# Large back-angle quasielastic scattering for 7Li+159Tb

Piyasi Biswas1,3222Present address:Shahid Matangini Hazra Government College
for Women, Tamluk, Chakshrikrishnapur, Kulberia, Purba Medinipur, West Bengal
- 721649, India A. Mukherjee1,3<EMAIL_ADDRESS>D. Chattopadhyay1
Saikat Bhattacharjee1,3 M.K. Pradhan1333Present address: Department of
Physics, Belda College, Belda, Paschim Medinipur, West Bengal - 721424, India
Md. Moin Shaikh1444Present address: Department of Physics, Chanchal College,
Chanchal, Malda, West Bengal - 732123, India Subinit Roy1,3 A.
Goswami1555Deceased P. Basu1666Retired S. Santra2,3 S.K. Pandit2,3 K.
Mahata2,3 A. Shrivastava2,3 1Saha Institute of Nuclear Physics, 1/AF
Bidhannagar, Kolkata-700064, India
2Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai-400085, India
3Homi Bhabha National Institute, Anushaktinagar, Mumbai-400094, India

###### Abstract

Quasielastic scattering excitation function at large backward angle has been
measured for the weakly bound system, 7Li+159Tb at energies around the Coulomb
barrier. The corresponding quasielastic barrier distribution has been derived
from the excitation function, both including and excluding the
$\alpha$-particles produced in the reaction. The centroid of the barrier
distribution obtained after inclusion of $\alpha$-particles was found to be
shifted higher in energy, compared to the distribution excluding the
$\alpha$-particles. The quasielastic data, excluding the $\alpha$-particles,
have been analyzed in the framework of continuum discretized coupled channel
calculations. The quasielastic barrier distribution for 7Li+159Tb, has also
been compared with the fusion barrier distribution for the system.

##  I Introduction

Heavy ion fusion at near-barrier energies is strongly affected by the internal
structure of the colliding nuclei and coupling to the direct nuclear
processes, like inelastic excitation and direct nucleon transfer. The coupling
of the relative motion to the internal degrees of freedom successfully
explained the sub-barrier fusion enhancement observed in heavy ion collisions
with respect to the one dimensional barrier penetration model calculations
Beck85 ; Das_An .

The coupling essentially modifies the effective interaction potential and in
turn splits the single, uncoupled fusion barrier into a distribution of
barriers. The fusion barrier distribution, $D_{fus}$ for a system can be
derived from the measured fusion excitation function as Row91 ,

$D_{fus}(E)=\frac{d^{2}}{dE^{2}}\left\lfloor{E\sigma_{fus}(E)}\right\rfloor$
(1)

where $\sigma_{fus}(E)$ is the fusion cross section for the system at the
center-of-mass energy, E. Over the past several years of research in heavy ion
collision, $D_{fus}(E)$ has evolved to be a powerful tool to decipher the
effects of coupling of various channels on sub-barrier fusion and hence probe
the reaction dynamics of nucleus-nucleus collision Das_An . Since extraction
of $D_{fus}(E)$ involves second derivative of $E\sigma_{fus}(E)$, obtaining a
meaningful barrier distribution requires very precisely measured fusion data.

A similar barrier distribution can also be extracted from large back-angle
quasielastic scattering excitation function Timmers95 . The quasielastic
scattering is defined as the sum of all direct processes, like elastic and
inelastic scattering and transfer processes. Fusion is related to transmission
through the barrier, whereas large back-angle quasielastic scattering is
related to reflection at the barrier. Because of the conservation of reaction
flux, these two processes may be considered as complementary to each other.
The quasielastic barrier distribution, $D_{qel}$ is obtained as Timmers95 ,

$D_{qel}(E)=-\frac{d}{dE}\left\lfloor\frac{d\sigma_{qel}}{d\sigma_{Ruth}}\left(E\right)\right\rfloor$
(2)

where (${d\sigma_{qel}}/{d\sigma_{Ruth}}$) is the ratio of quasielastic
scattering and Rutherford scattering differential cross sections at a fixed
back-angle. As $D_{qel}$ is derived from the first derivative, unlike
$D_{fus}$, the uncertainty associated with $D_{qel}$ is less than that
associated with $D_{fus}$.

It has been observed that for heavy ion collisions involving tightly bound
nuclei, where fusion is the most dominant reaction process at near-barrier
energies, $D_{fus}$ and $D_{qel}$ are very similar Das_An ; Timmers98 ; Hu98 ;
Si02 ; Lin07 . By contrast, for very heavy systems, where deep inelastic
processes become important, it has been argued by Zagrabaev Zag08  that the
quasielastic barrier distribution extracted from the sum of elastic and
inelastic backscattering processes represents the total reaction threshold
distribution and it differs from the distribution derived from the fusion
excitation function. For reactions, where cross sections of non-fusion
channels are comparable to fusion cross sections, a deviation of $D_{qel}$
from $D_{fus}$ is expected to be seen Mit07 ; Nts07 .

Similarly, for weakly bound systems the distribution $D_{qel}$ extracted only
from the sum of the contribution of elastic, inelastic and transfer processes
at large back-angle will provide information about the total reaction
threshold distribution and not about the fusion barriers. This is because of
the fact that for weakly bound systems, apart from other direct processes,
breakup, transfer induced breakup and incomplete fusion (ICF) are very
important processes competing with fusion at near-barrier energies. Several
experimental studies of quasielastic scattering excitation function at large
back-angle and corresponding barrier distribution have been reported for
various systems involving weakly bound stable projectiles Lin07 ; Mon09 ; Mu09
; Oto09 ; Zerv09 ; Zerv10 ; Jia10 ; Zerv12 ; Zadro13 ; Pal14 ; Moin15 ; Moin20
. In most of these works, the measured quasielastic scattering excitation
function and the corresponding $D_{qel}$ were analyzed within the framework of
coupled reaction channel (CRC) or continuum discretized coupled channel (CDCC)
models, while a few of these works compared $D_{qel}$ with $D_{fus}$. The
latter works showed that for weakly bound systems, $D_{qel}$ is in general
broader compared to $D_{fus}$. Also, the centroid of the distribution
$D_{qel}$, with quasielastic events defined as the sum of elastic and
inelastic scattering and transfer processes, is found to be shifted lower in
energy than that of $D_{fus}$. But the shift in energy between the centroids
of $D_{qel}$ and $D_{fus}$ is found to be different for different systems. For
6,7Li induced reactions with 64Ni (Z=28), though the peak of $D_{qel}$ is
observed to be shifted towards lower energy compared to $D_{fus}$ for 6Li+64Ni
Moin15 , the distributions $D_{qel}$ and $D_{fus}$ are seen to be similar for
7Li+64Ni Moin20 . For 6,7Li induced reactions with 208Pb (Z=82) Lin07 ,
$D_{qel}$ and $D_{fus}$ have been reported to be similar if breakup
contribution is included in the quasielastic scattering excitation function.
However, for 6,7Li induced reactions with 197Au (Z=79) Pal14 , distributions
$D_{qel}$ are seen to shift towards higher energies with respect to $D_{fus}$
after inclusion of breakup-$\alpha$ channel in the quasielastic scattering
excitation functions for both 6Li and 7Li cases. The observations reported for
6,7Li-induced reactions with 208Pb and 197Au are contradictory, though 197Au
nucleus lies very close to ${}^{208}Pb$. In the backdrop of this scenario, it
would be interesting to investigate the role of the structure of target nuclei
while comparing $D_{qel}$ with $D_{fus}$ in 6,7Li-induced reactions.

In this context, we carried out a measurement of large back-angle quasielastic
excitation function and the corresponding barrier distribution for the system
7Li+159Tb, at near-barrier energies, where 159Tb is a well-deformed target
nucleus. A preliminary analysis of these measurements was reported in Ref. WOC
. This work is a part of our systematic investigation of different reaction
mechanisms in 6,7Li+159Tb Mukh06 ; Pradhan11 ; Pradhan13 . Complete and
incomplete fusion excitation functions for 6,7Li+159Tb were reported in Refs.
Mukh06 ; Pradhan11 . Different processes contributing to the measured large
$\alpha$-yield in the reaction 6Li+159Tb were disentangled and reported in
Ref. Pradhan13 . The primary motivation of the present work is to investigate
the role of couplings to 7Li projectile and 159Tb target excitations on large
back-angle quasielastic scattering process within the framework of coupled
channel calculations.

The present paper is organized as follows: The experimental details, along
with the results are described in Sec. II. The measured quasielastic
scattering excitation function and the corresponding barrier distribution are
analyzed in the framework of coupled channel calculations in Sec. III. A
comparison of the $D_{qel}$ and $D_{fus}$ is discussed in Sec. IV. Finally,
Sec. V summarizes the work.

##  II Experimental details

###  II.I Experimental setup

The 7Li beams in the energy range 17-34 MeV, in steps of 1 MeV, from the 14UD
BARC-TIFR Pelletron Accelerator at Mumbai, India were used to bombard a self-
supporting 159Tb target foil of thickness $\approx$1.1 mg/cm2. The energies of
the incident beam were corrected for the loss of energy in the target material
at half-thickness of the target. To detect and identify the charged particles
produced in the reaction, a set of four $\Delta$E-E telescopes of Si-surface
barrier detectors were placed at $\pm$170∘ and $\pm$160∘ inside a scattering
chamber of diameter 1 m. The $\Delta$E-E telescopes were mounted at $\pm$170∘
and $\pm$160∘ primarily to check the consistency of the measured quasielastic
scattering events. The thicknesses of the detectors of each telescope were so
chosen that the charged particles lose part of their kinetic energies in the
first detector ($\Delta$E) and stop by depositing the residual energies (Eres)
in the second detector (E). However, the stop detectors used in the experiment
were not thick enough to stop the Z=1 particles. Two Si-surface barrier
detectors, of thicknesses 300 $\mu$m and 500 $\mu$m, were placed at $\pm$20∘
with respect to the beam direction for monitoring the beam and also for
normalization purposes. In front of each telescope and monitor, a collimator
was placed to define the solid angle.

