)}.$

###### Proof.

The proof follows that of [15, Lemma 2.4] and we also denote $C$ any large
constant depending only on the map $h_{D}$.

If $D$ is odd (respectively, even), we consider an interval $J$ in
$\mathbb{S}^{1}$ (respectively, in $I_{0}$) centered at $0$ (respectively, at
the negative fixed point $q$ of $h_{D}$). We have $h(I^{\prime\prime})\subset
J$ (respectively, $h_{D}^{2}(I^{\prime\prime})\subset J$), if
$I^{\prime\prime}$ is sufficiently small.

First, we consider $x\in I^{\prime\prime}$ with
$|x-\tilde{x}|\geq\alpha^{\frac{1}{2(D-1)}}$, let $k_{0}=k_{0}(y)$ (uniformly
bounded) be the smallest positive integer $j$ such that $h_{D}^{j}(y)$ escape
from $J$. Recall that for $D$ odd we have $\tilde{x}=1/2$ (respectively, for
$D$ even we have $\tilde{x}=0$). Note that $h_{D}(1/2)=0$ (respectively,
$h_{D}^{2}(0)=q$) is the fixed point of $h_{D}$ and $h_{D}^{\prime}(0)=\rho=2$
(respectively, $h_{D}^{\prime}(q)=\rho\leq 4$). We write $\tilde{q}=0$ for $D$
odd (respectively, $\tilde{q}=q$ for $D$ even).

We fix $\rho_{1}<\rho<\rho_{2}$ with $\rho_{1}>\rho_{2}^{1-\eta/D}$. Let
$\delta_{0}=|J|/2$. We take $J$ sufficiently small such that for every
$|y-\tilde{q}|<\delta_{0}$, we have

$\rho_{1}<|\partial_{x}f(\varphi(\tau,y))|<\rho_{2}.$

Given $(\theta,x)\in\mathbb{S}^{1}\times\mathcal{M}$ we denote
$d_{i}=|x_{\ell+i}-\tilde{q}|$, $i\geq 0$, $\ell=1$ if $D$ is odd
(respectively, $\ell=2$ if $D$ is even). We suppose
$\delta_{1}=|I^{\prime\prime}|/2>0$ and $\alpha$ small enough so that
$|x-\tilde{x}|<\delta_{1}$ implies

$d_{0}\leq C|x-\tilde{x}|^{D}+C\alpha<\delta_{0}.$

Now, let $(\theta,x)$ and $i\geq 1$ be such that $|x-\tilde{x}|<\delta_{1}$
and $d_{0},\dots,d_{i-1}<\delta_{0}$. Thus, $d_{i}\leq\rho_{2}d_{i-1}+C\alpha$
and by induction

$d_{i}\leq\rho_{2}^{i}d_{0}+C\alpha(1+\rho_{2}+\dots+\rho_{2}^{i-1})\leq\rho_{2}^{i}(C\alpha+C|x-\tilde{x}|^{D}).$
(5)

Suppose first that $|x-\tilde{x}|<2\sqrt[D]{\alpha}$. Then,
$|x-\tilde{x}|^{D}<2^{D}\alpha$ and from (5), we get $\displaystyle
d_{i}\leq\rho_{2}^{i}C\alpha$.

Let $\tilde{N}=\tilde{N}(\alpha)\geq 1$ be the minimum integer such that
$\rho_{2}^{\tilde{N}}C\alpha\geq\delta_{0}$ and then define
$N=\ell+\tilde{N}$. Hence, $d_{i}=|x_{\ell+i}-\tilde{q}|<\delta_{0}$ for all
$0\leq i\leq N-1$ and

$\displaystyle\prod_{j=0}^{N-1}|\partial_{x}f(\theta_{j},x_{j})|$
$\displaystyle=\prod_{j=0}^{\ell-1}|\partial_{x}f(\theta_{j},x_{j})|\cdot\prod_{j=0}^{\tilde{N}-1}|\partial_{x}f(\theta_{\ell+j},x_{\ell+j}|\geq\frac{1}{C}|x-\tilde{x}|^{D-1}\rho_{1}^{\tilde{N}}$
$\displaystyle\geq\frac{1}{C}|x-\tilde{x}|^{D-1}\rho_{2}^{(1-\eta/D)\tilde{N}}\geq\frac{1}{C}|x-\tilde{x}|^{D-1}\alpha^{-1+\eta/D}$
$\displaystyle\geq|x-\tilde{x}|^{D-1}\alpha^{-1+\eta/(D-1)}.$

Part a) is proved.

