# Over 2.7 Million Carbon-Enhanced Metal-Poor stars from BP/RP Spectra in Gaia
DR3

Madeline Lucey1, Nariman Al Kharusi1, Keith Hawkins1, Yuan-Sen Ting2,3, Nesar
Ramachandra4,5, Timothy C. Beers6,7, Young Sun Lee8, Adrian M. Price-Whelan9
and Jinmi Yoon10
1Department of Astronomy, The University of Texas at Austin, 2515 Speedway
Boulevard, Austin, TX 78712, USA
2Research School of Astronomy & Astrophysics, Australian National University,
Cotter Road, Weston Creek, ACT 2611, Canberra, Australia
3Research School of Computer Science, Australian National University, Acton
ACT 2601, Australia
4Computational Science Division, Argonne National Laboratory, 9700 South Cass
Avenue, Lemont, IL 60439, USA
5High Energy Physics Division, Argonne National Laboratory, 9700 South Cass
Avenue, Lemont, IL 60439, USA
6Department of Physics and Astronomy, University of Notre Dame, Notre Dame,
IN. 46556, USA
7JINA Center for the Evolution of the Elements (JINA-CEE), USA
8Department of Astronomy and Space Science, Chungnam National University,
Daejeon 34134, Republic of Korea
9Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue,
New York, NY 10010, USA
10Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218,
USA<EMAIL_ADDRESS>
(Accepted . Received ; in original form )

###### Abstract

Carbon-enhanced metal-poor (CEMP) stars make-up almost a third of stars with
[Fe/H]<–2, although their origins are still poorly understood. It is highly
likely that one type of CEMP star (CEMP-$s$ stars) is tied to mass-transfer
events taking place in binary stars, while another type (CEMP-no stars) has
been suggested to be enriched by the nucleosynthetic yields of the first
generations of stars. Historically, studies of CEMP stars have been explored
in the Galactic halo, but more recently they have also been detected in the
thick disk and bulge components of the Milky Way. Gaia DR3 has provided an
unprecedented sample of over 200 million low-resolution (R$\approx$ 50)
spectra from the BP and RP photometers. In this work, we use XGBoost to
classify these spectra and detect the largest all-sky sample of CEMP stars to
date. In total, we find 2,736,922 CEMP stars, with a contamination rate of 5%.
This sample spans from the inner to outer Milky Way with distances as close as
0.4 kpc from the Galactic center, and as far as $>$ 30 kpc. We also find that
0.28% of these stars are identified as non-single stars in Gaia DR3. By
providing the largest, uniformly analyzed sample of CEMP stars we can further
investigate the frequency of CEMP-s and CEMP-no stars throughout the Galaxy
and constrain the origins of these classes of stars.

###### keywords:

††pubyear: 2021††pagerange: Over 2.7 Million Carbon-Enhanced Metal-Poor stars
from BP/RP Spectra in Gaia DR3–Over 2.7 Million Carbon-Enhanced Metal-Poor
stars from BP/RP Spectra in Gaia DR3

##  1 Introduction

Stellar chemical abundances act as a fossil record of the interstellar medium
(ISM) from the time the star is formed given that a star’s atmospheric
abundances generally do not change over its lifetime. Therefore, it is
generally true that more metal-poor stars formed at earlier times when the
Universe contained fewer metals. Furthermore, the detailed chemical
composition of metal-poor stars can illuminate the early chemical evolution of
the Universe that resulted from the lives and deaths of the first generations
of stars.

Many studies of metal-poor stars have focused on the Galactic halo, where the
metallicity distribution function is dominated by metal-poor stars (e.g.,
Frebel & Norris, 2015). These studies have found a significant fraction of
metal-poor stars in the halo that exhibit high levels of carbon enhancement
([C/Fe] $>$ \+ 0.7) and have been dubbed carbon-enhanced metal-poor (CEMP)
stars (e.g., Beers et al., 1992; Beers & Christlieb, 2005; Christlieb et al.,
2008). This fraction increases with decreasing metallicity in the Galactic
halo, with carbon-enhanced stars making up 10–30% of stars with [Fe/H]<–2 and
$\approx$80% of stars with [Fe/H]<–4 (Lucatello et al., 2006; Lee et al.,
2013; Placco et al., 2014; Yoon et al., 2018). However, as shown in Arentsen
et al. (2022), it can be difficult to compare these fractions across CEMP
samples, given the various selection effects and systematic biases in the
abundance analysis.

Further analysis of CEMP stars have identified a number of subclasses. A
significant fraction of CEMP stars also exhibit enhancement in slow neutron-
capture ($s$-process) elements, and are thus called CEMP-$s$ stars (Beers &
Christlieb, 2005). There are also CEMP-$r$ stars, which show enhancements in
rapid neutron-capture ($r$-process) elements, CEMP-$r$/$s$. which exhibit
enhancements in both $r$\- and $s$-process elements (Gull et al., 2018), and
CEMP-$i$ stars, which show enhancement in intermediate neutron capture
(i-process) elements (Frebel, 2018). Lastly, there are CEMP-no stars which do
not exhibit $s$-, $r$-, or $i$-process enhancements (Norris et al., 2013).
CEMP-$r$(/$s$) and CEMP-$i$ stars are rare, while CEMP-no and CEMP-$s$ stars
are the most common.

It is thought that CEMP-$s$ stars, which are more common at [Fe/H]$>$–3.0, are
the result of chemical enrichment by mass-transfer events from
(post-)asymptotic giant branch (AGB) stars (Lugaro et al., 2012; Placco et
al., 2013). This hypothesis is strongly supported by the high rate of binarity
among CEMP-$s$ stars (McClure & Woodsworth, 1990; Preston & Sneden, 2001;
Lucatello et al., 2005; Bisterzo et al., 2010; Abate et al., 2015; Hansen et
al., 2016b; Jorissen et al., 2016). In fact, binarity rates as high as 82%
have been reported for CEMP-$s$ stars (Hansen et al., 2016b).

On the other hand, CEMP-no stars are not typically in binary systems, and
therefore are not likely to have experienced a mass-transfer event
(Starkenburg et al., 2014; Hansen et al., 2016a; Arentsen et al., 2019).
Hence, CEMP-no stars likely formed from an ISM that was already carbon
enhanced. Given their low metallicity and increasing frequency at lower
metallicities, it is thought that these stars are truly ancient stars, and
were primarily enriched by the first generation of stars. It has been
suggested that the first stars may have had high rotation rates, which would
lead to large carbon production (Chiappini et al., 2006; Meynet et al., 2006).
Furthermore, it is possible that the first stars exploded as faint supernovae,
which also overproduce carbon (Umeda & Nomoto, 2003; Nomoto et al., 2013;
Tominaga et al., 2014).

Initial studies indicate that the frequency of CEMP-$s$ and CEMP-no stars
varies throughout the Galaxy. Specifically, the number of CEMP stars appears
to increase with increasing distance from the Sun, although we note that these
studies are mostly focused on the Galactic halo (Carollo et al., 2012; Frebel
et al., 2006; Lee et al., 2017; Yoon et al., 2018). Furthermore, the relative
fraction of CEMP-no stars compared to CEMP-$s$ stars also increases at larger
distances (Carollo et al., 2014; Lee et al., 2019). Ultra faint dwarf galaxies
have shown similar fractions of CEMP-no stars as the Milky Way, but dwarf
spheroidal galaxies have a clear deficit of CEMP-no stars (Norris et al.,
2010; Lai et al., 2011; Frebel et al., 2014; Skúladóttir et al., 2015;
Salvadori et al., 2015). In a comparative study between Galactic halo and
dwarf galaxy CEMP stars, Yoon et al. (2019) suggests that the majority of
Galactic halo CEMP-no stars have been accreted from dwarf galaxies.
Furthermore, CEMP-no and CEMP-$s$ stars have been discovered in the metal-weak
thick disk (MWTD; Beers et al., 2017). Dietz et al. (2021) tentatively tags
the retrograde MWTD CEMP-no population to the Gaia-Enceladus system, while
suggesting the equivalent prograde population has both in-situ and ex-situ
origins.

There are fewer studies of metal-poor stars towards the center of the Galaxy
than compared to the Galactic halo. This is partly due to the difficulty of
targeting metal-poor stars in a region of the Galaxy that is dense with metal-
rich stars. Furthermore, high levels of extinction demand long exposure times
and large-aperture telescopes in order to achieve sufficient signal-to-noise
spectroscopic observations for metallicity measurements. Fortunately, the
advent of metallicity-sensitive photometric surveys (e.g., Skymapper and
Pristine; Bessell et al., 2011; Wolf et al., 2018; Starkenburg et al., 2017)
have led to studies of thousands of metal-poor inner Galaxy stars. Studies
using SkyMapper photometry have found a much lower fraction of CEMP stars in
the inner Galaxy compared to the Galactic halo (Howes et al., 2014, 2015,
2016; Lucey et al., 2022). However, metallicity estimates from Skymapper
photometry have proven to be biased against CEMP stars (Da Costa et al., 2019;
Chiti et al., 2020). Targeting with Pristine photometry, Arentsen et al.
(2021) found a CEMP frequency that is consistent with the Galactic halo for
stars with [Fe/H] <–3, but also found that it is much lower than the halo at
higher metallicities.