###  II.II Data analysis and Results

A typical two-dimensional $\Delta$E-Etot (where, Etot=$\Delta$E+Eres)
spectrum, at $E_{lab}$=26 MeV is shown in Fig. 1. The peak in the Z=3 band
arises from contributions due to elastic scattering of 7Li and inelastic
scattering from the excited states of 159Tb. The low-lying levels of 159Tb are
very closely spaced and so the inelastic excited states of the target could
not be separated from the elastic events. Besides, the Z=3 band may also
contain contribution from the inelastic excitation of the projectile, 7Li
Zadro13 . Moreover, it may also contain contribution from 6Lig.s., produced
via $n$-stripping of 7Li Zadro13 ; Pandit16 , since the events corresponding
to 6Li could not be separated from those of 7Li in the spectra. However, the
Z=3 band predominantly consists of elastic and inelastic events. So, for
nomenclature purpose, here we refer the Z=3 band as partial quasielastic
scattering band, which represents primarily the sum of elastic and inelastic
events. A one-dimensional projection of the Z=3 band was observed to show a
quasielastic peak with FWHM of $\approx$500 keV at energy, Elab= 29 MeV.

Figure 1: (Color online) Typical $\Delta$E-Etot spectrum for the 7Li+159Tb
reaction at Elab=26 MeV and $\theta_{lab}$=+160o

The Z=2 band corresponds to various events producing $\alpha$-particles in the
reaction. A one-dimensional projection of the Z=2 band shows a broad
$\alpha$-peak. The contribution of the $\alpha$-particles, emitted mostly at
energies corresponding to the beam velocity primarily originate from breakup
related processes, like no-capture breakup (NCBU) and ICF. The broad
$\alpha$-peak can also have contributions from the transfer of a single
nucleon and/ or a cluster of nucleons followed by the breakup process, and
thereby resulting in $\alpha$-particles. The processes that might contribute
to the $\alpha$\- particle cross-sections for the 7Li+159Tb reaction are:
(1) NCBU: Breakup of 7Li (B.U threshold for $\alpha+t$ is 2.47 MeV) into
$\alpha$ and $t$, either direct or sequential or both, where both the
fragments escape without any of them being captured by the target
(2) Triton-ICF: $t$ captured by the target following break up of 7Li into
$\alpha$ and $t$ or a one step $t$-transfer to the target
(3) Neutron stripping: Single $n$-stripping from 7Li will produce 6Li which
may break into $\alpha$ and $d$, if excited above its breakup threshold 1.47
MeV
(4) Deuteron stripping: $d$-stripping from 7Li will produce unbound 5He that
decays to $\alpha$ and a neutron
(5) Proton stripping: $p$-stripping from 7Li will produce unbound 6He which
then decays to $\alpha$ and two neutrons
(6) Proton pickup: $p$-pickup by 7Li will lead to 8Be which immediately decays
to two $\alpha$-particles. Since this is an inclusive measurement, each 8Be
will contribute two $\alpha$-particles to the total $\alpha$-yield. But, the
contribution of $\alpha$-particles from the $p$-pickup channel may be expected
to be very small in comparison to the total contribution of $\alpha$-particles
from other processes, like ICF ($t$+159Tb) Pandit16 . Hence, the extra
$\alpha$-particle contribution arising from the double counting of
$\alpha$-particles may be neglected in comparison to the total
$\alpha$-particle contribution for the reaction.

The Z=1 band in the figure shows a fall-back feature because the stop
detectors were not thick enough to stop the Z=1 particles. So the events
corresponding to Z=1 could not be used in the analysis.

The ratio of quasielastic to Rutherford cross-sections is given by the
expression,

$\begin{split}\frac{d\sigma_{qel}}{d\sigma_{Ruth}}\left(E,\theta_{tel}\right)=\left\lfloor\frac{N_{qel}(E,\theta_{tel})}{N_{m}(E,\theta_{m})}\right\rfloor
X\\\
\left\lfloor\frac{(d\sigma_{Ruth}/d\Omega)(E,\theta_{m})}{(d\sigma_{Ruth}/d\Omega)(E,\theta_{tel})}\right\rfloor\left(\frac{\Delta\Omega_{m}}{\Delta\Omega_{tel}}\right)\end{split}$
(3)

where,
Nqel(Nm) is the average yield in telescope (monitor) detector,
${\frac{d\sigma_{Ruth}}{d\Omega}}(E,\theta_{m}(\theta_{tel}))$ is the
calculated Rutherford scattering cross-section at the corresponding bombarding
energy E, at monitor angle $\theta_{m}$ (telescope angle $\theta_{tel}$), and
(${\frac{\Delta\Omega_{m}}{\Delta\Omega_{tel}}}$) is the solid angle ratio of
monitor to telescope detectors.
The ${\frac{\Delta\Omega_{m}}{\Delta\Omega_{tel}}}$ ratio for each of the four
telescope angles was determined from the measurements at the lowest bombarding
energies of 17, 18 and 19 MeV, where the elastic scattering is purely
Rutherford.

The ”partial” quasielastic counts $N_{qel}(E,\theta_{tel})$ at each bombarding
energy were obtained from the sum of elastic and inelastic counts in the Z=3
band in Fig. 1. As the measurements were done at angles close to 180∘,
centrifugal correction was incorporated to obtain the effective c.m. energies
($E_{eff}$). The results of the quasielastic events at $\pm$170∘ and $\pm$160∘
were converted to those for 180∘ by mapping to $E_{eff}$ using the relation
Timmers95 ,

$E_{eff}=\frac{2E_{c.m.}}{1+cosec\frac{\theta_{c.m.}}{2}}$ (4)

To check the consistency of the data, the quasielastic excitation functions
and barrier distributions were extracted using the data taken at $\pm$170∘ and
$\pm$160∘, and after appropriate centrifugal correction they were found to
agree fairly well each other. The good agreement between the measurements at
different angles gave us confidence in our data.

The ”partial” quasielastic scattering excitation function determined from the
Z=3 events and the corresponding quasielastic barrier distribution, Dqel,
extracted using Eq. (1) are shown by the solid circles ($\bullet$) in Figs.
2(a) and 2(b), respectively. The Ec.m. energies in the figures are essentially
$E_{eff}$ energies. The quasielastic cross sections in Fig. 2(a) were obtained
by averaging the cross sections for the two telescopes at $\pm$170∘, after
appropriate centrifugal corrections. The barrier distribution shown by the
solid circles ($\bullet$) in Fig. 2(b) was derived from the ”partial”
quasielastic excitation function, instead of the ”total” quasielastic cross
sections which would include all relevant reaction channels and not only
elastic and inelastic events. So the derived barrier distribution, shown in
Fig. 2(b) by the solid circles ($\bullet$), does not correspond to the fusion
barrier distribution, but rather reflect the reaction threshold distribution
Zag08 . The quasielastic scattering cross sections and the corresponding
barrier distribution, determined from the sum of Z=3 (elastic + inelastic) and
Z=2 ($\alpha$) events, are also shown in Figs. 2(a) and 2(b) by the solid
triangles ($\blacktriangle$). It is observed that the inclusion of
$\alpha$-particles in the definition of quasielastic events, shifts the
centroid of the barrier distribution higher by $\approx$800 keV than the
distribution corresponding to only Z=3 events. A similar observation has also
been reported for the system 7Li+197Au Pal14 .

Figure 2: (Color online) Comparison of a) quasielastic excitation function and
b) quasielastic barrier distribution for 7Li+159Tb, excluding ($\bullet$) and
including ($\blacktriangle$) the $\alpha$ particles produced in the reaction.

##  III Coupled Channel Calculations

In this section, our primary focus is to investigate the effects of couplings
between different reaction channels at near-barrier energies in the CRC
framework, and not the relation between $D_{fus}$ and $D_{qel}$. So, here we
considered the ”partial” quasielastic scattering cross sections derived only
from the Z=3 events. This is because the Z=1 particles could not be stopped in
the detectors and the different reaction channels contributing to the Z=2
events, discussed above, could not be disentangled in the present inclusive
measurement. The coupled channel calculations were carried out to analyze the
measured ”partial” quasielastic scattering excitation function and barrier
distribution for 7Li+159Tb, employing the code FRESCO (version FRES 2.9)
Thomp88 .

The primary input for the CRC calculations is the entrance channel optical
potential which consists of Coulomb potential plus the bare nuclear potential.
The bare nuclear potential parameters for a system are derived from an optical
model analysis of a set of measured elastic scattering differential cross
sections for the system. But measuring pure elastic scattering cross sections
for the system 7Li+159Tb is experimentally very difficult because of the very
closely spaced low-lying excited energy levels of 159Tb. The only elastic
scattering angular distribution data available in the literatute for 7Li+159Tb
are those of Ref. Patel15 . But these data have contributions from the low-
lying excited states of 159Tb. So, bare nuclear potential parameters could not
be obtained for the system 7Li+159Tb.