Suppose now $|x-\tilde{x}|\geq\sqrt[D]{\alpha}$. Then (5) gives
$d_{i}\leq\rho_{2}^{i}C|x-\tilde{x}|^{D}$. Let $\tilde{p}(x)$ be the minimum
integer such that $\rho_{2}^{\tilde{p}(x)}C|x-\tilde{x}|^{D}\geq\delta_{0}$.
Define $p(x)=\ell+\tilde{p}(x)$. Then, as before,

$\displaystyle\prod_{j=0}^{p(x)-1}|\partial_{x}f(\theta_{j},x_{j})|$
$\displaystyle\geq\frac{1}{C}|x-\tilde{x}|^{D-1}\rho_{1}^{\tilde{p}(x)}\geq\frac{1}{C}\left(\frac{\rho_{1}}{\rho_{2}^{(D-1)/D}}\right)^{\tilde{p}(x)}$
$\displaystyle\geq\frac{1}{C}\left(\frac{\rho_{2}^{1-\eta/D}}{\rho_{2}^{(D-1)/D}}\right)^{\tilde{p}(x)}\geq\frac{1}{C}\rho_{2}^{(1/D-\eta/D)\tilde{p}(x)}$
$\displaystyle\geq\frac{1}{\kappa}\rho_{2}^{p(x)/(D+1)}.$

The last inequality follows from the fact that $p(x)\gg 1$ uniformly as long
as $\delta_{1}\ll\delta_{0}$. So, if we take $\sigma_{1}=\rho_{2}^{1/(D+1)}$
the result follows. The proof is complete. ∎

######  Lemma 2.5.

There exist $\sigma_{2}>1$ and $C_{2}>0$ such that

$\prod_{j=0}^{k-1}\left\lvert\partial_{x}f(\theta_{j},x_{j})\right\rvert\geq
C_{2}\sqrt[D]{\alpha^{D-1}}\;\sigma_{2}^{k},$

for every $(\theta,x)$ with $\left\lvert
x_{0}-\tilde{x}\right\rvert,\left\lvert
x_{1}-\tilde{x}\right\rvert,\dots,\left\lvert
x_{k-1}-\tilde{x}\right\rvert\geq\sqrt[D]{\alpha}$. If in addition
$\left\lvert x_{k}-\tilde{x}\right\rvert<\delta_{1}$, then

$\prod_{j=0}^{k-1}\left\lvert\partial_{x}f(\theta_{j},x_{j})\right\rvert\geq
C_{2}\sigma_{2}^{k}.$

###### Proof.

We consider $D\geq 2$ and keep the notations of previous lemma.

Let us recall that $\delta_{1}=|I^{\prime\prime}|/2$. From the hypothesis on
$h_{D}$, if $\sigma_{0}=7/4$ then

$|(h_{D}^{n})^{\prime}(y)|\geq\sigma_{0}^{n},$

whenever $|y-\tilde{x}|,\dots,|y_{n-1}-\tilde{x}|\geq\delta_{1}$. By
continuity, supposing $\alpha$ small enough and reducing $\sigma_{0}$ if
necessary, if $(\tau,y)\in\mathbb{S}^{1}\times\mathcal{M}$ with
$|y_{0}-\tilde{x}|,\cdots,|y_{n-1}-\tilde{x}|\geq\delta_{1}$ then

$\prod_{j=0}^{n-1}|\partial_{x}f(\tau_{j},y_{j})|\geq\sigma_{0}^{n}.$ (6)

Let $(\theta,x)$ be as in the statement and let $j_{1}<\cdots<j_{s}$ be the
values of $j\in\\{0,\cdots,k-1\\}$ for which $|x_{j}-\tilde{x}|<\delta_{1}$.
If $s=0$ the result follows immediately from (6). So, let us suppose $s>0$.
When $|x_{k}-\tilde{x}|<\delta_{1}$ we set $j_{s+1}=k$. Denoting
$p_{i}=p(x_{j_{i}})$, $i=1,\cdots,s$, by Lemma 2.4 we have

$\prod_{j=j_{i}}^{j_{i}+p_{i}-1}|\partial_{x}f(\theta_{j},x_{j})|\geq\frac{1}{\kappa}\sigma_{1}^{p_{i}},$
(7)

for all $i<s$. Moreover, if $j_{s}+p_{s}\leq k$ then (7) also holds for $i=s$,
this is the case if $|x_{k}-\tilde{x}|<\delta_{1}$, as the definition of
$p(x)$ implies $j_{i}+p_{i}<j_{i+1}$. It follows from (6) that

$\prod_{j=0}^{j_{i}}|\partial_{x}f(\theta_{j},x_{j})|\geq\sigma_{0}^{j_{1}}\quad\text{
and
}\prod_{j=j_{i}+p_{i}}^{j_{i+1}-1}|\partial_{x}f(\theta_{j},x_{j})|\geq\sigma_{0}^{j_{i+1}-j_{i}-p_{i}},$
(8)

for all $i<s$ and, again, the second inequality remains valid for $i=s$ when
$|x_{k}-\tilde{x}|<\delta_{1}$. Now, we take
$\sigma_{2}=\min\\{\sigma_{0},\sigma_{1}\\}$ and we get

$\prod_{j=0}^{k-1}|\partial_{x}f(\theta_{j},x_{j})|\geq\sigma_{0}^{j_{1}}\prod_{i=1}^{s}\sigma_{1}^{p_{i}}\sigma_{0}^{j_{i+1}-j_{i}-p_{i}}\geq\sigma_{2}^{k},$

whenever $|x_{k}-\tilde{x}|<\delta_{1}$. The second part of the lemma is
proved.