Measuring and understanding the frequency and relative rates of CEMP-
no/CEMP-$s$ stars throughout the Galaxy will be crucial for shedding new light
on the origins and formation mechanisms of these stars, including whether or
not CEMP-no stars are true inheritors of the elements created by the first
stars. Given that the measured properties of CEMP samples have been shown to
vary across different samples (Arentsen et al., 2022), creating a uniformly
analyzed sample with limited selection effects across the Milky Way will be
essential for achieving this goal. The release of the Gaia BP/RP spectra in
DR3 presents a unique opportunity to discover the largest, all-sky sample of
CEMP stars to date. However, the BP/RP spectra have quite low resolution (R=
$\lambda/\Delta\lambda$ $\approx$ 50), and require unconventional methods for
analysis.

In this work, we present a novel method for detecting CEMP stars in the Gaia
BP/RP spectra with machine learning, specifically the XGBoost classification
algorithm, and apply it to the spectra released in DR3. In Section 2 we
describe the BP/RP spectra along with other data used in our analysis. We
introduce XGBoost, our chosen classification algorithm in Section 3. We
evaluate the accuracy and completeness of our classification in Section 4. In
Section 5, we interpret the XGBoost model. Finally, in Section 6 we present
the sample along with an investigation into the binarity rate and Galactic
distribution of these stars.

##  2 Data

Figure 1: Examples of synthetic mock BP and RP spectra (top) and the impact of
carbon enhancement on the spectra (bottom). Specifically, in the top panel we
compare stars of the same stellar parameters and reddening, except with
different carbon abundances. The BP and RP spectra are plotted separately,
with the BP spectra at lower wavelengths. The BP and RP spectra overlap at
$\approx$650 nm. The dotted lines have [C/Fe]=+0.5 while the solid lines have
[C/Fe]=+1.0. Starting from the dark blue spectrum, which has typical stellar
parameters for a metal-poor giant (Teff=4500 K, log $g$=2.5 and [Fe/H]=–2.0),
we change one stellar parameter or extinction value at a time to observe how
the spectra are impacted. The green spectra have increased extinction with
$\rm{A_{V}}$=9.0 mag. The light blue spectra are hotter, with Teff=5500 K. The
red spectra are more metal-rich, with [Fe/H]=0.0. In the bottom panel, we have
subtracted the dotted spectrum ([C/Fe]=+0.5) from the solid spectrum
([C/Fe]=+1.0) of the same stellar parameters and reddening. The colors of the
lines correspond to the spectra that were subtracted. The impact of carbon on
the spectra changes drastically with the stellar parameters; we therefore
require a flexible classification model in order to achieve low contamination
of our detected carbon-enhanced stars.

The Gaia mission has revolutionised Milky Way astronomy and beyond, primarily
by providing astrometric data for billions of stars (Gaia Collaboration et
al., 2016; Gaia Collaboration et al., 2022a). Simultaneously, the Gaia mission
has also been collecting low-resolution spectra (R$\approx$ 50) with the blue
photometer (BP) and red photometer (RP). These spectra will provide effective
temperature (Teff), surface gravity (log $g$), and metallicity ([M/H])
estimates (Liu et al., 2012), but have too low of resolution to provide
further elemental abundances (Gavel et al., 2021). Molecules, however, absorb
large bands of light, and therefore may be easier to detect in the BP/RP
spectra. Given the wavelength coverage of 3300–10500 Å (Carrasco et al.,
2021), we expect to be able to detect carbon-enhanced stars from the plethora
of carbon molecular bands in that range (e.g., $\rm{C_{2}}$ Swan bands at
$\approx$ 4500–6000 Å, and CN bands at $\approx$ 7000–10500 Å).

To explore the impact of carbon on the BP/RP spectra, we create mock synthetic
spectra. We employ the MARCS carbon-enhanced model atmosphere grids
(Gustafsson et al., 2008) and the TURBOSPECTRUM radiative transfer code
(Alvarez & Plez, 1998; Plez, 2012) to construct these spectra. We also use the
fifth version of the Gaia-ESO atomic linelist (Heiter et al., 2020) with the
addition of molecular lines for CH (Masseron et al., 2014), CN, NH, OH, MgH,
$\rm{C_{2}}$ (T. Masseron, private communication), SiH (Kurucz
linelists111http://kurucz.harvard.edu), and TiO, ZrO, FeH, CaH (B. Plez,
private communication). To apply the instrumental profile of the BP/RP
photometers, we use the DR3 calibrated passbands (Riello et al., 2021). The
true spectral resolution is a function of wavelength for both BP and RP
spectra, with the BP spectra ranging from R$\approx$100 at the blue edge to
R$\approx$30 at the red edge, and the RP spectra ranging from R$\approx$100 at
the blue edge to R$\approx$70 at the red edge. To simplify the calculation, we
assume a resolution of 50 for the BP spectra and 70 for the RP spectra. We
also model the impact of dust extinction using the
DUST_EXTINCTION222https://github.com/karllark/dust_extinction package with the
extinction curve from Fitzpatrick (2004), assuming $R_{V}$=3.1.

Figure 1 shows examples of a number of synthetic mock spectra, along with the
impact of carbon on the calculated flux. The top panel shows eight spectra, in
sets of two with the same stellar parameters, except one spectrum has
[C/Fe]=+0.5 (dotted line) and the other has [C/Fe]=+1.0 (solid line). We start
with the stellar parameters of a typical metal-poor giant star (Teff=4500 K,
log $g$=2.5 and [Fe/H]=–2.0), which is shown as the dark blue spectra. We then
change one parameter at a time to test how the carbon enhancement impacts
spectra of different stellar parameters and reddening. The green spectra have
$\rm{A_{V}}$=9.0 mag, the blue spectra have Teff=5500 K and the red spectra
have [Fe/H]=0.0. This is a worst case example, given that we expect very few
spectra released in Gaia DR3 to have $\rm{A_{V}}$ $\geq$9.0 mag, given that
most stars have G<17.6 mag. In the bottom panel, we subtract the spectrum with
[C/Fe]=+1.0 from the spectrum with the same stellar parameters, but with
[C/Fe]=+0.5.

The impact of carbon enhancement on the BP/RP spectra (Fig. 1) is very
dependent on the stellar parameters and reddening. The metal-rich spectra
(with [Fe/H]=0.0), have the largest difference between the carbon-enhanced and
carbon-normal stars. This is likely because they have the largest [C/H], with
[C/H]=+1.0 and [C/H]=+0.5 compared to the metal-poor spectra ([Fe/H]=–2.0)
with [C/H]=–1.0 and [C/H]=–1.5. Generally, when [C/H] is larger, the relative
impact of carbon molecules on stellar spectra for the same $\Delta$[C/H] is
larger. Compared to the the standard metal-poor giant (dark blue; Teff=4500
K), the hotter star (light blue; Teff=5500 K), is generally more impacted by
the carbon enhancement at bluer wavelengths. This is likely a consequence of
the overall higher flux at bluer wavelengths due to the hotter star’s bluer
blackbody peak. As the spectra are not continuum normalized, but merely
normalized by the maximum flux, the blackbody continuum does impact the
relative effect of carbon enhancement on the spectra. Furthermore, reddening
greatly reduces the flux, and therefore the strength of the carbon-enhancement
signal in the BP spectra. Fortunately, there are carbon molecules (e.g., CN
bands at $\approx$ 7000–10500 Å) that impact the RP spectra so we are still
able to detect reddened carbon-enhanced stars. These four combinations of
stellar parameters and reddening values has lead to significantly different
results for the impact of carbon enhancement on the BP/RP spectra. As we hope
to be able to detect carbon-enhanced stars across a wide range of stellar
parameters and reddenings, we require a complex model that can adapt to the
different signals of carbon enhancement. To balance complexity with
interpretability, we choose XGBoost as our algorithm for detecting carbon-
enhanced stars.

###  2.1 Training and Testing Sample

Figure 2: Color-magnitude diagram of our training sample which is taken from
the SEGUE (Lee et al., 2013) and AEGIS surveys (Yoon et al., 2018). The x-axis
is the Gaia $\rm{G_{BP}-G_{RP}}$ color; the y-axis is the absolute G
magnitude, calculated from the distance modulus using the Gaia apparent G
magnitude and parallax. The SEGUE sample, which primarily comprises main-
sequence turn-off and blue horizontal-branch stars, is shown in the left-hand
panel. In the second panel the AEGIS sample is shown, which includes giants
and dwarfs. The third panel shows all of the stars with [C/Fe]$>$+0.7 from
both the AEGIS and SEGUE surveys. These stars are primarily turn-off and giant
stars, with few dwarf stars. The far right panel shows our final training
sample with the AEGIS and SEGUE samples combined. The black dashed lines
correspond to the color and magnitudes cuts made on our training/testing
sample.