The elastic scattering angular distribution data Patel15  were therefore re-
analyzed in the present work, in the CDCC framework, to obtain a set of
properly adjusted cluster folding potentials for $\alpha$+159Tb and $t$+159Tb,
where the 7Li nucleus was considered to have a $\alpha$+$t$ cluster structure
with B.U. threshold of 2.47 MeV. The CDCC calculations were done using the
code FRESCO. The continuum of 7Li above the B.U. threshold of 2.47 MeV
consists of non-resonant and resonant states. It has been observed Cam16  that
in the CDCC calculations for elastic scattering angular distribution of the
relatively more weakly bound 6Li-induced reactions, couplings due to resonant
states are more dominant compared to the non-resonant ones. So, for the ease
of calculations, for the CDCC model space of 7Li we considered only the low-
lying non-resonant continuum states up to an excitation energy of 4.4 MeV of
7Li and the two resonant states 7/2- and 5/2- at 4.63 and 6.68 MeV,
respectively. Continuum states with angular momentum $l$=0, 1, 2 and 3 were
considered. The non-resonant continuum was discretized into momentum bins of
width $\Delta k$=0.2 fm-1, only up to $k_{max}$=0.4 fm-1. The binning of the
continuum with $l$=3 was suitably done so as to include the two resonant
states, 7/2- and 5/2- with average excitation energies of 2.16 and 4.21 MeV
relative to the B.U. threshold of 7Li and widths of 0.2 MeV and 3.0 MeV,
respectively DC18 . The widths taken were sufficient enough to accommodate the
main strength of the resonances. To calculate the bin wavefunctions, the
binding potential between $\alpha$ and $t$ for bound and resonant states of
7Li projectile were taken from Ref. DC18 . The wavefunction of the projectile-
target relative motion was expanded in partial waves up to Jmax=150 and it was
integrated numerically up to 140 fm, in steps of 0.05 fm. In addition to the
continuum of 7Li, the bound excited state of 7Li, having spin 1/2- and
Eex=0.477 MeV, with reduced transition probability B(E2$\uparrow$)= 8.3
$e^{2}fm^{4}$ Pal14  was included in the coupling scheme. Also, the two low-
lying excited states, 5/2+ state at 0.058 MeV and 7/2+ state at 0.137 MeV, of
159Tb were included in the coupling scheme Patel15 . The B(E2) values NNDC
for the corresponding transitions in 159Tb, used in the calculations are
listed in Table I. The Coulomb reduced matrix elements and the nuclear
deformation lengths for the coupled channel calculations were derived from the
B(E2) values, assuming rotational model for 159Tb. Adjusting the cluster
folded potentials of 7Li+159Tb, and fixing them at depth $V_{0}$=23.9 MeV,
radius parameter, $r_{0}$=1.2 fm and diffuseness, $a$=0.5 fm for
$\alpha$+159Tb and at $V_{0}$=29.9 MeV, $r_{0}$=1.2 fm and $a$=0.5 fm for
$t$+159Tb, gave a reasonable description of the elastic scattering angular
distribution data Patel15  at incident energy Elab=35 MeV. The imaginary parts
of the potentials were taken to be of Woods-Saxon form, with $W_{0}$=50 MeV,
$r_{0}$=1.0 fm and $a$=0.4 fm. All reorientation couplings have been
considered in the calculations. Figure 3 compares the angular distribution
data of Ref. Patel15 , at Elab= 35 MeV, with the CDCC calculated cross
sections. The solid line in the figure shows the calculated cross sections in
the above coupling scheme, where coupling to both resonant and non-resonant
parts of the continuum of 7Li was included. Fairly good agreement is observed
between the calculated and measured angular distribution cross sections. To
see the importance of the effect of coupling to non-resonant part of the
continuum of 7Li on the elastic scattering angular distribution, the same
calculations were repeated excluding the non-resonant continuum states of 7Li
in the above coupling scheme. The calculated elastic scattering angular
distribution at Elab= 35 MeV thereby obtained are shown by the dashed line in
Fig. 3. The very good agreement between the two calculations confirms that for
elastic scattering angular distribution of 7Li+159Tb, couplings due to
resonant states of 7Li are more dominant compared to the non-resonant ones.

Table 1: Reduced transition probabilities B(E2) NNDC  used in the coupled channel analysis for the calculations of the Coulomb matrix elements and nuclear deformation lengths for inelastic transitions in 159Tb Transition | B(E2; $J_{i}\rightarrow J_{f}$)
---|---
($J_{i}\rightarrow J_{f}$) | (e2b2)
$5/2\rightarrow 3/2$ | 1.87
$7/2\rightarrow 5/2$ | 1.25
$7/2\rightarrow 3/2$ | 0.72
$9/2\rightarrow 7/2$ | 0.61
$9/2\rightarrow 5/2$ | 1.13
$11/2\rightarrow 9/2$ | 0.56
$11/2\rightarrow 7/2$ | 1.47
$13/2\rightarrow 11/2$ | 0.32
$13/2\rightarrow 9/2$ | 1.74
Figure 3: (Color online) Elastic scattering angular distribution for the
7Li+159Tb system at Elab=35 MeV Patel15 . The solid and dashed lines represent
the CDCC calculations with and without coupling to non-resonant continuum
states of 7Li. Coupling to resonant states in the continuum of 7Li were
included in both the calculations. Figure 4: (Color online) Elastic scattering
angular distributions for the 7Li+159Tb system Patel15 , at different
bombarding energies. The solid lines represent the CDCC calculations.

Now that it is established that the effect of the resonant states of 7Li on
the elastic scattering angular distribution of 7Li+159Tb are dominant compared
to the non-resonant continuum states, in the present re-analysis of the
elastic scattering angular distribution data Patel15  coupling to non-resonant
continuum states was not considered in further calculations where continuum
coupling is used. Hereafter, for continuum coupling, only the two resonant
states, 7/2- at 4.63 MeV and 5/2- at 6.68 MeV, of 7Li were considered in the
CDCC coupling scheme. The CDCC calculations with the above coupling scheme,
excluding the non-resonant continuum of 7Li, and using the cluster folded
potentials for $\alpha$+159Tb and $t$+159Tb, as obtained above, the elastic
scattering angular distributions for 7Li+159Tb were calculated at Elab=26, 28,
30, 35 and 44 MeV. Figure 4 compares the angular distribution data of Ref.
Patel15  with the theoretical cross sections at different bombarding energies.
The solid lines in the figure show the calculated cross sections. Fairly good
agreement can be seen between the calculated and experimental elastic
scattering angular distribution cross sections.

Having reproduced the elastic scattering angular distribution data reasonably
well, the above coupling scheme was used to calculate the large back-angle
quasielastic scattering excitation function for 7Li+159Tb. The calculations
were performed with different coupling conditions:
(i) No coupling to the continuum of 7Li was considered. Only inelastic
coupling to low-lying excited states of 159Tb and bound state of 7Li at 0.477
MeV were included.
(ii) Only couplings to two resonant states, 7/2- and 5/2- at 4.63 and 6.68
MeV, in the continuum of 7Li were included. In this scheme, couplings to
neither target excited states, nor bound excited state of 7Li were considered.
(iii) Subsequently, calculations with couplings only to the above two resonant
states in 7Li continuum, the bound excited state of 7Li and the low-lying
excited states of 159Tb were done.
(iv) Finally, full CDCC calculations with couplings to both resonant and non-
resonant states of 7Li continuum, along with couplings to the bound excited
state of 7Li and the low-lying excited states of 159Tb were performed.
The results of these calculations are discussed below.

Figure 5: (Color online) Comparison of the measured partial quasielastic
excitation function for the 7Li+159Tb system compared with the coupled channel
calculations with different inelastic coupling conditions. See text for
details. Figure 6: (Color online) Comparison of the partial quasielastic
barrier distribution for the 7Li+159Tb system compared with coupled channel
predictions with different inelastic coupling conditions. See text for
details.

The calculations were first performed with coupling condition (i). The dot-
dashed lines in Figs. 5 and 6 represent the quasielastic scattering excitation
function and the corresponding barrier distribution, calculated with inelastic
coupling only up to the first excited state of 159Tb at 5/2+. The dotted lines
in the figures are the no-coupling calculations. It has already been mentioned
in Section II that the quasielastic peak for the Z=3 band had a FWHM of
$\approx$500 keV at Elab= 29 MeV. So, calculations including inelastic
excitation of 159Tb up to 13/2+ at 0.510 MeV in the coupling scheme, were
done. The dashed line in the Fig. 5 shows the quasielastic excitation
function, thereby calculated. The corresponding barrier distribution is
plotted in Fig. 6. No significant change is observed either in excitation
function or barrier distribution, if inelastic excited states of 159Tb beyond
5/2+ are included in the coupling scheme. This indicates that the 5/2+ state
of 159Tb is a strong contributor to the target inelastic coupling. Figures 5
and 6 show that inclusion of only inelastic excited states of 159Tb in the
coupling scheme fails to reproduce the quasielastic scattering excitation
function and barrier distribution for 7Li+159Tb. So, projectile excitation was
then considered, in addition to the target excitation, by including the bound
excited state of 7Li at 0.477 MeV in the coupling scheme. For comparison with
experimental results, the quasielastic scattering cross sections were
determined by adding the calculated elastic cross sections to the cross
sections of the inelastic states up to 13/2+ of 159Tb and the bound excited
state of 7Li, and are shown by the solid line in Fig. 5. The corresponding
barrier distribution is shown by the solid line in Fig. 6. The calculations
are still seen to underestimate the measured quasielastic excitation function
at higher energies and also fail to reproduce the experimental barrier
distribution.

Figure 7: (Color online) Effect of full coupling to the 7Li continuum in
addition to the inelastic coupling to 7Li and 159Tb bound excited states on
the quasielastic scattering excitation function for the 7Li+159Tb system. See
text for details. Figure 8: (Color online) Effect of full coupling to the 7Li
continuum in addition to the inelastic couping to 7Li and 159Tb bound excited
states on the quasielastic barrier distribution for the 7Li+159Tb system. See
text for details.

The CDCC calculations were then repeated with coupling condition (ii), i.e.,
couplings only to the two resonant states, $7/2^{-}$ and $5/2^{-}$ at 4.63 and
6.68 MeV, respectively, in the continuum of 7Li. The resulting quasielastic
excitation function and barrier distribution are shown by the dashed lines in
Figs. 7 and 8, respectively. Subsequently, the calculations were repeated with
coupling condition (iii), i.e., couplings to the above two resonant states in
the continuum of 7Li and also projectile and target inelastic couplings of
case (i). The results are shown by the dot-dot-dashed lines in the figures. It
can be seen that the inclusion of coupling to the resonant states of 7Li
continuum, in addition to the inelastic couplings to the bound excited state
of 7Li and the low-lying excited states of 159Tb better reproduces the
quasielastic scattering excitation function except at higher energies, but not
much change is observed in the barrier distribution. It is observed that the
effect of couplings to the channels included in coupling scheme (iii)
essentially reduces the height of the quasielastic barrier distribution and
also broadens the distribution, as compared to the no-coupling calculations.

Finally, full CDCC calculations were performed with coupling condition (iv),
i.e., couplings to both resonant and non-resonant states in continuum of 7Li
along with the inelastic couplings to 7Li bound excited state (1/2-, 0.477
MeV) and 159Tb low-lying excited states of case (i). The CDCC model space of
7Li was discretized into small bins of width $\Delta k$=0.2 fm-1 up to
$k_{max}$=0.8 fm-1. The resonant states were treated appropriately to avoid
double counting. Continuum states with angular momentum $l$=0, 1, 2 and 3 were
considered. For lower bombarding energies, the convergence is reached by
decreasing the upper limit of the excitation energy. Other details of the
scheme are discussed above in the CDCC calculations for elastic scattering
angular distribution. The results of the calculations are shown by the solid
lines in Figs. 7 and 8\. The inclusion of non-resonant part of the 7Li
continuum in the calculations significantly affects the height and location of
the centroid of the quasielastic barrier distribution, though no considerable
change is observed in the quasielastic excitation function. The height of the
barrier distribution is now almost similar to the experimental barrier
distribution, and the centroid of the distribution has also shifted
considerably towards the experimental barrier distribution.