For the first part, note that, even if (7) and (8) are not valid for $i=s$, we
have

$\prod_{j=j_{s}}^{k-1}|\partial_{x}f(\theta_{j},x_{j})|\geq(DA-\alpha)|x_{j_{s}}-\tilde{x}|^{D-1}\sigma_{0}^{k-j_{s}-1}\geq
C_{2}\sqrt[D]{\alpha^{D-1}}\sigma_{0}^{k-j_{s}-1},$

as a consequence of (6). The lemma is proved. ∎

Consider now $\eta=\log\sigma_{2}/(4\log 32)\leq 1/3$. Let $M=M(\alpha)$ be
the maximum integer such that $32^{M}\alpha<1$. The fact that
$\rho\leq\sup|h_{D}^{\prime}|\leq 4$ implies $M<N$. For any $r\geq 0$, we
denote

$J(r)=\\{x\in\mathbb{R}\colon|x-\tilde{x}|<\sqrt[D]{\alpha}e^{-r}\\}.$

The next lemma is a similar result of [15, Lemma 2.6] and its proof follows
closely the ideas and techniques in that paper.

######  Lemma 2.6.

There exist constants $C_{3}>0$ and $\beta>0$ such that, given an admissible
curve $\hat{Y}_{0}=\operatorname{graph}(Y_{0})$ and any
$r\geq\frac{1}{D-1}\left(\frac{1}{D}-\frac{2\eta}{D-1}\right)\log\frac{1}{\alpha}$,

$m\left(\\{\theta\in\mathbb{S}^{1}\colon\hat{Y}_{M}(\theta)\in\mathbb{S}^{1}\times
J(r-2)\\}\right)\leq C_{3}e^{-5\beta r}.$

The proof relies on the following lemma.

######  Lemma 2.7.

There are $H_{1},H_{2}\subset\\{1,\dots,d\\}$ with $\\#H_{1},\\#H_{2}\geq
d/16]$ such that $|Z_{j_{1}}-Z_{j_{2}}|\geq\alpha/100$ for all
$\theta\in\mathbb{S}^{1}$, $j_{1}\in H_{1}$, and $j_{2}\in H_{2}$.

###### Proof.

See [15, Lemma 2.7]. ∎

###### Proof of Lemma 2.6.

Let
$\hat{Y}_{j}(\theta)=\varphi^{j}(\theta,Y_{0}(\theta))=(g^{j}(\theta),Y_{j}(\theta))$.
We use $C$ to represent any large positive constant depending only on $h_{D}$.

As $\hat{Y}_{0}=\operatorname{graph}(Y_{0})$ is an admissible curve, we have
$\operatorname{osc}(Y_{0})\leq\alpha$ and we claim
$\operatorname{osc}(Y_{j})\leq 4\operatorname{osc}(Y_{j-1})+2\alpha$, where
$\operatorname{osc}(Y_{j})=\sup Y_{j}-\inf Y_{j}$. Indeed, note that

$\displaystyle\hat{Y}_{j}(\theta)$
$\displaystyle=\varphi^{j}(\theta,Y_{0}(\theta))=\varphi(\varphi^{j-1}(\theta,Y_{0}(\theta)))$
$\displaystyle=\varphi(g^{j-1}(\theta),Y_{j-1}(\theta))=(g^{j}(\theta),f(g^{j-1}(\theta),Y_{j-1}(\theta))).$

Hence, $Y_{j}(\theta)=f(g^{j-1}(\theta),Y_{j-1}(\theta))$ and so

$\operatorname{osc}(Y_{j})\leq\operatorname{osc}(\alpha\sin
2\pi(g^{j-1}(\theta)))+4\operatorname{osc}(Y_{j-1})+\alpha\leq
4\operatorname{osc}(Y_{j-1})+2\alpha.$

Thus, $\operatorname{osc}(Y_{j})\leq 2\alpha 4^{j}\leq 2(32^{-M}4^{j})$ and
$\displaystyle\operatorname{osc}(Y_{M})<2\alpha^{3/5}<\sqrt{\alpha}$.

If $|Y_{M}(\theta)-\tilde{x}|\geq\sqrt[D]{\alpha}$ for all $\theta$ the lemma
follows since
$\\{\theta\in\mathbb{S}^{1}\colon\hat{Y}_{M}(\theta)\in\mathbb{S}^{1}\times
J((r-2)(D-1)^{2})\\}=\emptyset$. So, let us suppose that for some
$\tau\in\mathbb{S}^{1}$ we have $|Y_{M}(\tau)-\tilde{x}|<\sqrt[D]{\alpha}$ and
thus

$|Y_{M}(\theta)-\tilde{x}|<2\sqrt[D]{\alpha}\quad(<\delta_{1})\text{ for every
}\theta\in\mathbb{S}^{1}.$ (9)

Let us denote $\mathcal{O}=\\{h^{i}(\tilde{x})\colon i\geq 1\\}$ and
$\delta_{j}(\theta)=\operatorname{dist}(Y_{j}(\theta),\mathcal{O})$. Similar
argument of (5) yields

$\delta_{j+i}(\theta)\leq C4^{i}(\alpha+|Y_{j}(\theta)-\tilde{x}|^{D}).$

for all $\theta$, $0\leq j\leq M-1$, and $1\leq i\leq M-j$.