In order to teach our model how to accurately detect carbon-enhanced stars, we
require a sample of stars that are already classified. To acquire this, we use
the spectroscopic catalogs from the Sloan Extension for Galactic Understanding
and Exploration (SEGUE; Yanny et al., 2009; Lee et al., 2013; Rockosi et al.,
2022) and AAOmega Evolution of Galactic Structure (AEGIS; Yoon et al., 2018)
surveys. The SEGUE survey was a sub-survey of the Sloan Digital Sky Survey
(SDSS-II and SDSS-III; Fukugita et al., 1996; Gunn et al., 1998, 2006; York et
al., 2000; Stoughton et al., 2002; Abazajian et al., 2003, 2004, 2005, 2009;
Pier et al., 2003; Adelman-McCarthy et al., 2006, 2007, 2008; Aihara et al.,
2011; Ahn et al., 2012), and obtained $\approx$500,000 low-resolution (R=2000)
optical spectra. However, these spectra are primarily obtained for stars in
the Northern Hemisphere, so we supplement this sample with stars from the
Southern AEGIS survey. The AEGIS survey observed $\approx$70,000 low-
resolution (R$\approx$1,300) optical spectra, from which both metallicity and
carbon abundance could be measured. For both of these surveys, we use the
measured carbon abundances that are not corrected for evolutionary effects
(e.g., the first and second dredge ups).

For the SEGUE survey data, we start with 52,273 stars from T. Beers (2022,
personal communication), of which 33,582 have [C/Fe]$>$+0.7. We find that
15,508 of the 52,273 stars have BP/RP spectra released in Gaia DR3, of which
8,885 ($\approx$57%) have [C/Fe]>+0.7. In preliminary tests, we found that our
algorithm is incapable of detecting carbon enhancement in hot stars so we
apply a cut in the $\rm{G_{BP}-G_{RP}}$ color. Specifically, we only include
stars with $\rm{G_{BP}-G_{RP}}$ $>$ 0.8 mag. This leaves us 1,279 carbon-
enhanced stars and 1,316 carbon-normal stars.

From the AEGIS Survey, we start with a total of 58,015 stars that have
measured carbon abundances, of which 1,476 have [C/Fe]$>$+0.7. We find that
36,169 of the total 58,015 stars have BP/RP spectra released in Gaia DR3. Of
those, only 876 have [C/Fe]>+0.7. After applying the color cut we have 461
carbon-enhanced stars and 29,503 carbon-normal stars.

In total, we employ 32,589 spectra for the training and testing, of which
1,740 have [C/Fe]>+0.7. We randomly select $\approx$40% of this data as our
testing sample, while the remaining $\approx$60% is used for training. Figure
2 shows color-magnitude diagrams of our training/testing sample. Specifically,
we show the Gaia DR3 $\rm{G_{BP}-G_{RP}}$ color on the x-axis and the absolute
G magnitude calculated using the Gaia parallax and apparent G magnitude on the
y-axis. The leftmost panel shows the 2,595 stars from the SDSS/SEGUE survey
that we use in the training, while second panel from the left shows the same
for the 29,964 of stars from the AEGIS survey. The AEGIS and SDSS samples have
significantly different distributions in the color-magnitude diagram, and
therefore evolutionary states. This is likely a result of the different
selection functions of the two surveys. The SDSS sample specifically targeted
bluer stars (including blue horizontal-branch stars) in order to obtain
photometric calibrating stars. In the rightmost panel we show the distribution
of the entire training set of 32,589 of stars. We use this distribution to
determine for which stars our algorithm will be able to detect carbon
enhancement, while avoiding extrapolation. The second panel from the right
shows the part of the sample that has [C/Fe] > +0.7. As the majority of the
carbon-enhanced sample comes from the SDSS survey, the distribution of these
stars in the color-magnitude diagram closely matches that of the SDSS sample.
It is possible that this may introduce a bias in our classification which we
can investigate by evaluating the precision and completeness of the
classification as a function of color and absolute G magnitude (see Section
4).

###  2.2 Gaia BP/RP Spectra

The Gaia BP/RP spectra are a unique data set, not only due to their wide
wavelength coverage, low-resolution (R$\approx$50), and unprecedented number
of stars, but also because the spectra have been released as a linear
combination of basis functions, specifically as Hermite function coefficients
(Carrasco et al., 2021). This was done because of the complexity of the Gaia
instrument, which has two wide fields of view and 14 detectors. To create the
calibrated mean spectra, multiple epochs of observations with different
instrumental conditions need to be combined. In this work, we use the
coefficients as the input data for our model rather than convert them to
sampled spectra, which results in some information loss (Carrasco et al.,
2021).

In total, the BP and RP spectra comprise 55 coefficients each, but also come
with a recommended truncation. This is possible because the coefficients have
been rotated to an optimized basis so that the bulk of the spectral
information is contained in the first few coefficients. A truncation is then
recommended based on the magnitude of the coefficients compared to their
corresponding uncertainties. For more details see Carrasco et al. (2021). As
XGBoost requires the input data to be vectors of the same length, we apply the
largest recommend truncation to avoid losing potentially useful information.
The largest recommended truncation is 55 for both BP and RP spectra (i.e., all
coefficients are relevant). Therefore we do not truncate the coefficients.
Because we do not want to include apparent magnitude information, we normalize
the coefficients by the maximum of the combined BP and RP coefficients.

The XGBoost algorithm cannot reliably extrapolate. Therefore, we ensure to
only classify stars that are similar to our training sample. Specifically, we
constrain our sample using the absolute G magnitude and $\rm{G_{BP}-G_{RP}}$
color. As shown in rightmost panel of Figure 2 in the black dotted lines, we
only classify stars that satisfy the following criteria:

  1. 1.

$0.8<\rm{G_{BP}-G_{RP}}<2.0$

  2. 2.

$\rm{G+5\rm{log_{10}(\varpi)}+5}<7.0$.

Although our training sample includes stars outside of this range, we chose to
restrict to these values where most of the sample resides. Note that we do not
place a lower limit on the absolute G magnitude, since stars with very low
absolute G magnitudes likely have small, uncertain parallaxes that cause an
overestimation in the brightness, and therefore underestimate in the absolute
G magnitude. As we do not want to introduce a selection bias by removing stars
with uncertain parallaxes, we chose to include them. In Section 2.1, we
investigate how the precision and completeness of our classification behaves
at the edges of this region where the training sample is less dense. In total,
we find 160,422,878 BP/RP spectra in Gaia DR3 that are within our color and
magnitude cuts.

##  3 XGBoost

Figure 3: General architecture of XGBoost for classification. In short,
XGBoost iteratively creates trees to fit the residuals from the prediction of
the previous tree. The first tree made gives a prediction of zero for each
spectrum. For the final output, the predictions of each tree are summed after
being mulitplied by the learning rate, $\eta$. This value is then input into
the sigmoid function, $\sigma(x)=1/(1+e^{-x})$, to calculate the final
probability, $\hat{y}$. If $\hat{y}>$0.5, then the star is classified as
carbon enhanced.

In order to detect carbon-enhanced stars across a wide range of stellar
parameters and reddenings, we require a flexible model. We chose to use
XGBoost, which is powerful but still easy to interpret. Furthermore, XGBoost
is optimized for efficiency, allowing fast training and inference. XGBoost is
quickly becoming a popular machine learning algorithm in astronomy, with
applications in a large variety of sub-fields (e.g., Hayden et al., 2020;
Machado Poletti Valle et al., 2021; Li et al., 2021; He et al., 2022; Pham &
Kaltenegger, 2022)

Figure 3 shows the general architecture of XGBoost for classification. For a
detailed description of the algorithm see Chen & Guestrin (2016). In short,
XGBoost sequentially builds decision trees to fit the residuals from the
previous tree. XGBoost continues to train trees until it reaches the maximum
number of trees set by the user or the residuals stop consistently shrinking.
The results from each tree are then summed together, weighted by the learning
rate, $\eta$. This value is then plugged into the sigmoid function,
$\sigma(x)=1/(1+e^{-x})$, to calculate the probability that the star is carbon
enhanced ($\hat{y}$). If $\hat{y}$ > 0.5, then the star is classified as
carbon enhanced.

To train the XGBoost algorithm, a number of hyperparameters need to be set.
Namely, we can set the maximum number of trees, the learning rate ($\eta$),
the percentage of the training sample and the percentage of input coefficients
to use for each tree, as well as the maximum depth of each tree. We can also
set limits on the purity of a sample for a given leaf to prevent overfitting.
To explore the parameter space and find the optimal set of hyperparameters, we
use RandomSearchCV from scikit-learn (Pedregosa et al., 2011).

##  4 Precision and Completeness

Figure 4:  The confusion matrix describing the precision and completeness of
our classification for the testing sample. The top-left box shows the number
of true positives, along with the precision, which is calculated by taking the
true positives divided by the total predicted positives (left column). The
number of false negatives are shown in the top-right box and the true
negatives are shown in the bottom-right box, respectively. We miss
$\approx$27% (false negatives divided by the sum of the true positives and
false negatives) of the carbon-enhanced stars in our testing sample.  Figure
5:  The false positive and true positive rates, as functions of absolute G
magnitude, $\rm{G_{BP}-G_{RP}}$ color, [C/Fe], and [Fe/H]. We show the
distribution of stellar parameters for the testing sample in a grayscale, with
darker areas corresponding to more stars. In general, the model tends to
struggle most with redder stars, which are likely cooler and more extincted.
Furthermore, the false positives tend to have +0.5 $\leq$[C/Fe]$\leq$+0.7,
while the false negatives tend to have +0.7$\leq$[C/Fe]$\leq$+1.0. Therefore,
our model likely only can interpret the carbon abundance to $\approx$+0.5 for
some stars.  Figure 6: The contamination rate and true positive percentage, as
a function of G magnitude and absolute Galactic latitude, as a proxy for
extinction. The gray bins show the arbitrarily scaled number density of stars
as a function of absolute Galactic latitude (top panel) and G magnitude
(bottom panel). For each panel, the scale for the true positive percentage
(dark blue) is on the left y-axis; the right y-axis shows the scale for the
contamination rate (green). As expected, we find that our classification
improves for bright stars, which likely have high signal-to-noise spectra, and
stars at high absolute Galactic latitude, which is consistent with low
extinction.