It needs to be pointed out that the effects due to couplings to transfer and
transfer induced breakup channels have not been explored here. Inclusion of
these couplings may better reproduce the quasielastic excitation function at
the higher energies. The small shift observed between the centroids of the
experimental and theoretical barrier distributions could be due to the
reaction channels, especially transfer induced breakup, not included in the
calculations.

##  IV Comparison of $D_{qel}$ and $D_{fus}$

A comparison of the $D_{qel}$, including and excluding the $\alpha$ particle
contribution, with the $D_{fus}$ may shed some light on the importance of the
$\alpha$ particle contribution in defining the quasielastic scattering events
for weakly bound systems.

To compare $D_{qel}$ with $D_{fus}$, an attempt was made to extract the
$D_{fus}$ from the measured complete fusion (CF) excitation function for
7Li+159Tb Mukh06 ; Broda75 . Unfortunately, $D_{fus}$ could not be extracted
from the reported fusion cross sections Mukh06 ; Broda75 , because only a few
data points were available for differentiation. Therefore, a rough comparison
of the experimental $D_{qel}$ was made with the theoretical $D_{fus}$,
extracted from the calculated fusion cross sections which reproduced the
measured CF cross sections Mukh06 ; Broda75 .

Following Ref. Mukh06 , the fusion cross sections were calculated using the
coupled channels code, CCFULL CCFULL  with Akyuz Winther potential and all
other parameters as mentioned in the reference. In addition to the coupling
scheme used in Ref. Mukh06 , in the present work coupling to the bound excited
state of the projectile 7Li, having spin $1/2^{-}$ and $E_{ex}$=0.477 MeV was
also included using the rotational scheme Pal14 . Figure 9 compares the
experimental CF cross sections Mukh06 ; Broda75  with the calculated fusion
cross sections. The dotted curve shows the no coupling calculations Mukh06 .
The dot-dot-dashed line (CC1) shows the coupled channels calculation
considering rotational coupling to six excited states of 159Tb Mukh06 . The
dashed line (CC2) shows the coupled channels calculations including rotational
coupling to six excited states of 159Tb and also rotational coupling to the
bound excited state of 7Li. The measured complete fusion cross sections for
7Li+159Tb at above barrier energies are known to be suppressed by a factor of
0.74 compared to the fusion cross sections obtained from coupled channel
calculations Mukh06 . The solid line in Fig. 9 shows the CC2 cross sections,
after being scaled by a factor of 0.74, and will be referred hereafter as the
calculated CF excitation function. The fusion barrier distributions, $D_{fus}$
corresponding to calculated CC2 and CF cross sections were then obtained using
equation (1).

Figure 9: (Color online) Complete fusion cross sections for the 7Li+159Tb
system Mukh06 . The dotted curve shows the no coupling calculations. The dot-
dash-dashed (CC1) and dashed (CC2)lines are the coupled channels calculations
performed with the code CCFULL CCFULL . The solid line shows the coupled
channels calculations (CC2) scaled by the factor 0.74. Figure 10: (Color
online) Comparison of the barrier distribution obtained from the quasielastic
excitation function with and without inclusion of $\alpha$ particles for the
system 7Li+159Tb. The dotted and the dashed lines represent the $D_{fus}$
extracted from the coupled channel calculated fusion cross section without and
with (CC2) coupling, respectively. The solid line shows the theoretical CF
$D_{fus}$, obtained by scaling the CC2 cross sections by a factor of 0.74. The
theoretical $D_{fus}$ values have been normalized by the factor 1/($\pi
R_{b}^{2}$) to compare with $D_{qel}$. See text for details.

To compare $D_{qel}$ with $D_{fus}$, the $D_{fus}$ values thus obtained were
normalized by 1/$\pi R_{b}^{2}$, where the barrier radius $R_{b}$ was taken
from Ref. Mukh06 . Figure 10 shows a comparison of the experimental $D_{qel}$,
including and excluding the contribution of the $\alpha$-particles, with the
theoretical $D_{fus}$ (normalized). The dotted and the dashed lines represent
the $D_{fus}$ extracted from the calculated fusion cross sections without and
with coupling (CC2), respectively. The solid line shows the results obtained
when $D_{fus}$ (normalized) is derived from the calculated CF cross sections
Pal14 . It can be seen from the figure that the peak of $D_{qel}$ excluding
the contribution of $\alpha$-particles (shown by the symbol $\bullet$) lies at
an energy slightly lower than that of $D_{fus}$ calculated from the CC2 cross
sections and shown by the dashed line. This observation is consistent with
those of Refs. Lin07  and Zag08 .

The experimental $D_{qel}$ including the contribution of the
$\alpha$-particles and shown by the symbol $\blacktriangle$ in the figure is
found to agree reasonably well with the calculated CF $D_{fus}$, except a
small mismatch at the higher energies. Similar observation has also been
reported for the system 7Li+208Pb Lin07 . This indicates that the agreement of
$D_{qel}$, including the contribution of $\alpha$-particles, and $D_{fus}$
might be independent of the structure of target nuclei. The similarity of CF
barrier distribution with the QEL barrier distribution, including the alpha
contribution, for 7Li-induced reactions might be understood in the following
way.
For weakly bound systems, quasielastic scattering cross sections
($\sigma_{qel}$), are given by,

$\sigma_{qel}=1-(\sigma_{CF}+\sigma_{ICF})$ (5)

where $\sigma_{CF}$ and $\sigma_{ICF}$ are CF and ICF cross sections,
respectively. For 7Li+159Tb reaction, $t$-capture process is the dominant ICF
contributor with $\alpha$-capture process being relatively less significant
Mukh06 ; an observation also reported for 7Li+${}^{124}Sn$ Par18 . So, for
7Li-induced reactions, $\sigma_{ICF}$ $\approx$ $\sigma_{t-capture}$.
It has also been reported Pandit17  that for 7Li-induced reactions,

$\sigma_{t-capture}=\sigma_{\alpha}-\sigma_{\alpha-CN},$ (6)

where $\sigma_{\alpha}$ and $\sigma_{\alpha-CN}$ represent the inclusive
$\alpha$-yield and the contribution of $\alpha$-particles originating from the
decay of the compound nucleus (CN) produced in the CF process, respectively.
But the CN formed in the fusion of 7Li with 159Tb and other heavy mass targets
decays predominantly by $xn$ evaporation at near-barrier energies Mukh06 ;
Pandit17 , and hence $\sigma_{\alpha-CN}$ is expected to be negligible for
such systems. Therefore, for 7Li-induced reactions with heavy mass targets,
$\sigma_{ICF}$ $\approx$ $\sigma_{t-capture}$ $\approx$ $\sigma_{\alpha}$,
which in conjunction with eqn. (5) gives

$\sigma_{qel}+\sigma_{\alpha}\approx 1-\sigma_{CF}$ (7)

This perhaps explains, why $D_{qel}$ obtained from the sum of quasielastic and
$\alpha$-contributions reasonably agrees with the CF barrier distribution for
7Li+159Tb and 7Li+208Pb Lin07 .

However, before reaching any conclusion, one needs to carry out simultaneous
measurement of large back-angle quasielastic scattering and fusion excitation
functions, and compare the corresponding experimental barrier distributions
for more 7Li-induced reactions.

##  V Summary

The quasielastic scattering excitation function at large backward angle has
been measured for the system 7Li+159Tb at energies around the Coulomb barrier
and the corresponding barrier distribution has been extracted. The
quasielastic scattering excitation function and the corresponding barrier
distribution were determined, both with and without the contribution of
$\alpha$-particles. The centroid of the quasielastic barrier distribution is
seen to shift towards a higher energy with the inclusion of the contribution
of $\alpha$\- particles. This corroborates the observations of Zagrabaev Zag08
for weakly bound systems.

The experimental ”partial” quasielastic scattering cross sections and the
barrier distribution determined only from the Z=3 events have been compared
with the coupled channel calculations. As proper bare potential was not
available in the literature for the system 7Li+159Tb, the elastic scattering
angular distribution data of Ref. Patel15  were re-analyzed in the CDCC
framework to obtain a set of of properly adjusted cluster folded potentials
for $\alpha$+159Tb and $t$+159Tb so as to reproduce the elastic scattering
angular distribution data.

Using the cluster folded potentials for $\alpha$-159Tb and $t$-159Tb thus
determined, the coupled channel calculations were done, including different
couplings at a time, in the CDCC framework to see their effects on the
measured quasielastic scattering excitation function and corresponding barrier
distribution separately. The inelastic coupling scheme that included the low-
lying excited states of 159Tb and the bound excited state of 7Li, fails to
reproduce the experimental quasielastic excitation function and the barrier
distribution for 7Li+159Tb. The quasielastic scattering excitation function
could be reproduced reasonably well, except at the higher energies, by
including coupling to the continuum of 7Li, in addition to the above inelastic
couplings. Though the height of the quasielastic barrier distribution could be
reproduced reasonably well, a small shift between the centroids of the
experimental and theoretical barrier distribution was observed. The shift
might be attributed to the effects of other reaction channels, especially
transfer and transfer induced breakup processes not considered in the work.

As experimental $D_{fus}$ could not be obtained, the calculated fusion cross
sections which reproduced the measured fusion cross sections of 7Li+159Tb
Mukh06  were used to derive the $D_{fus}$. The $D_{fus}$, thereby obtained was
compared with the experimental $D_{qel}$, both including and excluding the
contribution of the $\alpha$-particles. This comparison indicates that the
distribution $D_{qel}$ including the contribution of $\alpha$-particles, is
considerably similar to CF $D_{fus}$ for 7Li+159Tb system. Similar observation
was also reported for 7Li+208Pb Lin07 , thus showing that the similarity
between CF $D_{fus}$ and $D_{qel}$, including $\alpha$-yield, might be
independent of the structure of target nuclei. It has been argued that this
similarity of CF barrier distribution with the barrier distribution obtained
from the sum of quasielastic and $\alpha$-particle contributions, probably
lies in the fact that the $t$-capture process is the dominant ICF process in
7Li-induced reactions with heavy mass targets. In order to have a better
understanding of the role of $\alpha$-contributing channels on $D_{fus}$ and
$D_{qel}$ in reactions with weakly bound projectiles, more simultaneous
measurements of fusion and quasielastic barrier distributions are needed for
7Li-induced reactions with various target nuclei, and also for reactions
induced by other weakly bound projectiles.