If $|Y_{j_{0}}(\tau)-\tilde{x}|\leq\sqrt[D]{\alpha}$ for some
$\tau\in\mathbb{S}^{1}$ and some $0\leq j_{0}\leq M-1$ then
$\delta_{M}(\tau)\leq
C4^{M-j_{0}}(\alpha+|Y_{j_{0}}(\tau)-\tilde{x}|^{D})<C4^{M}\alpha<C\sqrt{\alpha}$,
contradicting (9). Hence

$|Y_{j}(\theta)-\tilde{x}|>\sqrt[D]{\alpha}\text{ for any
}\theta\in\mathbb{S}^{1}\text{ and }0\leq j\leq M-1.$ (10)

In addition, note that the above reasoning together with (9) gives

$4^{M-j}|Y_{j}(\theta)-\tilde{x}|^{D}\geq\frac{1}{C}\text{ for all
}\theta\in\mathbb{S}^{1}\text{ and }0\leq j\leq M-1,$ (11)

here we are taking $C\gg 1/\operatorname{dist}(\tilde{x},\mathcal{O})$.

To derive uniform bound for the distortion of $\partial_{x}f$ on iterates of
$\hat{Y}_{0}$, note that given $0\leq j\leq M-1$,
$(\theta_{j},x_{j}),(\tau_{j},y_{j})\in\hat{Y}_{j}$, and $1\leq i\leq M-j$ we
have

$\displaystyle\left|\frac{\partial_{x}f^{i}(\theta_{j},x_{j})}{\partial_{x}f^{i}(\tau_{j},y_{j})}\right|$
$\displaystyle=\prod_{m=j}^{j+i-1}\left|\frac{\partial_{x}f(\theta_{m},x_{m})}{\partial_{x}f(\tau_{m},y_{m})}\right|$
(12)
$\displaystyle=\prod_{m=j}^{j+i-1}\left|1+\frac{\partial_{x}f(\theta_{m},x_{m})-\partial_{x}f(\tau_{m},y_{m})|}{\partial_{x}f(\tau_{m},y_{m})}\right|.$

We have

$\displaystyle|\partial_{x}f(\theta_{m},x_{m})-\partial_{x}f(\tau_{m},y_{m})|$
$\displaystyle\leq D(D-1)A\operatorname{osc}Y_{m}\leq D(D-1)A2\alpha 4^{m}.$

On one hand, if $y_{m}\in\mathcal{M}\setminus I^{\prime\prime}$ then
$|\partial_{x}f(\tau_{m},y_{m})|\geq 7/4$. Hence, assuming $\alpha$ small
enough,

$\frac{|\partial_{x}f(\theta_{m},x_{m})-\partial_{x}f(\tau_{m},y_{m})|}{|\partial_{x}f(\tau_{m},y_{m})|}\leq
C\alpha 4^{m}<\sqrt{\alpha}.$

On the other hand, if $y_{m}\in I^{\prime\prime}$ using (11) we get

$|\partial_{x}f(\tau_{m},y_{m})|=DA|y_{m}-\tilde{x}|^{D-1}\geq\frac{DA}{\left(\frac{1}{C}4^{M-m}\right)^{(D-1)/D}},$

which implies

$\frac{|\partial_{x}f(\theta_{m},x_{m})-\partial_{x}f(\tau_{m},y_{m})|}{|\partial_{x}f(\tau_{m},y_{m})|}\leq
C\alpha 4^{M}<\sqrt{\alpha}.$

Therefore, from (12), using the fact that
$M\approx\log\frac{1}{\tilde{\alpha}}$ and assuming $\alpha$ small enough, we
have

$\left|\frac{\partial_{x}f^{i}(\theta_{j},x_{j})}{\partial_{x}f^{i}(\tau_{j},y_{j})}\right|\leq\prod_{m=j}^{j+i-1}(1+\sqrt{\alpha})\leq(1+\sqrt{\alpha})^{2i}\leq
e^{2M\sqrt{\alpha}}\leq 2.$

We just proved that given any $0\leq j\leq M-1$ and $1\leq i\leq M-j$

$\left|\frac{\partial_{x}f^{i}(\theta_{j},x_{j})}{\partial_{x}f^{i}(\tau_{j},y_{j})}\right|\leq
2\text{ for every }(\theta_{j},x_{j}),(\tau_{j},y_{j})\in\hat{Y}_{j}.$

We fix an arbitrary $\hat{y}\in\hat{Y}_{0}$ and let
$\lambda_{j}=|\partial_{x}f^{M-j}(\varphi^{j}(\hat{y}))|$. From Lemma 2.5
(also recall (10) and (9)), we have $\lambda_{j}\geq C_{2}\sigma_{2}^{M-j}$
for $0\leq j\leq M-1$. On the other hand, the previous inequality gives

$\frac{1}{2}\frac{\lambda_{j}}{\lambda_{i+j}}\leq|\partial_{x}f^{i}(\theta_{j},x_{j})|\leq
2\frac{\lambda_{j}}{\lambda_{i+j}}\quad\text{ for all
}(\theta_{j},x_{j})\in\hat{Y}_{j}.$ (13)

Now, see [4, Proof of Proposition 5.2], assume

$\displaystyle r>\left(\frac{D-1}{D}+2\eta\right)\log\frac{1}{\alpha}.$

Then

$\frac{D-1}{D}\log\frac{1}{\alpha}<r\left(1+\frac{2D}{D-1}\eta\right)^{-1}<r\left(1-\frac{\eta}{D-1}\right).$

Thus,

$\frac{1}{\alpha^{\frac{D-1}{D}}}<e^{r(1-\eta/(D-1))}.$

From Corollary 2.3, for an admissible curve
$\hat{X_{0}}=\operatorname{graph}X_{0}$, we have, for every $j\geq 1$,

$\displaystyle m(\\{\theta\in\mathbb{S}^{1}\colon\hat{X_{j}}(\theta)\in$
$\displaystyle J(r-2)\\})\leq
C_{1}\sqrt{\frac{\sqrt[D]{\alpha}e^{-(r-2)}}{\alpha}}$
$\displaystyle=C_{1}\sqrt{\frac{e^{-(r-2)}}{\alpha^{\frac{D-1}{D}}}}\leq
e^{2}C_{1}e^{-\frac{\eta}{D-1}r}.$

Take $\beta=\eta/(5(D-1)$ in order to get the result in this case.