In order to estimate the true positive rate (i.e., precision) and completeness
of our sample of newly identified CEMP stars, we use our testing sample
(described in Section 2.1) where we already know the observed carbon abundance
from the catalogs of Lee et al. (2013) and Yoon et al. (2018). As we constrain
the full set of spectra that we classify by the absolute G magnitude and
$\rm{G_{BP}-G_{RP}}$ color of the training/testing sample (see Section 2.2),
we can be confident that the precision and completeness estimates are
representative of the full sample. In addition, we have stellar parameters and
carbon abundances for these stars, which allows us to study how the precision
and completeness behaves as a function of various parameters, including
observational effects.

Figure 4 shows the confusion matrix which shows the number of true and false
positives along with the number of true and false negatives for our entire
testing sample. The boxes are colored by the number of stars that fall in that
part of the table. Stars which are known to be carbon enhanced from the SEGUE
and AEGIS data are in the top row while the known carbon normal stars are in
the bottom row. If the XGBoost algorithm classifies the star as carbon
enhanced it goes in the first column, while stars classified as carbon-normal
go in the second column. Therefore, the top-left box and the bottom-right box
contain stars that are correctly classified, while the bottom-left box and
top-right box contain stars that were falsely classified. From these numbers,
we can calculate the precision and the completeness percentage. To calculate
the precision, we divide the number of true positives (488) by the summation
of the false positives (24) and true positives. We measure a precision of 95%.
We also calculate our completeness percentage by dividing the number of true
positives by the summation of the true positives and the false negatives
(181). We achieve a completeness of 73%.

Figure 5 shows 2D maps of the false positive (left) and true positive rates
(right) as a function of absolute G magnitude, and $\rm{G_{BP}-G_{RP}}$ color
(top), as well as [C/Fe] and [Fe/H] (bottom). In these plots, we also show the
underlying density distribution of stars as gray hexagonal bins. The left
panels show the false positive rate, which is calculated by taking the number
of false positives divided by the number of true carbon-normal stars in that
bin. The right panels show the true positive rate, which is calculated by
dividing the number of true positives (i.e., correctly identified CEMP stars)
by the total number of true CEMP stars in that bin.

In the top row of Figure 5, we do not see a trend in the false positive and
true positive rate with the $\rm{G_{BP}-G_{RP}}$ color or absolute G
magnitude. In the top right panel, we see that at the edges of our
distribution the true positive percentage highly varies. These is merely
because of the few true positives in these bins.

In the bottom row of Figure 5, we see a slight trend in the false positive and
true positive rates with the [C/Fe] abundance. Specifically, we see the false
positive rate is higher for stars with +0.5<[C/Fe]<+0.7, and the true positive
rate is lowest for stars with +0.7<[C/Fe]<+1.0. This indicates the our
classification is most inaccurate for stars with [C/Fe]$\approx$+0.7, which is
to be expected given that is is unlikely we could measure a [C/Fe] abundance
from these extremely low-resolution (R$\approx$50) spectra that is more
precise than $\approx$0.2 dex, at best.

Figure 6 shows the true positive percentage and contamination rate, as a
function of the absolute Galactic latitude (top panel) and G magnitude (bottom
panel). Here, we use the absolute Galactic latitude as a proxy for the
extinction, given that stars with low absolute Galactic latitudes will be in
the disk, and be more impacted by extinction than stars with high absolute
Galactic latitudes. The gray histogram shows the arbitrarily scaled underlying
distribution of the testing sample for these parameters. Here the
contamination rate is calculated by taking the number of false positives
divided by the total number of stars classified as CEMP stars in each
parameter bin. The green line shows the contamination rate with the scaling
provided by the right y-axis, while the dark blue lines shows the true
positive percentage (calculated in the same way as Figure 5), with
corresponding scaling on the left y-axis.

As expected, we find that the classification performs better at high absolute
Galactic latitudes where extinction is low. Specifically, we see that the true
positive percentage increases from $\approx$70% at Galactic latitudes <|10|∘
to $\approx$79% at Galactic latitudes >|80|∘. We also see that the
contamination rate drops from $\approx$10% to $\approx$0% over the same range.
Furthermore, we find that the classification also improves with brighter G
magnitude, as expected, since the G magnitudes is directly related to the
signal-to-noise for these spectra. We find that the true positive percentage
increases from $\approx$70% at G of $\approx$ 17 mag to $\approx$100% at G
=12.5 mag. Over the same range, we find that the contamination rate drops from
$\approx$5% to $\approx$0%. Therefore, we recommend using bright, high
absolute Galactic latitude stars to retrieve a higher fidelity sub-sample for
our provided catalog of carbon-enhanced candidate stars.

##  5 Model Interpretation

Figure 7:  The SHAP values, as a function of [C/Fe] (x-axis), for the four
spectral coefficients that are most correlated with the carbon abundance. Each
point corresponds to an individual star in our testing sample and are colored
by the given coefficient’s value. We show a horizontal black dashed line at
SHAP value=0 and a vertical black dashed line at [C/Fe]=+0.7. Large positive
SHAP values indicate that the measured coefficient value greatly increased the
inferred probability of the star being carbon enhanced. Therefore, if the
model uses the carbon information in the coefficients, we expect that stars
with [C/Fe]>+0.7 to have high positive SHAP values.

As we have used a data-driven method to classify stars in this work, we want
to ensure that the final model matches our physical intuition. It is also
important to ensure that the model is not using any confounding variables
(e.g., extinction or metallicity) that could be correlated with carbon
enhancement but are not a direct measurement of the true carbon abundance.
Typically, model interpretation is done by determining the importance of each
input feature for the final inference and comparing to expectations from
physical models. Here, we investigate the importance of two key Hermite
coefficients and how they relate to the true carbon abundance.

We measure the impact of each Hermite basis coefficient on the classification
using SHAP (Shapley Additve exPlanations) values (Lundberg & Lee, 2017). SHAP
values allow us to explore the importance of individual features, as a
function of various parameters, since each star’s coefficients are assigned
individual values. The SHAP values are defined so that their summation plus
the average predicted value ($\phi_{0}$) is equal to the predicted value
($\hat{y}$) . Explicitly,

$\hat{y}=\phi_{0}+\sum_{i=1}^{M}\phi_{i}$ (1)

where $\phi_{0}$ is the average value of $\hat{y}$ for all of the spectra,
$\phi_{i}$ is the SHAP value for coefficient $i$, and $M$ is the total number
of coefficients per spectrum. Therefore, each SHAP value directly measures the
impact of the selected coefficient on the inference of $\hat{y}$ for a given
star. We refer the interested reader to Lundberg & Lee (2017) for further
details on the calculation of SHAP values.

To decide which spectral coefficients contain the majority of information
about the carbon abundance, we calculate the Pearson Correlation coefficient
for each coefficient and the carbon abundance for the entire testing sample.
We find that the second, third, fifth and sixth BP coefficients have the
highest absolute Pearson Correlation coefficients. Therefore we check to
ensure that our XGBoost model gives high SHAP values for these coefficients
for carbon-enhanced stars.

We calculate the SHAP values for all of the spectra in our testing sample.
Figure 7 shows how the SHAP values relate to the carbon abundance for the four
selected coefficients that have the largest Pearson Correlation coefficients.
Specifically, we have the [C/Fe] on the x-axis and the SHAP values on the
y-axis. We show the points colored by the coefficient’s value. As large
positive SHAP values indicate that the given coefficient increased the
probability that the star is carbon enhanced, it is expected that stars with
[C/Fe]>+0.7 should have higher SHAP values than carbon-normal stars. We find
that the XGBoost model is able to pick up this sensitivity, in that the SHAP
values for these coefficients are large and positive for carbon-enhanced
stars. Although there are some coefficients were the SHAP value is negative
for a carbon-enhanced star, these stars may still be classified as carbon-
enhanced based on the value of other coefficients. Similarly, these are many
carbon-normal stars that have positive SHAP values, but the combined effect of
all of the other coefficients effectively decrease the probability of the star
being carbon enhanced, so that the false positive rate is not exceedingly
high. Given the positive correlation between [C/Fe] and the SHAP value, it is
likely that XGBoost model is using the carbon information in the spectra to
determine whether a given star is carbon enhanced rather than using another
confounding variable.