###### Acknowledgements.

We would like to thank the Pelletron crew for the smooth operation of the
accelerator. Thanks are also due to Mr. Sujib Chandra Chatterjee of Saha
Institute of Nuclear Physics for his technical support during the experiment.

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# Dimension drop of harmonic measure for some finite range random walks on
Fuchsian Schottky groups

Ernesto García  Pablo Lessa

###### Abstract

We prove that the harmonic measures of certain finite range random walks on
Fuchsian Schottky groups, have dimension strictly smaller than the Hausdorff
dimension of the corresponding limit set.

##  1 Introduction

###  1.1 A dimension drop conjecture

Let $G$ be the group of isometries of the hyperbolic plane $\mathbb{H}^{2}$
and $\Gamma<G$ be a non-elementary discrete subgroup.

Let $\Lambda$ be the limit set of $\Gamma$, which is a closed $\Gamma$
invariant subset of the visual boundary $\partial\mathbb{H}^{2}$ of
$\mathbb{H}^{2}$. We denote by $\dim(\Lambda)$ the Hausdorff dimension of
$\Lambda$ with respect to the visual distance based at any point
$o\in\mathbb{H}^{2}$.

For any finitely supported probability measure $\mu$ whose support generates
$\Gamma$ as a semi-group there exists a unique $\mu$-stationary probability
$\nu$ on $\partial\mathbb{H}^{2}$. That is, $\nu$ satisfies

$\nu=\sum\limits_{g\in\Gamma}\mu(g)g_{*}\nu,$

where $g_{*}\nu$ denotes the push-forward of $\nu$ by $g$. Since any
$\Gamma$-invariant compact set admits a $\mu$-stationary measure, it follows
by uniqueness that the support of $\nu$ must be contained in $\Lambda$.

The measure $\nu$ is also the harmonic measure of any random walk of the form
$x_{n}=g_{1}\cdots g_{n}o$ where $o\in\mathbb{H}^{2}$ and
$g_{1},\ldots,g_{n},\ldots$ are i.i.d. random elements with common
distribution $\mu$. By this we mean that it is the distribution of the random
limit point $x_{\infty}=\lim\limits_{n\to+\infty}x_{n}$.

Furthermore, $\nu$ is exact dimensional (see [Tan19], and also [LP08], [HS17],
and [Led83]), i.e. there exists a non-negative real number $\dim(\nu)$ such
that

$\dim(\nu)=\lim\limits_{r\downarrow 0}\frac{\log(\nu(B(\xi,r)))}{\log(r)},$

for $\nu$-a.e. $\xi\in\partial\mathbb{H}^{2}$, where $B(\xi,r)$ denotes the
ball of radius $r$ centered at $\xi$ with respect to the visual metric based
at some point $o\in\mathbb{H}^{2}$.

The dimension $\dim(\nu)$ can be characterized as the infimum of the Hausdorff
dimensions of Borel sets with full $\nu$-measure (see for example [You82,
Proposition 2.1]). Since $\nu$ is supported on $\Lambda$ one has
$\dim(\nu)\leq\dim(\Lambda)$. We conjecture that equality is never attained
for finitely supported $\mu$.

######  Conjecture 1 (Dimension drop conjecture).

Let $\mu$ be a finitely supported probability on $G$ whose support generates
(as a semi-group) a discrete non-elementary subgroup $\Gamma$, and let $\nu$
be the unique $\mu$-stationary measure on the limit set $\Lambda$ of $\Gamma$.
Then $\dim(\nu)<\dim(\Lambda)$.

The conjecture is motivated by the observation that, in other contexts, the
harmonic measure of a random walk ‘built from local information’ has a
dimension drop with respect to the natural geometric measure on the boundary.

Two contexts where analogous statements have been proven are the harmonic
measure of a Jordan domain bounded by a curve with Hausdorff dimension
strictly larger than $1$ (see [Mak85]). And the harmonic measure for the
simple random walk on a Galton-Watson tree (see [LPP95]).

Another motivation for Conjecture 1 is to extend the scope of the following
well known conjecture of singularity with respect to visual measure on the
boundary (see [DKN09a, Conjecture 1.21], and [KLP11]) to the case where
$\Lambda$ is not the entire visual boundary, allowing it to be investigated in
a larger family of examples.

######  Conjecture 2 (Singularity conjecture).

If $\mu$ is a finite support probability whose support generates (as a semi-
group) a Fuchsian group $\Gamma$ then the unique $\mu$ stationary measure
$\nu$ on $\partial\mathbb{H}^{2}$ is singular with respect to the class of
visual measures.

We notice that Conjecture 2 has been studied in other contexts such as, higher
dimensional hyperbolic space [RT21], Teichmüller space [GMT17], and higher
rank semi-simple Lie groups [KLP11]. In all these contexts analogous
statements to Conjecture 1 seem to warrant further investigation.

Conjecture 2 is known to be true when $\Gamma$ is not co-compact but has
finite co-volume (see [GLJ93], [BHM11], [DKN09b], [GMT15]).

Some progress has been made in the case when $\Gamma$ is co-compact. In
[CLP21] and [Kos21] the conjecture is verified for nearest neighbor random
walks on tilings by regular polygons. A general result for symmetric random
walks on the fundamental group of the surface of genus two is obtained in
[KT22]). For any finite support probability measure on the fundamental group
of a closed surface, dimension drop for the harmonic measure associated to
discrete and faithful representations of the group outside of a compact subset
of Teichmüller space was established in [AGG+22].

Dimension drop of harmonic measure was established for finite range random
walks on discrete groups $\Gamma$ which are not virtually free and whose
boundary is endowed with a distance coming from a word metric in [GMM18,
Theorem 1.1]. However, their result does not apply in the current setting (see
[GMM18, Remark 1.1]).

If $\mu$ is not assumed to be finitely supported then Conjecture 1 is false.
In the co-compact case a measure $\mu$ whose stationary measure is continuous
can be constructed via the Furtenberg-Lyons-Sullivan discretization procedure
(see [LS84]) or by more general results of Connell and Muchnik [CM07]. In the
convex co-compact case it has been shown that the probability can be chosen
with finite exponential moment (see [LNP21, Appendix]).

Conjecture 2 is false if the support of $\mu$ generates a dense group (see
[Bou12] and [BPS12]).

Conjecture 1 should be contrasted with the classical case of self-similar sets
defined by iterated function systems satisfying the open set condition. In
that case a probability $\mu$ on the generators of the iterated function
system can be found such that the $\mu$-stationary measure realizes the
Hausdorff dimension of the self-similar set (see [Mor46] and [Hut81]).

###  1.2 A partial result for Schottky groups

The purpose of this article is to prove some special cases of Conjecture 1
when the discrete group $\Gamma$ is a Schottky group in two generators.

To state our result we will need to discuss the expression of elements of the
group $\Gamma$ in terms of words in a free generator. For this purpose we fix
the finite alphabet $\Sigma=\\{a,b,a^{-1},b^{-1}\\}$ and let $\Sigma^{*}$
denote the set of finite words on this alphabet, including the empty word
which we denote by $\varepsilon$.

We fix the rewriting rules
$R=\\{(aa^{-1},\varepsilon),(a^{-1}a,\varepsilon),(bb^{-1},\varepsilon),(b^{-1}b,\varepsilon)\\}$
which are the standard presentation for the free group with generators
$\\{a,b\\}$. Each word $w\in\Sigma^{*}$ has a unique reduced form
$\overline{w}$ which is obtained by successively replacing any occurrence of
the left-hand side of a pair in $R$ with the empty word.

The set of reduced words $\Sigma^{*}_{R}$ is a free group with generators
$a,b$ when endowed with the product $w\cdot w^{\prime}=\overline{ww^{\prime}}$
(where $ww^{\prime}$ is the concatenation of $W$ and $w^{\prime}$).

We now consider a probability measure $\mu$ on $\Sigma^{*}_{R}$ whose support
$\operatorname{supp}(\mu)$ is finite and generates $\Sigma^{*}_{R}$ as a semi-
group. And denote the set of words which are a prefix of some word in
$\operatorname{supp}(\mu)$ (this includes the empty word $\varepsilon$) by
$\operatorname{Prefix}(\operatorname{supp}(\mu))$.

Suppose $\rho:\Sigma^{*}_{R}\to\Gamma$ is an isomorphism from $\Sigma^{*}_{R}$
to a Schottky group in $G$. By this we mean that there exists family of half-
planes $\\{H_{x}:x\in\Sigma\\}$ whose closures in
$\mathbb{H}^{2}\cup\partial\mathbb{H}^{2}$ are pairwise disjoint and such that
$\rho(x)H_{y}\subset H_{x}$ for all $x$ and all
$y\in\Sigma\setminus\\{x^{-1}\\}$. See Figure 1.

We let $\nu$ denote the $\rho_{*}\mu$-stationary measure on the limit set
$\Lambda$ of $\Gamma$.

Our main theorem is the following:

######  Theorem A (Dimension drop for some finite range random walks on
Schottky groups).

In the setting above suppose that $a$ occurs as the last letter of exactly one
element of $\text{Prefix}(\operatorname{supp}(\mu))$. Then
$\dim(\nu)<\dim(\Lambda)$.

The theorem applies to nearest neighbor random walks, i.e. when
$\operatorname{supp}(\mu)=\Sigma$. Examples with arbitrarily large cardinality
may be obtained defining $\operatorname{supp}(\mu)$ as the set of all non-
empty reduced words with length at most $n$, which either do not contain the
letter $a$, or are of the form $aw$ where $w$ does not contain $a$. For $n=2$
this yields

$\operatorname{supp}(\mu)=\\{b,a^{-1},b^{-1},b^{2},ba^{-1},a^{-1}b,a^{-2},a^{-1}b^{-1},b^{-1}a^{-1},b^{-2}\\}\cup\\{a,ab,ab^{-1}\\}.$

Figure 1: Half-planes $H_{x}$ for $x\in\Sigma$ associated to the Schottky
group $\rho(\Sigma^{*}_{R})$, in the Poincaré disk model of $\mathbb{H}^{2}$.

##  2 Hidden Markov property of harmonic measure

We let $\Sigma^{\omega}$ denote the set of infinite words and
$\Sigma^{\omega}_{R}$ the subset of infinite reduced words (i.e. every prefix
is reduced). We use $w[i]$ for the $i-$th letter of a word (infinite or
finite) so $w=w[1]w[2]\cdots$, and set $w[i:j]=w[i]w[i+1]\cdots w[j]$.