Now consider

$\displaystyle\frac{1}{D-1}\left(\frac{1}{D}-\frac{2\eta}{D-1}\right)\log\frac{1}{\alpha}\leq
r\leq\left(\frac{D-1}{D}+2\eta\right)\log\frac{1}{\alpha}.$

It is enough to proof the lemma for
$r_{0}=\displaystyle\frac{1}{D-1}\left(\frac{1}{D}-\frac{2\eta}{D-1}\right)\log\frac{1}{\alpha}$.
For the other $r^{\prime}$s the result follows by replacing $\beta$ by
$\displaystyle\beta\frac{1/D-2\eta/(D-1)}{(D-1)^{2}(1/D+2\eta/(D-1))}$.

We fix $K=400e^{2(D-1)^{2}}$ and consider positive integers
$t_{1}<t_{2}<\dots\leq M$ defined by $t_{1}=1$ and

$t_{i+1}=\min\\{s\colon t_{i}<s\leq M\text{ and }\lambda_{t_{i}}\geq
2K\lambda_{s}\\}\quad\text{ (if it exists)}.$

We set $\displaystyle k=k(r_{0})=\max\\{i\colon\lambda_{t_{i}}\geq
2Ke^{-r_{0}(D-1)^{2}}/\alpha^{\frac{D-1}{D}}\\}$.

We claim that there is a constant $\gamma_{1}>0$ such that
$k(r_{0})\geq\gamma_{1}r_{0}$. Indeed, we have $\lambda_{t_{i}}\leq
2K\lambda_{t_{i+1}-1}\leq 8K\lambda_{t_{i+1}}$ for all $i$ and so

$\lambda_{t_{k+1}}\geq C_{2}\sigma_{2}^{M-1}(8K)^{-k},$ (14)

and, by definition,

$\lambda_{t_{k+1}}\leq 2Ke^{-r_{0}(D-1)^{2}}/\alpha^{(D-1)/D}.$ (15)

From (14) and (15) we obtain

$C_{2}\sigma_{2}^{M-1}(8K)^{-k}\leq 2Ke^{-r_{0}(D-1)^{2}}/\alpha^{(D-1)/D},$

which implies

$k\log(8K)\geq
r_{0}(D-1)^{2}+M\log\sigma_{2}-\frac{D-1}{D}\log\frac{1}{\alpha}+C.$

It follows from the definition of $\eta$ and $M$ that $\displaystyle
M\log\sigma_{2}\geq 4\eta\log\frac{1}{\alpha}$. Hence

$\displaystyle k\log(8K)$ $\displaystyle\geq
r_{0}(D-1)^{2}-\left(\frac{D-1}{D}-4\eta\right)\log\frac{1}{\alpha}+C$
$\displaystyle\geq
r_{0}(D-1)^{2}-\left(\frac{\frac{D-1}{D}-4\eta}{\frac{D-1}{D}-2\eta}\right)(D-1)^{2}r_{0}+C$
$\displaystyle\geq\left(1-\frac{\frac{D-1}{D}-4\eta}{\frac{D-1}{D}-2\eta}\right)(D-1)^{2}r_{0}+C\geq
2(D-1)^{2}\eta r_{0},$

proving the claim for $\gamma_{1}=2(D-1)^{2}\eta/\log(8K)$.

As in the proof of [15, Lemma 2.6], for each
$\tilde{l}=(l_{1},\dots,l_{M})\in\\{1,\dots,d\\}^{M}$ we denote by
$\omega(\tilde{l})$ the only element $\omega\in\mathcal{P}_{M}$ satisfying
$g^{i}(\omega)\subset[\tilde{\theta}_{l_{i}-1},\tilde{\theta}_{l_{i}}),i=1,\dots,M$.
Given $1\leq j\leq M$ we let
$\hat{Y}_{j}=\operatorname{graph}(Y_{j}(\tilde{l}))=\varphi^{j}(\hat{Y}_{0}(\omega(\tilde{l})))$.
We say that $\tilde{l}$ and $\tilde{m}$ are _incompatible_ if

$|Y_{M}(\tilde{l},\theta)-Y_{M}(\tilde{m},\theta)|\geq
4e^{(2-r)(D-1)^{2}}\sqrt[D]{\alpha}\text{ for all }\theta\in\mathbb{S}^{1}.$

In this case $\hat{Y}_{M}(\tilde{l})$ and $\hat{Y}_{M}(\tilde{m})$ can not
both intersect a same vertical segment $\\{\theta\\}\times J((r-2)(D-1)^{2})$.
By Lemma 2.7 there are
$H_{1}^{\prime},H_{1}^{\prime\prime}\subset\\{1,\dots,d\\}$ with
$\\#H_{1}^{\prime},\\#H_{1}^{\prime\prime}\geq[d/16]$ such that given
$l_{1}^{\prime}\in H_{1}^{\prime}$ and $l_{1}^{\prime\prime}\in
H_{1}^{\prime\prime}$ we have