##  6 Properties of the CEMP Sample

Figure 8: The behavior of the occurrence and binarity rate of carbon-enhanced
stars as a function of metallicity ([M/H]) provided by the Gaia DR3 GSPPHOT
pipeline. We also plot the arbitrarily scaled metallicity distribution of our
CEMP sample in gray. In dark blue we show the fraction of stars with [M/H]<$x$
that are classified as carbon enhanced out of the total number of BP/RP
spectra that we classify in a given metallicity bin. In green we plot the
fraction of carbon-enhanced (CE) stars that are labeled as non-single in Gaia
DR3. We also plot the fraction of non-single stars for the carbon-normal (non-
CE) population in light blue for comparison. It is likely the metallicities
are overestimated for our CEMP sample.  Figure 9:  The Galactic distribution
of our carbon-enhanced (CE) sample. Specifically, in the left panel, we show
the distribution of stars in the Galactic coordinates X and Y. The right panel
shows the distribution in the Galactic X and Z coordinates. The Galactic
center is located at (0,0,0) kpc while the Sun is at (8.3,0,0) kpc. The
distribution of our CEMP sample is very halo-like and is quite extended rather
than staying confined to low $Z$ as one would expect for a disk population.
Figure 10:  The Galactic distribution of our carbon-enhanced (CE) sample
compared to the non-carbon-enhanced (non-CE) sample. Specifically, in the left
panel, we show the reverse cumulative density distributions of the distance
from the Galactic plane (|Z|) for both samples, while the right panel shows
the density distributions of the distance ($\rm{R_{GC}}$) from the Galactic
center. We also show the Sun’s position at 8.3 kpc as a black dashed line. The
carbon-enhanced stars are generally farther from the Galactic center and the
Galactic plane than the non-carbon-enhanced stars.

Out of the $\approx$160 million stars that we classify, we find 2,736,922 CEMP
stars. This is the largest, homogeneously identified sample of CEMP stars to
date. In this section, we briefly investigate a few properties of this sample,
including the binarity rate and the Galactic distribution.

For each star in Gaia DR3, a flag indicating whether the star is an
astrometric, spectroscopic, or eclipsing binary is provided (Gaia
Collaboration et al., 2022b). We use these flags and the reported
metallicities from the Gaia GSPPHOT pipeline (Andrae et al., 2022), along with
our classification, to explore the relationship between metallicity and
binarity for carbon-enhanced stars. In Figure 8, we plot the occurrence and
binarity rate of CE stars as a function of metallicity. In addition, we show
the metallicity distribution of our sample of CEMP stars in a gray histogram,
arbitrarily scaled. Given that carbon-enhanced stars were not accounted for,
it is likely that the [M/H] from the GSPPHOT pipeline and the iron abundance
of these stars are actually much lower.

Specifically, in dark blue, we show the ratio of the number of stars
classified as CE to the total number of BP/RP spectra we analyzed for
[M/H]<$x$. In other words, the dark blue line gives the fraction of CEMP stars
for stars with [M/H]$<$x. Compared to results from the literature, we find a
fraction that is roughly an order of magnitude lower (Lucatello et al., 2006;
Lee et al., 2013; Placco et al., 2014; Yoon et al., 2018; Arentsen et al.,
2022). This may be a result of low completeness in that we are could be
classifying many CEMP stars as carbon-normal. From our testing sample we
estimated our completeness as 73%, but these results may indicate it it closer
to 10%. Another possibility is that there may be inaccuracies with the [M/H]
measurements from the Gaia DR3 GSPPHOT pipeline. It is quite unexpected that
the CEMP fraction for stars with [M/H]$<$0 would be higher than for
[M/H]$<$-2.5. This is likely a result of the [M/H] for CEMP stars being
overestimated since carbon-enhanced stars were not included in the models used
for the analysis in the GSPPHOT pipeline. Furthermore, the GSPPHot pipeline
may be over estimating the number of metal-poor stars observed which would
cause our CEMP fraction to be underestimated.

In addition, we also show the fraction of non-single stars among the CE
(green) and non-CE (light blue) populations. As expected from the literature
(Hwang et al., 2021), the binary fractions generally increase with increasing
[M/H]. It is interesting that the fraction of CEMP in binaries is largely than
the fraction of carbon-normal stars in binaries at high metallicities. This
may be a result of CEMP-$s$ stars having a higher binarity rate than carbon-
normal stars (McClure & Woodsworth, 1990; Preston & Sneden, 2001; Lucatello et
al., 2005; Bisterzo et al., 2010; Abate et al., 2015; Hansen et al., 2016b;
Jorissen et al., 2016). Although these stars have [M/H]$>$-0.5 from the
GSPPHOT pipeline, it is likely that their iron abundance is actually much
lower. This effect is likely increased for CEMP-$s$ stars because of the high
abundance of $s$-process elements.

Figure 9 shows the Galactic distribution of the CEMP sample. To calculate the
Galactic positions of each CEMP star we detect, we simply invert the reported
Gaia eDR3 parallax. We choose not to use distances that are calculated using
photometry since it is unknown how the carbon enhancement will bias these
results. As we want to compare to the Galactic distribution of carbon-normal
stars, we want to ensure we do not introduce such a bias. We show the
distribution of our CEMP sample in Galactic X and Y coordinates in the right
panel, while the left panel shows the distribution in Galactic X and Z
coordinates. The Galactic center is located at (0,0,0) kpc with the Sun
located at (8.3,0,0) kpc (Reid et al., 2014).

Consistent with a Galactic halo population, our CEMP sample is extended to far
distances from the Galactic center. It is important to note that some of the
largest distances may be unreliable given that the fractional parallax
uncertainty for faint stars can be quite large. Given that the majority of
BP/RP spectra released in DR3 have G<17.6 mag and assuming an absolute G
magnitude of -2.5 mag (roughly the tip of the red giant branch), we expect to
have detected CEMP stars at distances of up to $\approx$30 kpc. We find that
45,423 stars in our CEMP sample that have parallaxes corresponding to
distances from the Galactic center $>$30 kpc. Therefore, it is likely that we
have detected CEMP stars in other galaxies.

In addition, we find there is a dearth of stars at low distances from the
plane ($Z$), especially towards the Galactic center. Although there are fewer
than expected for a disk or bulge population, we do still find a number of
stars in the inner Galaxy, with distances as low as $\approx$0.4 kpc from the
Galactic Center. However, is is difficult to determine how our observed
distribution is impacted by the selection function of our algorithm and Gaia
DR3. Further work is required to fully understand and model the selection
function of our sample which will enable us to draw strong conclusions from
population statistics.

We also investigate the relative Galactic distribution of CEMP stars compared
to carbon-normal stars in Figure 10. Specifically, in the left panel, we show
the reverse cumulative density distribution of the distance from the Galactic
plane (|Z|) in green for CEMP stars and in blue for carbon-normal stars. We
find that the CEMP stars tend to be farther from the Galactic plane than the
carbon-normal stars. This is consistent with previous work, which found the
frequency of CEMP stars to increase with increasing distance from the Sun
(Carollo et al., 2012; Frebel et al., 2006; Lee et al., 2013, 2017; Yoon et
al., 2018). In addition, the right panel shows the density distribution of the
distance from the Galactic center for the CEMP stars (green) and the carbon-
normal stars (blue). We also mark the distance of the Sun from the Galactic
center as a black dashed line at 8.3 kpc. In general, we find that the CEMP
stars are more distant from the Sun than the carbon-normal stars. Principally,
we find a higher density of CEMP stars in the outer regions of the Galaxy,
with distances from the Galactic center $>$10 kpc. This indicates these stars
may originate in the outer Galactic halo, as suggested by Yoon et al. (2018,
2019).

##  7 Summary

The origins of CEMP stars are poorly understood, even though they comprise
$\approx$30% of stars with [Fe/H]<–2 (Lucatello et al., 2006; Lee et al.,
2013; Placco et al., 2014; Yoon et al., 2018; Arentsen et al., 2022). A
significant fraction of CEMP stars have enhancements in $s$-process elements
and are called CEMP-s stars (Beers & Christlieb, 2005). These stars are
thought to receive their overabundant carbon and $s$-process material from a
mass-transfer event with their binary companion, which has evolved to or past
the AGB (Lugaro et al., 2012; Placco et al., 2013). On the other hand, CEMP
stars without neutron-capture enhancements are thought to have been primarily
enriched by material from the first generation of stars (Umeda & Nomoto, 2003;
Chiappini et al., 2006; Meynet et al., 2006; Nomoto et al., 2013; Tominaga et
al., 2014). However, there are many remaining questions about these unique
stars, including why they seem to be less frequent in the central regions of
our Galaxy, where we expect the highest concentration of ancient stars to
reside (Howes et al., 2014, 2015, 2016; Arentsen et al., 2021; Lucey et al.,
2022). As suggested by Yoon et al. (2019), this dearth of CEMP stars may be
caused by the dilution of CEMP stars in more massive subsystems (e.g., dwarf
galaxies) with prolonged star formation histories, which could be the origin
of metal-poor stars in the inner Galaxy.

In this work, we leverage the data from the all-sky Gaia survey to increase
the number of known CEMP stars from $\approx$50,000 to over 2.7 million. We
accomplish this by leveraging the $\sim$200 million BP/RP spectra made
available in Gaia DR3. Using the XGBoost algorithm for classification, we
achieve a contamination rate of 5% and completeness of 73%. We ensure that the
XGBoost algorithm matches our physical intuition and primarily performs the
classification using spectral features that are correlated with the carbon
abundance.

We briefly investigate a few of the properties of our sample including the
binarity rate and the Galactic distribution. Specifically, we use the non-
single stars flag provided in Gaia DR3 and compare the binarity rate of CEMP
as a function of metallicity to carbon-normal stars. We also look at the
distributions of the distances from the Galactic plane and the Galactic center
of the CEMP stars compared to the carbon-normal stars. In general, we find
that the the CEMP stars are father from the Galactic plane. In addition, we
find they are generally farther from the Sun with a stronger tail towards
larger radii.