The set $\Sigma^{\infty}=\Sigma^{*}\cup\Sigma^{\omega}$ is a compact Polish
space under letter-wise convergence, i.e. $w_{n}\to w$ if and only if for each
$i\in\\{1,\ldots,|w|\\}$ one has $w_{n}[i]=w[i]$ for all $n$ large enough. The
subset
$\Sigma^{\infty}_{R}=\Sigma^{*}_{R}\cup\Sigma^{\omega}_{R}\subset\Sigma^{\infty}$
is compact.

In the setting of Theorem A let $g_{1},g_{2},\ldots$ be i.i.d. with common
distribution $\mu$. The limits

$g_{\infty}=\lim\limits_{n\to+\infty}g_{1}g_{2}\cdots g_{n},$

and

$\overline{g_{\infty}}=\lim\limits_{n\to+\infty}\overline{g_{1}g_{2}\cdots
g_{n}},$

exist almost surely and belong to $\Sigma^{\omega}$ and $\Sigma^{\omega}_{R}$
respectively. We let $\nu_{\Sigma^{\omega}}$ denote the distribution of
$g_{\infty}$ and $\nu_{\Sigma^{\omega}_{R}}$ the distribution of
$\overline{g_{\infty}}$.

We define $\pi:\Sigma^{*}\to\Sigma\cup\\{\varepsilon\\}$ as the projection
associating to each non-empty word its last letter, and satisfying
$\pi(\varepsilon)=\varepsilon$.

######  Theorem 1 (Reduced infinite words are hidden markov).

There exists a Markov chain $x_{1},x_{2},\ldots$ on
$\operatorname{Prefix}(\operatorname{supp}(\mu))\setminus\\{\varepsilon\\}$
such that

$\lim\limits_{n\to+\infty}\pi(x_{1})\pi(x_{2})\cdots\pi(x_{n}),$

exists, belongs to $\Sigma^{\omega}_{R}$ almost surely, and has distribution
$\nu_{\Sigma^{\omega}_{R}}$.

Theorem 1 implies in particular that $\nu_{\Sigma^{\omega}_{R}}$ is a hidden
Markov measure. This result follows from the recent work on coloured random
walks [BD20]. The construction we give for the Markov lift is roughly
equivalent to the discussion in [BD20, Section 1.2] though our proof is
different. Since we need the specific statement above for what follows, we
present the details in the present section.

The hidden Markov property of $\nu_{\Sigma^{\omega}_{R}}$ is also implicit in
formulas for harmonic measure in terms of matrix products in [Lal93]. However,
the number of states of the Markov pre-image implied by Lalley’s formulas is
large, and his construction would not suffice for our proof of Theorem A.

For nearest neighbor random walks on free groups the Markov property of
harmonic measure is well known and has been extended to certain types of free
products and other tree-like graphs (see [Led01b], [GW86], [PW19], [Mai05],
and [MM04]).

Our result implies for example, that when
$\operatorname{supp}(\mu)=\\{ab,a^{-1},b^{-1}\\}$, then
$\nu_{\Sigma^{\omega}_{R}}$ is Markov. Other examples of random walks which
are not nearest-neighbor but have Markov harmonic measures can be obtained by
using stopping times on a nearest-neighbor walk as in [For17].

Since the image of a Markov measure is rarely Markov (see for example [BR58],
[Kel82], and [GL05])) the case where $\nu_{\Sigma^{\omega}_{R}}$ is hidden
Markov but not Markov also occurs (and in some sense should be generic).

An example is obtained by setting
$\mu(\\{a\\})=p_{1},\mu(\\{a^{2}\\})=p_{2},\mu(\\{b\\})=p_{3}$ where
$0<\min(p_{1},p_{2},p_{3})<p_{1}+p_{2}+p_{3}=1$. To see this, observe that
infinite words in $\\{a,a^{2},b\\}$ are already reduced and let $[w]$ denote
the set of such words with $w$ as a prefix. So we get

$\frac{\nu_{\Sigma^{\omega}_{R}}([ab])}{\nu_{\Sigma^{\omega}_{R}}([a])}=\frac{p_{1}p_{3}}{p_{1}+p_{2}}<\frac{p_{1}^{2}p_{3}+p_{2}p_{3}}{p_{1}^{2}+p_{1}p_{2}+p_{2}}=\frac{\nu_{\Sigma^{\omega}_{R}}([a^{2}b])}{\nu_{\Sigma^{\omega}_{R}}([a^{2}])}.$

A natural context for the results of this section seems to be that of finite
complete rewriting systems, i.e. finite sets of reduction rules with the
property that applying available reductions in any order one eventually
arrives at a unique reduced form for each word.

For example, the plain groups studied in [BD20], [Mai05], and [MM04], admit a
finite complete rewriting system where every rule substitutes a pair of
consecutive letters either with the empty string or a single letter.

There exist finite presentations of surface groups which are complete re-
writing systems (see [Che86] and [Her94]). It seems of interest, in view of
Conjecture 2, to explore whether the harmonic measure on the set of infinite
reduced words for these presentations have the hidden Markov property.

###  2.1 Infinite non-reduced words

As an initial step towards the proof of Theorem 1 we give an explicit
construction for the following:

######  Lemma 1 (Non-reduced infinite words are hidden markov).

There exists a Markov chain $y_{1},y_{2},\ldots$ on
$\operatorname{Prefix}(\operatorname{supp}(\mu))$ such that

$\lim\limits_{n\to+\infty}\pi(y_{1})\pi(y_{2})\cdots\pi(y_{n}),$

exists, belongs to $\Sigma^{\omega}$ almost surely, and has distribution
$\nu_{\Sigma^{\omega}}$.

####  2.1.1 Weighted prefix graph

We will construct a weighted graph out of
$\operatorname{Prefix}(\operatorname{supp}(\mu))$, see Figure 2 for an
example.

We say $w$ is a one step prefix of $w$ and write
$w^{\prime}\overset{1}{\prec}w$, if $w=w^{\prime}x$ for some $x\in\Sigma$.
More generally we write $w^{\prime}\prec w$ if $w^{\prime}$ is a prefix of
$w$.

######  Definition 1 (Weighted prefix graph).

By the prefix graph associated to $\mu$ we mean the graph with vertex set
$\operatorname{Prefix}(\operatorname{supp}(\mu))$, where a single directed
edge is added from $w^{\prime}$ to $w$ whenever
$w^{\prime}\overset{1}{\prec}w$, and also from $w$ to $\varepsilon$ for each
$w\in\operatorname{supp}(\mu)$.

To each edge of the form $w\overset{1}{\prec}w^{\prime}$ we associate the
weight

$W(w,w^{\prime})=\sum\limits_{w^{\prime}\prec
w^{\prime\prime}}\mu(\\{w^{\prime\prime}\\}),$

and to each edge of the form $w\rightarrow\varepsilon$ the weight
$W(w,\varepsilon)=\mu(\\{w\\})$.

We now prove two basic properties of the weighted prefix graph which we need
to prove Lemma 1.

######  Proposition 1.

There exists $N$ such that every path of length $N$ in the prefix graph passes
through $\varepsilon$ at least once.

###### Proof.

Each step along a path that does not arrive at $\varepsilon$ increases word
length. The maximal possible word length is
$\max\\{|w|:w\in\operatorname{supp}(\mu)\\}$ which is finite since
$\operatorname{supp}(\mu)$ is a finite set. ∎

######  Proposition 2 (Incoming and outgoing weights are equal).

With the above definition one has

$\sum\limits_{y}W(y,x)=\sum\limits_{y}W(x,y),$

for all $x\in\operatorname{Prefix}(\operatorname{supp}(\mu))$.

###### Proof.

If $x=\varepsilon$ the left-hand side is $1$ since there is a term
$\mu(\\{w\\})$ for each $w\in\operatorname{supp}(\mu)$. The right-hand side is

$\sum\limits_{x\in\Sigma}\sum\limits_{x\prec w}\mu(\\{w\\}),$

which is the same sum grouped by first letter and therefore is also $1$.

For $x\neq\varepsilon$ one has $x^{\prime}\overset{1}{\prec}x$ for a unique
(possibly empty) word $x^{\prime}\in S^{\prime}$. Hence, the left-hand side is

$\sum\limits_{x\prec w}\mu(\\{w\\}).$

If $x\notin\operatorname{supp}(\mu)$ then the right-hand side is

$\sum\limits_{x\overset{1}{\prec}x^{\prime\prime}}\sum\limits_{x^{\prime\prime}\prec
w}\mu(\\{w\\}),$

which, is the sum above regrouped by the first letter after $x$.

If $x\in\operatorname{supp}(\mu)$ then the two sums differ by $\mu(x)$ which
appears in the right-hand side as $W(x,\varepsilon)$. This concludes the proof
of the claim. ∎

####  2.1.2 Proof of Lemma 1

To conclude the proof of Lemma 1 we consider the Markov chain on the weighted
prefix graph whose transition probabilities are proportional to the given
weights.

######  Lemma 2 (Markov chain on the weighted prefix graph).

Let $y_{1},y_{2},\ldots$ be a Markov chain on the weighted prefix graph
starting at $\varepsilon$ and with transition probabilities given by

$P(x,y)=\frac{W(x,y)}{\sum\limits_{z}W(x,z)}.$

Then $\lim\limits_{n\to+\infty}\pi(y_{1})\pi(y_{2})\cdots\pi(y_{n})$, has
distribution $\nu_{\Sigma^{\omega}}$.

###### Proof.

Let $\tau_{0}=1$ and inductively define
$\tau_{n+1}=\min\\{k>\tau_{n}:y_{k}=\varepsilon\\}$ for $n=0,1,\ldots$. By the
strong Markov property the sequence $y_{\tau_{1}-1},y_{\tau_{2}-1},\ldots$ is
i.i.d.. It suffices to show that $y_{\tau_{1}-1}$ has distribution $\mu$ to
prove the claim.

For this purpose fix $w\in\operatorname{supp}(\mu)$, set $n=|w|$, and observe
using Proposition 2 that

$\displaystyle\mathbb{P}\left(y_{\tau_{1}-1}=w\right)$
$\displaystyle=P(\varepsilon,w[1])\cdots P(w[n-1],w[n])P(w[n],\varepsilon)$
$\displaystyle=\frac{W(\varepsilon,w[1])}{\sum\limits_{x\in\Sigma}W(\varepsilon,x)}=\mu(\\{w\\}).\qed$

Figure 2: The weighted prefix graph associated to a probability $\mu$ with
support $\operatorname{supp}(\mu)=\\{a,a^{2},b\\}$.