$|Y_{1}(l_{1}^{\prime},l_{2},\dots,l_{M},\theta)-Y_{1}(l_{1}^{\prime\prime},l_{2},\dots,l_{M},\theta)|\geq\frac{\alpha}{100}$

for all $\theta\in\mathbb{S}^{1}$ and $l_{2},\dots,l_{M}$. Then, by (13), the
definition of $K$, and the fact that $1\leq k(r_{0})$,

$\displaystyle|Y_{M}(l_{1}^{\prime},l_{2},\dots,l_{M},$
$\displaystyle\theta)-Y_{M}(l_{1}^{\prime\prime},l_{2}\dots,l_{M},\theta)|\geq\frac{\lambda_{1}}{2}\frac{\alpha}{100}\geq$
$\displaystyle\geq\frac{Ke^{-r_{0}(D-1)^{2}}}{\alpha^{\frac{D-1}{D}}}\frac{\alpha}{100}=4e^{(2-r_{0})(D-1)^{2}}\sqrt[D]{\alpha},\text{
for all }\theta\in\mathbb{S}^{1},$

that is, $(l_{1}^{\prime},l_{2},\dots,l_{M})$ and
$(l_{1}^{\prime\prime},l_{2},\dots,l_{M})$ are incompatible for every
$l_{2},\dots,l_{M}$. Furthermore, we claim that all pair of form
$(l_{1}^{\prime},l_{2},\dots
l_{t_{2}-1},l_{t_{2}}^{\prime},\dots,l_{M}^{\prime})$ and
$(l_{1}^{\prime\prime},l_{2},\dots
l_{t_{2}-1},l_{t_{2}}^{\prime\prime},\dots,l_{M}^{\prime\prime})$ are
incompatible. Indeed, note that, as a consequence of (13) and the definition
of $t_{2}$, we have

$\displaystyle|Y_{t_{2}}(l_{1}^{\prime},l_{2}\dots,l_{M},\theta)-Y_{t_{2}}(l_{1}^{\prime\prime},l_{2}\dots,l_{M},\theta)|\geq\frac{\lambda_{1}}{2\lambda_{t_{2}}}\frac{\alpha}{100}\geq
K\frac{\alpha}{100}=4e^{2(D-1)^{2}}\alpha,$

for all $\theta\in\mathbb{S}^{1}$.

On the other hand,

$\displaystyle|Y_{t_{2}}(l_{1}^{\prime},l_{2},\dots,$ $\displaystyle
l_{t_{2}-1},l_{t_{2}}^{\prime},\dots,l_{M}^{\prime})(\theta)-Y_{t_{2}}(l_{1}^{\prime},l_{2},\dots,l_{t_{2}-1},l_{t_{2}},\dots,l_{M})(\theta)|\leq$
$\displaystyle\leq\operatorname{osc}(\varphi(\hat{Y}_{t_{2}-1}(l_{1}^{\prime},l_{2},\dots,l_{t_{2}-1})))\leq
8\alpha,$

for every $\theta\in\mathbb{S}^{1}$, and similarly for
$Y_{t_{2}}(l_{1}^{\prime\prime},\dots)$. Thus, as $\lambda_{t_{2}}\leq
k(r_{0})$, we have

$\displaystyle|Y_{M}(l_{1}^{\prime},l_{2},$ $\displaystyle\dots
l_{t_{2}-1},l_{t_{2}}^{\prime},\dots,l_{M}^{\prime},\theta)-Y_{M}(l_{1}^{\prime\prime},l_{2},\dots
l_{t_{2}-1},l_{t_{2}}^{\prime\prime},\dots,l_{M}^{\prime\prime},\theta)|\geq$
$\displaystyle\geq\frac{\lambda_{t_{2}}}{2}(4e^{2(D-1)^{2}}-16)\alpha\geq\frac{Ke^{-r_{0}(D-1)^{2}}}{\alpha^{\frac{D-1}{D}}}(4e^{2(D-1)^{2}}-16)\alpha=$
$\displaystyle=400e^{(2-r_{0})(D-1)^{2}}(4e^{2(D-1)}-16)\sqrt[D]{\alpha}\geq
4e^{(2-r_{0})(D-1)^{2}}\sqrt[D]{\alpha},$

for all $\theta\in\mathbb{S}^{1}$, proving the claim. Proceeding as above for
each $t_{i}$, we get that each segment $\\{\theta\\}\times J((r-2)(D-1)^{2})$
intersects at most $d^{M-k(r_{0})}\cdot(d-[d/16])^{k(r_{0})}$ admissible
curves $\hat{Y}_{M}(\tilde{l})$, see the proof of [15, Lemma 2.6] for details.
The fact that $\displaystyle M\leq\text{const }\log\frac{1}{\alpha}$ implies
that

$|(g^{M})^{\prime}|\geq(d-\alpha)^{M}\geq\text{const}d^{M}.$

Therefore,

$\displaystyle
m\left(\\{\theta\colon\hat{Y}_{M}(\theta)\in\mathbb{S}^{1}\times
J((r-2)(D-1)^{2})\\}\right)$
$\displaystyle\leq\frac{d^{M}((d-[d/16])/d)^{k(r_{0})}}{(d-\alpha)^{M}}$
$\displaystyle\leq\text{const}\left(\frac{99}{100}\right)^{\gamma_{1}r_{0}}.$

The lemma follows by taking
$\displaystyle\beta=\frac{\gamma_{1}}{5}\log\left(\frac{100}{99}\right)$. ∎

##  3 Proof of Theorem A

To conclude the proof we continue following closely the proof of [15]. We
provide the details when the critical point is of order $D\geq 2$ (note that
taking $D=2$, the proof is the same). See [15], for more details.