In future work, we plan to look at the orbital properties of these stars as
well as the rate of $s$-process enhancement. We plan to perform high-
resolution follow-up of many of these targets in order to confirm our
contamination rate, measure radial velocities, and determine $s$-process
abundances. We plan to specifically follow-up targets towards the central
region of the Galaxy where the number of known CEMP stars is much lower.
Following future Gaia releases, which will include many millions more BP/RP
spectra, we expect to continue this work and again increase the known number
of CEMP stars by orders of magnitude.

## Acknowledgements

This project was developed in part at the Gaia Fête, held at the Flatiron
institute Centre for Computational Astrophysics in 2022 June. We’d like to
thank David Hogg and Jason Hunt for helpful discussions and computational
resources. We also thank Gary Da Costa for observational data from the AEGIS
survey which made this work possible.

This material is based upon work supported by the National Science Foundation
Graduate Research Fellowship under Grant No. 000392968. T.C.B. acknowledges
partial support for this work from grant PHY 14-30152; Physics Frontier
Center/JINA Center for the Evolution of the Elements (JINA-CEE), awarded by
the US National Science Foundation. Y.S.T. acknowledges financial support from
the Australian Research Council through DECRA Fellowship DE220101520. Work at
Argonne National Laboratory was supported by the U.S. Department of Energy,
Office of High Energy Physics. Argonne, a U.S. Department of Energy Office of
Science Laboratory, is operated by UChicago Argonne LLC under contract no. DE-
AC02-06CH11357. N.R acknowledges the Laboratory Directed Research and
Development (LDRD) funding from Argonne National Laboratory, provided by the
Director, Office of Science, of the U.S. Department of Energy under Contract
No. DE-AC02-06CH11357. Y.S.L. acknowledges support from the National Research
Foundation (NRF) of Korea grant funded by the Ministry of Science and ICT
(NRF-2021R1A2C1008679).

This work has made use of data from the European Space Agency (ESA)
missionGaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data
Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the
institutionsparticipating in the Gaia Multilateral Agreement.

Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the
Participating Institutions, the National Science Foundation, and the U.S.
Department of Energy Office of Science. The SDSS-III web site is
http://www.sdss3.org/.

SDSS-III is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS-III Collaboration including the
University of Arizona, the Brazilian Participation Group, Brookhaven National
Laboratory, Carnegie Mellon University, University of Florida, the French
Participation Group, the German Participation Group, Harvard University, the
Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA
Participation Group, Johns Hopkins University, Lawrence Berkeley National
Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for
Extraterrestrial Physics, New Mexico State University, New York University,
Ohio State University, Pennsylvania State University, University of
Portsmouth, Princeton University, the Spanish Participation Group, University
of Tokyo, University of Utah, Vanderbilt University, University of Virginia,
University of Washington, and Yale University.

This research made use of Astropy,333http://www.astropy.org a community-
developed core Python package for Astronomy (Astropy Collaboration et al.,
2013, 2018). Other software used includes IPython (Pérez & Granger, 2007),
matplotlib (Hunter, 2007), numpy (Harris et al., 2020), scikit-learn
(Pedregosa et al., 2011) and scipy (Virtanen et al., 2020).

## Data Availability

The data underlying this article are publicly available through the Gaia
Archive.444https://gea.esac.esa.int/archive/ The generated catalog of CEMP
stars is available in the online supplementary material and CDS upon
acceptance.