###  2.2 Proof of Theorem 1

Let $y_{1},y_{2},\ldots$ be as in Lemma 1 and define
$S=\operatorname{Prefix}(\operatorname{supp}(\mu))\setminus\\{\varepsilon\\}$.

Given a reduced word $w$ we define the stopping time

$\tau_{w}=\min\\{n:\overline{\pi(y_{1})\cdots\pi(y_{n})}=w\\}.$

Let $f:S\to[0,1]$ be defined by

$f(s)=\mathbb{P}\left(\tau_{\pi(s)}<+\infty,y_{\tau_{\pi(s)}}=s\right).$

Similarly let $g:S\times S^{\prime}\to[0,1]$ be defined by

$g(s,s^{\prime})=\mathbb{P}^{s}\left(\tau_{\pi(s)\pi(s^{\prime})}<+\infty,y_{\tau_{\pi(s)\pi(s^{\prime})}}=s^{\prime}\right),$

where $\mathbb{P}^{s}$ denotes the probability for the Markov chain
conditioned to start at $s$.

Finally let $h:S\to[0,1]$ be defined by

$h(s)=\mathbb{P}^{s}\left(\pi(s)\prec\overline{\pi(y_{1})\cdots\pi(y_{n})}\text{
for all }n\text{ large enough}\right).$

######  Definition 2 (Reduced prefix graph).

We observe that $g(s,s^{\prime})>0$ if and only if there exists a path
$s=s_{0}\rightarrow s_{1}\rightarrow\cdots\rightarrow s_{n}=s^{\prime}$ in the
weighted prefix graph such that

$\overline{\pi(s_{0})\cdots\pi(s_{n})}=\pi(s)\pi(s^{\prime})$

and furthermore

$\overline{\pi(s_{0})\cdots\pi(s_{k})}\neq\pi(s)\pi(s^{\prime}),$

for all $k<n$.

By the reduced prefix graph we mean the set
$S=\operatorname{Prefix}(\operatorname{supp}(\mu))\setminus\\{\varepsilon\\}$
with a single directed edge between $s$ and $s^{\prime}$ when the above
property holds.

The following result concludes the proof of Theorem 1.

######  Lemma 3.

Let $r=\lim\limits_{n\to+\infty}\overline{\pi(y_{1})\cdots\pi(y_{n})}$ and set
$\tau_{n}=\min\\{k:\overline{\pi(y_{1})\cdots\pi(y_{k})}=r[1:n]\\}$ for
$n=1,2,\ldots$.

Then the sequence $x_{1}=y_{\tau_{1}},x_{2}=y_{\tau_{2}},\ldots$ is a Markov
chain and furthermore

$\mathbb{P}\left(x_{1}=s_{1},\ldots,x_{n}=s_{n}\right)=f(s_{1})g(s_{1},s_{2})\cdots
g(s_{n-1},s_{n})h(s_{n}),$ (1)

for all $s_{1},\ldots,s_{n}\in S$.

###### Proof.

It suffices to prove equation 1, since this implies that $x_{1},\ldots,x_{n}$
is Markov with initial distribution $p(s)=f(s)h(s)$ and transtion
probabilities $P(s,s^{\prime})=h(s)^{-1}g(s,s^{\prime})h(s^{\prime})$.

For this purpose we let $w=\pi(s_{1})\cdots\pi(s_{n})$ and write

$\mathbb{P}\left(x_{1}=s_{1},\ldots,x_{n}=s_{n}\right)=\mathbb{P}\left(F_{1}\cap\cdots
F_{n}\cap G\right)=\mathbb{P}\left(F_{n}\cap G\right),$ (2)

where $F_{i}=\\{\tau_{w[1:k]}<+\infty,y_{\tau_{w[1:k]}}=s_{k}\text{ for all
}k\leq i\\}$ and $G=\\{w\prec r\\}$.

We observe that conditioned on $F_{n}$ the event $G$ is equivalent to

$\left\\{\pi(s_{n})\prec\overline{\pi(y_{\tau_{w}})\cdots\pi(y_{k+\tau_{w}})}\text{
for all }k\text{ large enough}\right\\}.$

Therefore, by the strong Markov property we obtain

$\mathbb{P}\left(F_{n}\cap
G\right)=\mathbb{P}\left(F_{n}\right)\mathbb{P}\left(G|F_{n}\right)=\mathbb{P}\left(F_{n}\right)h(s_{n}).$
(3)

For each $i=1,\ldots,n$ we observe that conditioned on $F_{i}$ the event
$F_{i+1}$ is equivalent to there being some finite $k$ such that
$\overline{\pi(y_{\tau_{w[1:i]}})\cdots\pi(y_{k+\tau_{w[1:i]}})}=\pi(s_{i})\pi(s_{i+1})$
and the minimal $k$ with this property satisfying
$y_{k+\tau_{w[1:i]}}=s_{i+1}$.

Once again by the strong Markov property we obtain

$\mathbb{P}\left(F_{i+1}\right)=\mathbb{P}\left(F_{i}\right)\mathbb{P}\left(F_{i+1}|F_{i}\right)=\mathbb{P}\left(F_{i}\right)g(s_{i},s_{i+1}).$
(4)

Hence we have

$\displaystyle\mathbb{P}\left(x_{1}=s_{1},\ldots,x_{n}=s_{n}\right)$
$\displaystyle=\mathbb{P}\left(F_{n}\cap G\right)$
$\displaystyle=\mathbb{P}\left(F_{n}\right)\mathbb{P}\left(G|F_{n}\right)$
$\displaystyle=\mathbb{P}\left(F_{n-1}\right)\mathbb{P}\left(F_{n}|F_{n-1}\right)h(s_{n})$
$\displaystyle=\cdots$
$\displaystyle=\mathbb{P}\left(F_{1}\right)g(s_{1},s_{2})\cdots
g(s_{n-1},s_{n})h(s_{n}),$

the equality $\mathbb{P}\left(F_{1}\right)=f(s_{1})$ is direct from the
definition of $f$, which concludes the proof. ∎

##  3 Symbolic coding and thermodynamic formalism

Recall that there exists family of half-planes $\\{H_{x}:x\in\Sigma\\}$ whose
closures in $\mathbb{H}^{2}\cup\partial\mathbb{H}^{2}$ are disjoint and such
that for all $x$ one has $\rho(x)H_{y}\subset H_{x}$ for all
$y\in\Sigma\setminus\\{x^{-1}\\}$.

We fix a basepoint
$o\in\mathbb{H}^{2}\setminus\bigcup\limits_{x\in\Sigma}H_{x}$ and let
$\operatorname{dist}$ denote the hyperbolic distance. See Figure 1.

Given a finite reduced word $w\in\Sigma^{*}_{R}$ we denote the set of infinite
reduced words having $w$ as a prefix by

$[w]=\\{w^{\prime}\in\Sigma^{\omega}_{R}:w\prec w^{\prime}\\}.$

######  Theorem 2.

In the setting of Theorem A either $\dim(\nu)<\dim(\Lambda)$ or there exists a
constant $C>1$ such that

$C^{-1}\exp(-\delta\operatorname{dist}(o,\rho(\omega)o))\leq\nu_{\Sigma^{\omega}_{R}}\left([w]\right)\leq
C\exp(-\delta\operatorname{dist}(o,\rho(\omega)o)),$

for all $w\in\Sigma^{*}_{R}$, where $\delta=\dim(\Lambda)$.

###  3.1 Coding of the limit set

We recall basics facts on the symbolic coding of $\Lambda$, see [Lal89,
Section 9].

The mapping $\pi_{\Lambda}:\Sigma^{\omega}_{R}\to\Lambda$ defined by

$\pi_{\Lambda}(w)=\lim\limits_{n\to+\infty}\rho(w[1:n])o,$

is a Hölder homeomorphism when $\Lambda$ is endowed with the visual distance
$\operatorname{dist}_{o}$ based at $o$ and $\Sigma^{\omega}$ with the distance

$\operatorname{dist}(w,w^{\prime})=\exp\left(-\min\\{n:w[n]\neq
w^{\prime}[n]\\}\right).$

We define $f:\Sigma^{\omega}_{R}\to\mathbb{R}$ as

$f(w)=\log|F^{\prime}(\pi_{\Lambda}(w))|,$

where $F:\Lambda\to\Lambda$ restricted to
$\overline{H_{x}}\cap\partial\mathbb{H}^{2}$ is $\rho(x^{-1})$ for each
$x\in\Sigma$, and the derivative $F^{\prime}$ is taken with respect to the
constant speed parametrization with respect to visual distance based at $o$.

###  3.2 Thermodynamical formalism

The mapping $F$ is conjugate via $\pi_{\Lambda}$ to the left shift
$\sigma:\Sigma^{\omega}_{R}\to\Sigma^{\omega}_{R}$ defined by
$\sigma(w)[i]=w[i+1]$.

The Hausdorff dimension $\delta=\dim(\Lambda)$ is characterized (see [Bow79,
Section 4]) as the unique value of $s\geq 0$ such that $P(s)=0$ where

$P(s)=\sup\limits_{m}\left\\{h(m)-s\int f(w)dm(w)\right\\},$

and the supremum is taken over all shift-invariant probability measure $m$,
and $h(m)$ denote topological entropy of $m$. There is a unique shift-
invariant probability measure $m_{\delta}$ attaining the supremum above.

Recall, that the Gibbs measure on $\Sigma^{\omega}_{R}$ corresponding to a
Hölder potential $\psi:\Sigma^{\omega}_{R}\to\mathbb{R}$ is the unique shift-
invariant probability such that for some $C>1$ one has

$C^{-1}\exp\left(-nP+\sum\limits_{k=0}^{n-1}\psi(\sigma^{k}(w^{\prime}))\right)\leq
m([w])\leq
C\exp\left(-nP+\sum\limits_{k=0}^{n-1}\psi(\sigma^{k}(w^{\prime}))\right),$

for all $w\in\Sigma^{*}_{R}$ and all $w^{\prime}\in[w]$, where $P$ is the
pressure of $\psi$, which is equal to $0$ in the case of $m_{\delta}$.