Let us recall some ingredients and notations. For $n\geq 1$ be sufficiently
large and fixed, we define $m\geq 1$ by $m^{2}\leq n<(m+1)^{2}$ and take
$l=m-M$, where $M=M(\alpha)$ is as above. Note that $l\approx
m\approx\sqrt{n}$ as long as $n\gg\log\frac{1}{\alpha}$. Let $\hat{X}_{0}$ be
an arbitrary admissible curve. Given $1\leq\nu\leq n$ and
$\omega_{\nu+l}\in\mathcal{P}_{\nu+l}$, we set
$\gamma=\varphi^{\nu}(\hat{X}_{0}|\omega_{\nu+l})$. We say that $\nu$ is

  * •

a In-situation for $\theta\in\omega_{\nu+l}$ if
$\gamma\cap(\mathbb{S}^{1}\times J(0))\neq\emptyset$ but
$\gamma\cap(\mathbb{S}^{1}\times J(m))=\emptyset$;

  * •

a IIn-situation for $\theta\in\omega_{\nu+l}$ if
$\gamma\cap(\mathbb{S}^{1}\times J(m))\neq\emptyset$.

This setting is exactly the same of [15], let us remind some consequences for
completeness. It follows from Lemma 2.1 that $\gamma$ is the graph of a
function defined on $g^{\nu}(\omega_{\nu+l})\in\mathcal{P}_{l}$ and whose
derivative is bounded above by $\alpha$. So, the diameter in the $x$-direction
is bounded by $\alpha(d-\alpha)^{-l}\ll\sqrt[D]{\alpha}\;e^{-m}$. This means
that whenever $\nu$ is a IIn-situation for $\omega_{\nu+l}$ then
$\gamma\subset(\mathbb{S}^{1}\times J(m-1))$.

Let $B_{2}(n)=\\{\theta\in\mathbb{S}^{1}\colon\text{ some }1\leq\nu\leq
n\text{ is a II}_{n}\text{-situation for }\theta\\}$. From Corollary 2.3, we
obtain

$m(B_{2}(n))\leq
nC_{1}\sqrt{\frac{|J(m-1)|}{\alpha}}\leq\text{const}\;\alpha^{-(D-1)/(2D)}ne^{-m/2}\leq\text{const}\;e^{-\sqrt{n}/4}.$
(16)

Thus, from this point we focus on values $\theta$ having no IIn-situations in
$[1,n]$. Let $1\leq\nu_{1}<\dots<\nu_{s}\leq n$ be the In-situation of
$\theta$. The definition of $N$ implies $\nu_{i+1}\geq\nu_{i}+N$ for every
$i$; in particular $(s-1)N\leq n$.

For each $\nu=\nu_{i}$ we fix $r=r_{i}\in\\{1,\dots,m\\}$ minimum such that
$\gamma\cup(\mathbb{S}^{1}\times J(r))=\emptyset$. Then, by Lemma 2.4 and
definition of $J(r)$

$\displaystyle\prod_{\nu_{i}}^{\nu_{i}+N-1}\left|\partial_{x}f(\hat{X}_{j}(\theta))\right|$
$\displaystyle\geq(\sqrt[D]{\alpha}e^{-r_{i}})^{D-1}\alpha^{-1+\frac{\eta}{D-1}}$
$\displaystyle=e^{-(D-1)r_{i}}\alpha^{\frac{D-1}{D}-1+\frac{\eta}{D-1}}$
$\displaystyle=e^{-(D-1)r_{i}}\alpha^{-\frac{1}{D}+\frac{\eta}{D-1}},$

for each $1\leq i<s$.

Lemma 2.5 gives

$\prod_{1}^{\nu_{1}-1}\left|\partial_{x}f(\hat{X}_{j}(\theta))\right|\geq
C_{2}\sigma_{2}^{\nu_{1}-1}\quad\text{ and
}\quad\prod_{\nu_{i}+N}^{\nu_{i+1}-1}\left|\partial_{x}f(\hat{X}_{j}(\theta))\right|\geq
C_{2}\sigma_{2}^{\nu_{i+1}-\nu_{i}-N},$

for every $1\leq i<s$ and also

$\displaystyle\prod_{\nu_{s}}^{n}\left|\partial_{x}f(\hat{X}_{j}(\theta))\right|$
$\displaystyle\geq(DA-\alpha)|x_{\nu_{s}}-\tilde{x}|^{D-1}C_{2}(\sqrt[D]{\alpha}^{D-1}\sigma_{2}^{n-\nu_{s}}$
$\displaystyle\geq\text{const}\;(\sqrt[D]{\alpha})^{2(D-1)}e^{-(D-1)r_{s}}\sigma_{2}^{n-\nu_{s}}.$