## References

  * Abate et al. (2015) Abate C., Pols O. R., Karakas A. I., Izzard R. G., 2015, A&A, 576, A118
  * Abazajian et al. (2003) Abazajian K., et al., 2003, AJ, 126, 2081
  * Abazajian et al. (2004) Abazajian K., et al., 2004, AJ, 128, 502
  * Abazajian et al. (2005) Abazajian K., et al., 2005, AJ, 129, 1755
  * Abazajian et al. (2009) Abazajian K. N., et al., 2009, ApJS, 182, 543
  * Adelman-McCarthy et al. (2006) Adelman-McCarthy J. K., et al., 2006, ApJS, 162, 38
  * Adelman-McCarthy et al. (2007) Adelman-McCarthy J. K., et al., 2007, ApJS, 172, 634
  * Adelman-McCarthy et al. (2008) Adelman-McCarthy J. K., et al., 2008, ApJS, 175, 297
  * Ahn et al. (2012) Ahn C. P., et al., 2012, ApJS, 203, 21
  * Aihara et al. (2011) Aihara H., et al., 2011, ApJS, 193, 29
  * Alvarez & Plez (1998) Alvarez R., Plez B., 1998, A&A, 330, 1109
  * Andrae et al. (2022) Andrae R., et al., 2022, arXiv e-prints, p. arXiv:2206.06138
  * Arentsen et al. (2019) Arentsen A., Starkenburg E., Shetrone M. D., Venn K. A., Depagne É., McConnachie A. W., 2019, A&A, 621, A108
  * Arentsen et al. (2021) Arentsen A., et al., 2021, MNRAS, 505, 1239
  * Arentsen et al. (2022) Arentsen A., Placco V. M., Lee Y. S., Aguado D. S., Martin N. F., Starkenburg E., Yoon J., 2022, arXiv e-prints, p. arXiv:2206.04081
  * Astropy Collaboration et al. (2013) Astropy Collaboration et al., 2013, A&A, 558, A33
  * Astropy Collaboration et al. (2018) Astropy Collaboration et al., 2018, AJ, 156, 123
  * Beers & Christlieb (2005) Beers T. C., Christlieb N., 2005, ARA&A, 43, 531
  * Beers et al. (1992) Beers T. C., Preston G. W., Shectman S. A., 1992, AJ, 103, 1987
  * Beers et al. (2017) Beers T. C., et al., 2017, ApJ, 835, 81
  * Bessell et al. (2011) Bessell M., Bloxham G., Schmidt B., Keller S., Tisserand P., Francis P., 2011, Publications of the Astronomical Society of the Pacific, 123, 789
  * Bisterzo et al. (2010) Bisterzo S., Gallino R., Straniero O., Cristallo S., Käppeler F., 2010, MNRAS, 404, 1529
  * Carollo et al. (2012) Carollo D., et al., 2012, ApJ, 744, 195
  * Carollo et al. (2014) Carollo D., Freeman K., Beers T. C., Placco V. M., Tumlinson J., Martell S. L., 2014, ApJ, 788, 180
  * Carrasco et al. (2021) Carrasco J. M., et al., 2021, A&A, 652, A86
  * Chen & Guestrin (2016) Chen T., Guestrin C., 2016, arXiv e-prints, p. arXiv:1603.02754
  * Chiappini et al. (2006) Chiappini C., Hirschi R., Meynet G., Ekström S., Maeder A., Matteucci F., 2006, A&A, 449, L27
  * Chiti et al. (2020) Chiti A., Frebel A., Jerjen H., Kim D., Norris J. E., 2020, ApJ, 891, 8
  * Christlieb et al. (2008) Christlieb N., Schörck T., Frebel A., Beers T. C., Wisotzki L., Reimers D., 2008, A&A, 484, 721
  * Da Costa et al. (2019) Da Costa G. S., et al., 2019, MNRAS, 489, 5900
  * Dietz et al. (2021) Dietz S. E., Yoon J., Beers T. C., Placco V. M., Lee Y. S., 2021, ApJ, 914, 100
  * Fitzpatrick (2004) Fitzpatrick E. L., 2004, in Witt A. N., Clayton G. C., Draine B. T., eds, Astronomical Society of the Pacific Conference Series Vol. 309, Astrophysics of Dust. p. 33 (arXiv:astro-ph/0401344)
  * Frebel (2018) Frebel A., 2018, Annual Review of Nuclear and Particle Science, 68, 237
  * Frebel & Norris (2015) Frebel A., Norris J. E., 2015, ARA&A, 53, 631
  * Frebel et al. (2006) Frebel A., et al., 2006, ApJ, 652, 1585
  * Frebel et al. (2014) Frebel A., Simon J. D., Kirby E. N., 2014, ApJ, 786, 74
  * Fukugita et al. (1996) Fukugita M., Ichikawa T., Gunn J. E., Doi M., Shimasaku K., Schneider D. P., 1996, AJ, 111, 1748
  * Gaia Collaboration et al. (2016) Gaia Collaboration et al., 2016, A&A, 595, A2
  * Gaia Collaboration et al. (2022a) Gaia Collaboration Vallenari A., Brown A. G. A., Prusti T., et al. 2022a, A&A
  * Gaia Collaboration et al. (2022b) Gaia Collaboration et al., 2022b, arXiv e-prints, p. arXiv:2206.05595
  * Gavel et al. (2021) Gavel A., Andrae R., Fouesneau M., Korn A. J., Sordo R., 2021, A&A, 656, A93
  * Gull et al. (2018) Gull M., et al., 2018, ApJ, 862, 174
  * Gunn et al. (1998) Gunn J. E., et al., 1998, AJ, 116, 3040
  * Gunn et al. (2006) Gunn J. E., et al., 2006, AJ, 131, 2332
  * Gustafsson et al. (2008) Gustafsson B., Edvardsson B., Eriksson K., Jørgensen U. G., Nordlund Å., Plez B., 2008, A&A, 486, 951
  * Hansen et al. (2016a) Hansen T. T., Andersen J., Nordström B., Beers T. C., Placco V. M., Yoon J., Buchhave L. A., 2016a, A&A, 586, A160
  * Hansen et al. (2016b) Hansen T. T., Andersen J., Nordström B., Beers T. C., Placco V. M., Yoon J., Buchhave L. A., 2016b, A&A, 588, A3
  * Harris et al. (2020) Harris C. R., et al., 2020, Nature, 585, 357
  * Hayden et al. (2020) Hayden M. R., et al., 2020, arXiv e-prints, p. arXiv:2011.13745
  * He et al. (2022) He X.-J., Luo A. L., Chen Y.-Q., 2022, MNRAS, 512, 1710
  * Heiter et al. (2020) Heiter U., et al., 2020, arXiv e-prints, p. arXiv:2011.02049
  * Howes et al. (2014) Howes L. M., et al., 2014, MNRAS, 445, 4241
  * Howes et al. (2015) Howes L. M., et al., 2015, Nature, 527, 484
  * Howes et al. (2016) Howes L. M., et al., 2016, MNRAS, 460, 884
  * Hunter (2007) Hunter J. D., 2007, Computing in Science & Engineering, 9, 90
  * Hwang et al. (2021) Hwang H.-C., Ting Y.-S., Schlaufman K. C., Zakamska N. L., Wyse R. F. G., 2021, MNRAS, 501, 4329
  * Jorissen et al. (2016) Jorissen A., et al., 2016, A&A, 586, A158
  * Lai et al. (2011) Lai D. K., Lee Y. S., Bolte M., Lucatello S., Beers T. C., Johnson J. A., Sivarani T., Rockosi C. M., 2011, ApJ, 738, 51
  * Lee et al. (2013) Lee Y. S., et al., 2013, AJ, 146, 132
  * Lee et al. (2017) Lee Y. S., Beers T. C., Kim Y. K., Placco V., Yoon J., Carollo D., Masseron T., Jung J., 2017, ApJ, 836, 91
  * Lee et al. (2019) Lee Y. S., Beers T. C., Kim Y. K., 2019, ApJ, 885, 102
  * Li et al. (2021) Li C., et al., 2021, MNRAS, 506, 1651
  * Liu et al. (2012) Liu C., Bailer-Jones C. A. L., Sordo R., Vallenari A., Borrachero R., Luri X., Sartoretti P., 2012, MNRAS, 426, 2463
  * Lucatello et al. (2005) Lucatello S., Tsangarides S., Beers T. C., Carretta E., Gratton R. G., Ryan S. G., 2005, ApJ, 625, 825
  * Lucatello et al. (2006) Lucatello S., Beers T. C., Christlieb N., Barklem P. S., Rossi S., Marsteller B., Sivarani T., Lee Y. S., 2006, ApJ, 652, L37
  * Lucey et al. (2022) Lucey M., et al., 2022, MNRAS, 509, 122
  * Lugaro et al. (2012) Lugaro M., Karakas A. I., Stancliffe R. J., Rijs C., 2012, ApJ, 747, 2
  * Lundberg & Lee (2017) Lundberg S. M., Lee S.-I., 2017, Advances in neural information processing systems, 30
  * Machado Poletti Valle et al. (2021) Machado Poletti Valle L. F., Avestruz C., Barnes D. J., Farahi A., Lau E. T., Nagai D., 2021, MNRAS, 507, 1468
  * Masseron et al. (2014) Masseron T., et al., 2014, A&A, 571, A47
  * McClure & Woodsworth (1990) McClure R. D., Woodsworth A. W., 1990, ApJ, 352, 709
  * Meynet et al. (2006) Meynet G., Ekström S., Maeder A., 2006, A&A, 447, 623
  * Nomoto et al. (2013) Nomoto K., Kobayashi C., Tominaga N., 2013, ARA&A, 51, 457
  * Norris et al. (2010) Norris J. E., Wyse R. F. G., Gilmore G., Yong D., Frebel A., Wilkinson M. I., Belokurov V., Zucker D. B., 2010, ApJ, 723, 1632
  * Norris et al. (2013) Norris J. E., et al., 2013, ApJ, 762, 28
  * Pedregosa et al. (2011) Pedregosa F., et al., 2011, Journal of Machine Learning Research, 12, 2825
  * Pérez & Granger (2007) Pérez F., Granger B. E., 2007, Computing in Science and Engineering, 9, 21
  * Pham & Kaltenegger (2022) Pham D., Kaltenegger L., 2022, MNRAS,
  * Pier et al. (2003) Pier J. R., Munn J. A., Hindsley R. B., Hennessy G. S., Kent S. M., Lupton R. H., Ivezić Ž., 2003, AJ, 125, 1559
  * Placco et al. (2013) Placco V. M., Frebel A., Beers T. C., Karakas A. I., Kennedy C. R., Rossi S., Christlieb N., Stancliffe R. J., 2013, ApJ, 770, 104
  * Placco et al. (2014) Placco V. M., Frebel A., Beers T. C., Stancliffe R. J., 2014, ApJ, 797, 21
  * Plez (2012) Plez B., 2012, Turbospectrum: Code for spectral synthesis, Astrophysics Source Code Library (ascl:1205.004)
  * Preston & Sneden (2001) Preston G. W., Sneden C., 2001, AJ, 122, 1545
  * Reid et al. (2014) Reid M. J., et al., 2014, ApJ, 783, 130
  * Riello et al. (2021) Riello M., et al., 2021, A&A, 649, A3
  * Rockosi et al. (2022) Rockosi C. M., et al., 2022, ApJS, 259, 60
  * Salvadori et al. (2015) Salvadori S., Skúladóttir Á., Tolstoy E., 2015, MNRAS, 454, 1320
  * Skúladóttir et al. (2015) Skúladóttir Á., Tolstoy E., Salvadori S., Hill V., Pettini M., Shetrone M. D., Starkenburg E., 2015, A&A, 574, A129
  * Starkenburg et al. (2014) Starkenburg E., Shetrone M. D., McConnachie A. W., Venn K. A., 2014, MNRAS, 441, 1217
  * Starkenburg et al. (2017) Starkenburg E., et al., 2017, MNRAS, 471, 2587
  * Stoughton et al. (2002) Stoughton C., et al., 2002, AJ, 123, 485
  * Tominaga et al. (2014) Tominaga N., Iwamoto N., Nomoto K., 2014, ApJ, 785, 98
  * Umeda & Nomoto (2003) Umeda H., Nomoto K., 2003, Nature, 422, 871
  * Virtanen et al. (2020) Virtanen P., et al., 2020, Nature Methods, 17, 261
  * Wolf et al. (2018) Wolf C., et al., 2018, Publications of the Astronomical Society of Australia, 35, e010
  * Yanny et al. (2009) Yanny B., et al., 2009, AJ, 137, 4377
  * Yoon et al. (2018) Yoon J., et al., 2018, ApJ, 861, 146
  * Yoon et al. (2019) Yoon J., Beers T. C., Tian D., Whitten D. D., 2019, ApJ, 878, 97
  * York et al. (2000) York D. G., et al., 2000, AJ, 120, 1579


$\alpha_{N}=2\left(2^{-N}e^{N\exp(\beta_{N})}-1\right)$

with $\beta_{N}$ distinct algebraic numbers converging to $\ln\ln 2$ quickly
enough. For example, we can fix:

$\beta_{N}:=\left\lfloor 10^{N^{2}}\ln\ln 2\right\rfloor 10^{-N^{2}}$

And in this case $\alpha_{N}=O(N\cdot 10^{-N^{2}})$, which is quick enough. ∎

We have constructed a chaotic and _nonlinear_ dynamical system on the circle
with some prescribed Lyapunov exponent. With our choice of $\alpha_{N}$, we
have

$\widehat{\lambda}(x_{N})=e^{\left\lfloor 10^{N^{2}}\ln\ln 2\right\rfloor
10^{-N^{2}}}.$

In particular,

$\text{dim}_{\mathbb{Q}}\text{Vect}_{\mathbb{Q}}\left\\{\widehat{\lambda}(x)\
|\ x\text{ is periodic}\right\\}=\infty$

which was what we wanted to construct.

###  B.2 A nonlinear solenoid

In this section we construct an explicit nonlinear perturbation of the usual
solenoid. Denote by
$\mathbb{T}:=\mathbb{R}/\mathbb{Z}\times\overline{\mathbb{D}}$ the full torus.
Define $F:\mathbb{T}\rightarrow\mathbb{T}$ by the formula

$F(\theta,x,y):=\left(f(\theta),\frac{1}{4}x+\frac{1}{4\pi}\cos(2\pi\theta),\frac{1}{4}y+\frac{1}{4\pi}\sin(2\pi\theta)\right)$

where $f(\theta)=2\theta+g(\theta)$ is the function defined previously. Notice
that $F$ is a diffeomorphism onto its image. In particular, we can use it to
glue $\mathbb{T}$ with itself along its boundary, which allows us to see $F$
as a diffeomorphism of a genuine closed 3-manifold ([Bo78], chapter 1) which
countains $\mathbb{T}$.