######  Lemma 4.

If $m$ is a Gibbs measure on $\Sigma^{\omega}_{R}$ then either $m=m_{\delta}$
or $\dim(\pi_{*}m)<\dim(\Lambda)$.

###### Proof.

From [PU10, Theorem 8.1.4] one has that $\pi_{*}m$ is exact dimensional with
dimension

$\dim(\pi_{*}m)=\frac{h(m)}{\int f(w)dm(w)}.$

Since $m_{\delta}$ is the unique shift-invariant measure attaining the
supremum $P(\delta)=0$, for all $m\neq m_{\delta}$ one has

$h(m)-\delta\int f(w)dm(w)<P(\delta)=0.$

Hence, $\dim(\pi_{*}m)<\delta$ as claimed. ∎

For $\xi\in\partial\mathbb{H}^{2}$ we denote by
$b_{\xi}:\mathbb{H}^{2}\times\mathbb{H}^{2}\to\mathbb{R}$ the Busemann
function defined by

$b_{\xi}(x,y)=\lim\limits_{t\to+\infty}\operatorname{dist}(\alpha(t),x)-\operatorname{dist}(\alpha(t),y),$

where $\alpha:[0,+\infty)\to\mathbb{H}^{2}$ is any geodesic ray ending at
$\xi$ parametrized by arclength (see [Dal11, Theorem 1.18]).

Directly from the definition it follows that
$b_{\xi}(x,z)=b_{\xi}(x,y)+b_{\xi}(y,z)$ for all $x,y,z\in\mathbb{H}^{2}$.
Furthermore one has $|b_{\xi}(x,y)|\leq\operatorname{dist}(x,y)$.

We will need the following additional property for what follows:

######  Lemma 5.

For all $g\in G$ one has $|(g^{-1})^{\prime}(\xi)|=\exp(b_{\xi}(o,go))$ where
$g^{\prime}$ is the derivative with respect to the constant speed
parametrization with respect to the visual distance on
$\partial\mathbb{H}^{2}$ based at $o$.

###### Proof.

The proof is by direct calculation in the Poincaré disk model where
$\mathbb{H}^{2}$ is identified with $\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$
and $\partial\mathbb{H}^{2}$ with the boundary circle $\partial\mathbb{D}$.
Setting $o=0$, the parametrization of $\partial\mathbb{D}$ is by Euclidean
arclength.

We fix an orientation preserving isometry of $g$ which in this model is of the
form

$g(z)=\frac{az+b}{\overline{b}z+\overline{a}},$

with $|a|^{2}-|b|^{2}=-1$. By direct calculation

$g^{-1}(z)=\frac{\overline{a}z-b}{-\overline{b}z+a}.$

We calculate $|(g^{-1})^{\prime}(\xi)|=1/|a-\overline{b}\xi|^{2}$ while one
has (see [BH99, pg. 273] and $b_{\xi,x}$ defined there corresponds to
$-b_{\xi}(o,x)$ in our notation)

$\exp(b_{\xi}(o,g(o)))=\frac{1-|g(0)|^{2}}{|g(0)-\xi|^{2}}=\frac{1-|b|^{2}/|\overline{a}|^{2}}{|b/\overline{a}-\xi|^{2}}=1/|b-\overline{a}\xi|^{2}.$

Since $|\xi|=1$ we have

$|a-\overline{b}\xi|=|\overline{\overline{b}\xi-a}|=|b\overline{\xi}-\overline{a}|=|\overline{\xi}(b-\overline{a}\xi)|=|b-\overline{a}\xi|.\qed$

We now establish the estimates for $m_{\delta}$ needed for Theorem 2.

######  Lemma 6.

There exists $C>1$ such that

$C^{-1}\exp(-\delta\operatorname{dist}(o,\rho(w)o))\leq m_{\delta}([w])\leq
C\exp(-\delta\operatorname{dist}(o,\rho(w)o)),$

for all $w\in\Sigma^{*}_{R}$.

###### Proof.

From the Gibbs property of $m_{\delta}$ we obtain

$m_{\delta}([w])\approx|(F^{|w|})^{\prime}(\pi_{\Lambda}(\tilde{w}))|^{-\delta}=|(\rho(w)^{-1})^{\prime}\xi|^{-\delta}$

where $F^{k}$ denotes the $k$-th iterate of $F$, $\approx$ means up to a
multiplicative constant independent of $w$, $\tilde{w}\in[w]$, and
$\xi=\pi_{\Lambda}(\tilde{w})$.

By Lemma 5 we have

$|(\rho(w)^{-1})^{\prime}\xi|^{-\delta}=\exp(-\delta b_{\xi}(o,\rho(w)o)).$

Now take $R>0$ sufficiently large so that all geodesics joining $H_{x}$ to
$H_{y}$ for distinct $x,y\in\Sigma$ intersect $B(o,R)$. For this value of $R$
one has that the geodesic ray from $o$ to $\xi$ must intersect
$B(o,\rho(w)o)$. It follows that

$|b_{\xi}(o,\rho(w)o)-\operatorname{dist}(o,\rho(w))|\leq 2R.\qed$

###  3.3 Proof of Theorem 2

From [Led01a, Proposition 3.2] there exists a continuous positive density
$\varphi$ such that $\varphi\nu_{\Sigma^{\omega}_{R}}$ is a Gibbs measure.

Assume that $\dim(\nu)=\dim(\Lambda)$. Since
$\nu=\pi_{*}\nu_{\Sigma^{\omega}_{R}}$ from Lemma 4 it follows that
$\varphi\nu_{\Sigma^{\omega}_{R}}=m_{\delta}$. In particular, since $\varphi$
is bounded away from zero and infinity there exists $C^{\prime}>1$ such that

$(C^{\prime})^{-1}m_{\delta}([w])\leq\nu_{\Sigma^{\omega}_{R}}([w])\leq
C^{\prime}m_{\delta}([w]),$

for all reduced words $w$. Combining this with Lemma 6 concludes the proof.

##  4 Proof of Theorem A

We denote the translation length of an isometry $g$ of $\mathbb{H}^{2}$ by

$\ell(g)=\lim\limits_{n\to+\infty}\frac{1}{n}\operatorname{dist}(o,g^{n}o).$

We say a word $w\in\Sigma^{*}_{R}$ is cyclically reduced if its first and last
letters are not inverses.

######  Lemma 7.

Either $\dim(\nu)<\dim(\Lambda)$ or for all cyclically reduced
$w\in\Sigma^{\omega}_{R}$ one has

$\lim\limits_{n\to+\infty}\left(\nu_{\Sigma^{\omega}_{R}}([w^{n}])\right)^{\frac{1}{n}}=\exp(-\delta\ell(\rho(w))).$

###### Proof.

This is a direct corollary of Theorem 2. ∎

######  Lemma 8.

Either $\dim(\nu)<\dim(\Lambda)$ or for all cyclically reduced
$aw_{1},aw_{2}\in\Sigma^{*}_{R}$ one has

$\ell(\rho(aw_{1}aw_{2}))=\ell(\rho(aw_{1}))+\ell(\rho(aw_{2})).$

###### Proof.

Given $w\in\Sigma^{*}_{R}$ we denote by $A(w)$ the set of $w$-paths, which is
to say $\alpha\in A(w)$ if $\alpha=(s_{0}\rightarrow\cdots\rightarrow s_{n})$
is a path in the reduced prefix graph such that
$\pi(s_{0})\cdots\pi(s_{n})=w$. For such paths we write
$g(\alpha)=g(s_{0},s_{1})\cdots g(s_{n-1},s_{n})$, and
$g_{+}(\alpha)=g(\alpha)h(s_{n})$.

By hypothesis there is a unique element
$s_{a}\in\operatorname{Prefix}(\operatorname{supp}(\mu))$ such that
$\pi(s_{a})=a$. This implies that if $\alpha\in A(w)$ and $w[i]=a$ then the
$i$-th vertex visited by $\alpha$ is $s_{a}$.

Using this observation and Lemma 3 we calculate for $aw$ cyclically reduced

$\displaystyle\nu_{\Sigma^{\omega}_{R}}([(aw)^{n}])=f(s_{a})\left(\sum\limits_{\alpha\in
A(awa)}g(\alpha)\right)^{n-1}\sum\limits_{\alpha\in A(aw)}g_{+}(\alpha).$

By Lemma 7 this implies

$\exp(-\delta\ell(\rho(aw)))=\sum\limits_{\alpha\in A(awa)}g(\alpha),$

whenever $aw$ is cyclically reduced.

In particular we have

$\displaystyle\exp(-\delta\ell(\rho(aw_{1}aw_{2})))$
$\displaystyle=\sum\limits_{\alpha\in A(aw_{1}aw_{2}a)}g(\alpha)$
$\displaystyle=\sum\limits_{\alpha\in
A(aw_{1}a)}g(\alpha)\sum\limits_{\alpha\in A(aw_{2}a)}g(\alpha)$
$\displaystyle=\exp(-\delta\ell(\rho(aw_{1})))\exp(-\delta\ell(\rho(aw_{2}))),$

which concludes the proof. ∎

If it were the case that $\dim(\nu)=\dim(\Lambda)$, by Lemma 8 setting $w=aa$
and $w^{\prime}=ab$ one would have
$\ell(\rho(ww^{\prime}))=\ell(\rho(w))+\ell(\rho(w^{\prime}))$.

Notice that $\rho(w)$ is a translation along a geodesic beginning in
$H_{a^{-1}}$ and ending in $H_{a}$ while the axis of $\rho(w^{\prime})$ begins
in $H_{b^{-1}}$ and ends in $H_{a}$.

If the axes intersect then
$\ell(\rho(ww^{\prime}))<\ell(\rho(w))+\ell(\rho(w^{\prime}))$. If they do
not, they must be at a positive distance, and since both translations are
toward the endpoint in $H_{a}$ we would have
$\ell(\rho(ww^{\prime}))>\ell(\rho(w))+\ell(\rho(w^{\prime}))$ (see for
example [FF21, Theorem 2.1]).

In both cases
$\ell(\rho(ww^{\prime}))\neq\ell(\rho(w))+\ell(\rho(w^{\prime}))$, from which
we deduce that $\dim(\nu)<\dim(\Lambda)$.

##  5 Acknowledgments

The authors would like to thank Françoise Dal’bo, Sébastien Gouëzel and
François Ledrappier for several helpful conversations. And Steve Lalley, Jean
Mairesse, Pablo Shmerkin, and Wolfgang Woess for directing us towards relevant
literature.

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