Altogether this yields the following lower bound for
$\log\prod_{j=1}^{n}\left|\partial_{x}f(\hat{X}_{j}(\theta))\right|$:

$\displaystyle(n-(s-1)N)\log\sigma_{2}+\sum_{i-1}^{s}$
$\displaystyle\left(\left(\frac{1}{D}-\frac{\eta}{D-1}\right)\log\frac{1}{\alpha}-(D-1)r_{i}\right)$
$\displaystyle-s\;\text{const}-\frac{2D-1}{D}\log\frac{1}{\alpha}.$

We consider $G=\left\\{i\colon
r_{i}\geq\frac{1}{D-1}\left(\frac{1}{D}-\frac{2\eta}{D-1}\right)\log\frac{1}{\alpha}\right\\}$
(note that $G$ depends on $\theta$) and then

$\displaystyle\sum_{i-1}^{s}\left(\left(\frac{1}{D}-\frac{\eta}{D-1}\right)\log\frac{1}{\alpha}-(D-1)r_{i}\right)$
$\displaystyle\geq-\sum_{i\in G}(D-1)r_{i}+\eta s\log\frac{1}{\alpha}$
$\displaystyle\geq-(D-1)\sum_{i\in G}r_{i}+\gamma_{2}Ns,$

for some $\gamma_{2}>0$ independent of $\alpha$ or $n$ (because
$N\approx\text{ const }\log(1/\alpha))$. Thus, we have

$\displaystyle\log\prod_{j=1}^{n}\left|\partial_{x}f(\hat{X}_{j}(\theta))\right|$
$\displaystyle\geq(D+1)cn-(D-1)\sum_{i\in
G}r_{i}-s\;\text{const}-\frac{2D-1}{D}\log\frac{1}{\alpha}$ $\displaystyle\geq
Dcn-(D-1)\sum_{i\in G}r_{i},$

where $c=\frac{1}{D+1}\min\\{\gamma_{2},\log\sigma_{2}\\}$ and we use
$n\gg\log\frac{1}{\alpha}\approx N\gg 1$.

Now we introduce

$B_{1}(n)=\\{\theta\in\mathbb{S}^{1}\colon\sum_{i\in G}r_{i}\geq cn\\}$

and set $E_{n}=B_{1}(n)\cup B_{2}(n)$. Then

$\log\prod_{j=1}^{n}\left|\partial_{x}f(\hat{X}_{j}(\theta))\right|\geq
cn\text{ for every }\theta\in\mathbb{S}^{1}\setminus E_{n}.$

Considering (16), it remains to show that $m(B_{1}(n))\leq\text{const
}e^{-\gamma\sqrt{n}}$, for some $\gamma>0$. This is done using Lemma 2.6 and a
large deviations argument. For completeness, we provide a sketch of the proof,
for details see [15, Section 2]. Let $0\leq q\leq m-1$ be fixed and denote
$G_{q}=\\{i\in G\colon\nu_{i}\equiv q\mod m\\}$. Let $m_{q}=\max\\{j\colon
mj+q\leq n\\}$ (note $m_{q}\approx m\approx\sqrt{n}$) and for each $0\leq
j\leq m_{q}$ we let $\hat{r}_{j}=r_{i}$ if $mj=q=\nu_{i}$, for some $i\in
G_{q}$, and $\hat{r}_{j}=0$ otherwise. Notice that $G_{q}$ and the
$\hat{r}_{j}$ are functions of $\theta$. So, we write

$\Omega_{q}(\rho_{0},\cdots,\rho_{m_{q}})=\\{\theta\in\mathbb{S}^{1}\setminus
B_{2}(n)\colon\hat{r}_{j}=\rho_{j}\text{ for }0\leq j\leq m_{j}\\}$

where for each $j$ either $\rho_{j}=0$ or
$\rho_{j}\geq\displaystyle\frac{1}{D-1}\left(\frac{1}{D}-\frac{2\eta}{D-1}\right)\log\frac{1}{\alpha}$;
we also assume the $\rho_{j}$ not to be simultaneously zero. For $0\leq j\leq
m_{q}$ and $\omega_{mj+q+l}\in\mathcal{P}_{mj+q+l}$,
$\hat{Y}_{0}=\varphi^{mj+q+l}(\hat{X}_{0}|\omega{mj+q=l})$ is an admissible
curve and we have defined $l$ in such a way that $mj+q+l=m(j+1)+1-M$.
Therefore, we can apply Lemma 2.6 to obtain

$m(\\{\theta\in\omega_{mj+q+l}\colon\hat{r}_{j+1}=\rho\\})\leq
C_{*}C_{3}e^{-5\beta\rho},$

for all
$\displaystyle\rho\geq\frac{1}{D-1}\left(\frac{1}{D}-\frac{2\eta}{D-1}\right)\log\frac{1}{\alpha}$.
Here $C_{*}$ is a uniform upper bound for the metric distortion of the
iterates of $g$. Thus, we are in position to apply the large deviation
argument as in [15] to finish the proof.

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Vanderlei Horita<EMAIL_ADDRESS>
Departamento de Matemática, IBILCE/UNESP

Rua Cristóvão Colombo 2265

15054-000 S. J. Rio Preto, SP, Brazil