Then, $F(\mathbb{T})\subset\overset{\circ}{\mathbb{T}}$, and so
$T^{n}(\mathbb{T})$ is a strictly decreasing sequence of compact sets. The
intersection $S:=\bigcap_{n\geq 0}F^{n}(\mathbb{T})$ is called a (nonlinear)
solenoid. The set $S$ is an attractor for $F$. Since $F$ is a smooth
perturbation of the usual solenoid, which is a structurally stable Axiom A
diffeomorphism (this follows for example from Theorem 1 in [IPR10]), our
dynamical system is still Axiom A. Moreover, it has codimension 1 stable
lamination, as it contract in the $(x,y)$ variables.

Indeed, at a given point $p=(\theta,x,y)\in S$, we see that the Jacobian of
$F$ is

$\text{Jac}_{F}(p)=\begin{pmatrix}2+g^{\prime}(\theta)&0&0\\\
-\frac{1}{2}\sin(2\pi\theta)&1/4&0\\\
\frac{1}{2}\cos(2\pi\theta)&0&1/4\end{pmatrix}$

In particular, the subspace $E^{s}(p):=\\{(0,h,k)\ |\
h,k\in\mathbb{R}\\}\subset T_{p}\mathbb{T}=\mathbb{R}^{3}$ is independant of
$p$, and we check that $(dF)_{p}(E^{s}(p))\subset E^{s}(F(p))$ is a
contracting linear map. So we have found the stable direction.

This allows us to compute the derivative in the unstable direction at $p$.
Indeed, since

$(dF)_{p}:E^{s}(p)\oplus E^{u}(p)\rightarrow E^{s}(F(p))\oplus E^{u}(F(p))$

also sends $E^{u}(p)$ into $E^{u}(F(p))$, we can compute the determinant of
the jacobian by making the unstable derivative appear like so:

$\det(dF)_{p}=\partial_{u}F(p)\times\det((dF)_{\ |E^{s}})$

And we already know that

$\det((dF)_{\ |E^{s}})=\det(I/4)=1/16$

and that

$\det(dF)_{p}=(2+g^{\prime}(\theta))/16.$

Hence:

$\partial_{u}F(p)=f^{\prime}(\theta).$

Notice that the bunching condition (B) is satisfied.

Finally, we are going to construct some periodic orbits for $F$ and compute
their unstable Lyapunov exponents. If $p\in S$ is a periodic point for $F$,
then it is clear that its angular coordinate is periodic for $f$.
Reciprocally, let $\theta_{0}$ be a periodic point for $f$: there exists a
integer $n_{0}$ such that $f^{n_{0}}(\theta_{0})=\theta_{0}$. Then, the map
$F^{n_{0}}$ satisfy

$F^{n_{0}}(\\{\theta_{0}\\}\times\overline{\mathbb{D}})\subset\\{\theta_{0}\\}\times\mathbb{D}.$

and is contracting on this disk. Hence, there exists a unique associated fixed
point $p_{0}\in\\{\theta_{0}\\}\times\mathbb{D}$.

This allows us to exhibit some periodic orbits and compute the associated
Lyapunov exponents. Let $N\geq 2$ and consider $\theta_{N}:=\frac{1}{2^{N}-1}$
the periodic point for $f$ constructed in the last subsection. Let $p_{N}$ be
the unique associated fixed point, noted

$p_{N}=(\theta_{N},x_{N},y_{N})\in S.$

Then we have proved that its associated unstable Lyapunov exponent is

$\widehat{\lambda}(p_{N})=\widehat{\lambda}(\theta_{N})=e^{\beta_{N}}.$

In particular, we have

$\dim_{\mathbb{Q}}\text{Vect}_{\mathbb{Q}}\widehat{\lambda}\left(S_{\text{per}}\right)=\infty$

where $S_{\text{per}}:=\\{p\in S\ |\ p\ \text{is}\ F\ \text{periodic}.\\}$.
Hence, Theorem 1.2 applies. For example, the SRB measure and the measure of
maximal entropy enjoys polynomial Fourier decay in the unstable direction.
Indeed, let $\chi$ be a Hölder map with localized support at a point $p_{0}$
on the solenoid. Let $\mu$ be an equilibrium state. Then, the Fourier
transform of $\chi d\mu$ write, for $\xi\in\mathbb{R}^{3}$:

$\widehat{\chi d\mu}(\xi)=\int_{S}e^{-2i\pi
x\cdot\xi}\chi(x)d\mu(x)=\int_{S}e^{-it\phi_{v}}\chi d\mu$

where $v:=\xi/|\xi|$, $t=2\pi|\xi|$ and $\phi_{v}(x):=x\cdot v$. Fix a
direction $v$ such that $v$ is not orthogonal to $E^{u}(p_{0})$. Then, if
$\chi$ has small enough support, $|\partial_{u}\phi_{v}|>0$ around the support
of $\chi$. It follows from Theorem 1.2 that $\widehat{\chi
d\mu}(tv)\underset{t\rightarrow\infty}{\longrightarrow}0$ at a polynomial
rate. By Remark 4.1, the convergence to zero is even uniform on any cone in
the unstable direction.

## References

  * [ABV14] V. Araújo, O. Butterley, P. Varandas _Open sets of Axiom A flows with Exponentially Mixing Attractors (with Erratum)_ , Proceedings of the American Mathematical Society, 144(7), 2971-2984 (2016) arXiv:1402.6654v3
  * [Ba90] A. Baker, _Transcendental number theory_ , Cambridge Mathematical Library (2nd ed.), Cambridge University Press, ISBN: 978-0-521-39791-9, MR 0422171
  * [BD17] J. Bourgain, S. Dyatlov, _Fourier dimension and spectral gaps for hyperbolic surfaces_ , Geom. Funct. Anal. 27, 744–771 (2017), arXiv:1704.02909
  * [Be16] P. Berger, _Properties of the maximal entropy measure and geometry of Hénon attractors_ doi:10.48550/arXiv.1202.2822
  * [Bl96] C. Bluhm, _Random recursive construction of Salem sets_ Ark. Mat. 34 (1996), 51-63.
  * [Bo75] R. Bowen, _Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms_ (New edition of) Lect. Notes in Math. 470, Springer, 1975. ISBN : 978-3-540-77605-5
  * [Bo78] R. Bowen, _On Axiom A diffeomorphisms_ CBMS Regional Conference Series in Mathematics Volume: 35; 1978; ISBN: 978-0-8218-3438-1
  * [Bo10] J. Bourgain, _The discretized sum-product and projection theorems_ JAMA 112, 193-236 (2010) doi:10.1007/s11854-010-0028-x
  * [BS02] M. Brin, G. Stuck, _Introduction to Dynamical Systems._ 2002\. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511755316.
  * [Cl20] V. Climenhaga _SRB and equilibrium measures via dimension theory_ arXiv:2009.09260
  * [CP15] S. Crovisier, R. Potrie _Introduction to partially hyperbolic dynamics_ Lecture notes for School on Dynamical Systems, ICTP, Trieste, 2015.
  * [Do98] D. Dolgopyat, _On decay of correlations in Anosov flows_ ,
Ann. of Math. (2) 147 (2) (1998) 357–390.

  * [DV21] D. Daltro, P. Varandas _Exponential decay of correlations for gibbs measures on attractors of axiom A flows_ arXiv:2104.11839
  * [FH20] R. Fraser, K. Hambrook _Explicit Salem sets in $\mathbb{R}^{n}$_ preprint, arXiv:1909.04581
  * [Gr09] B. Green, _Sum-product phenomena in $\mathbb{F}_{p}$: a brief introduction_
Notes written from a Cambridge course on Additive Combinatorics, 2009.
arXiv:0904.2075

  * [GRH21] A. Gogolev, F. Rodriguez Hertz _Smooth rigidity for very non-algebraic Anosov diffeomorphisms of codimension one_ preprint, arXiv:2105.10539
  * [Ha89] B. Hasselblatt, _Regularity of the Anosov splitting and A new description of the Margulis measure_ PhD thesis.
  * [Ha97] B. Hasselblatt _Regularity of the Anosov splitting II_ Ergodic Theory and Dynamical Systems, 17 (1997), no. 1, 169–172 doi: 10.1017/S0143385797069757
  * [Ham86] S. Hammel, _A dissipative map of the plane - a model for optical bistability_
Phd thesis. hdl.handle.net/10150/188149

  * [Ha17] K. Hambrook, _Explicit Salem sets in $\mathbb{R}^{2}$._ Adv. Math. 311 (2017), 634–648.
  * [HP69] M. W. Hirsch, C. C. Pugh _Stable manifolds for hyperbolic sets_ Bulletin of the American Mathematical Society, 75(1), 149-152, (January 1969)
  * [IPR10] J. Iglesias, A. Portela, A. Rovella _C1 stable maps: Examples without saddles._ Fundamenta Mathematicae, 2010 doi:208. 10.4064/fm208-1-2
  * [JS16] T. Jordan, T. Sahlsten _Fourier transforms of Gibbs measures for the Gauss map_
Math. Ann., 364(3-4), 983-1023, 2016. arXiv:1312.3619

  * [Ka66] J. P. Kahane, _Images d’ensembles parfaits par des séries de Fourier Gaussiennes_ Acad. Sci. Paris 263 (1966), 678-681.
  * [Kau81] R. Kaufman, _On the theorem of Jarnik and Besicovitch_ , Acta Arith. 39 (1981), 265-267.
  * [KH95] A. Katok, B. Hasselblatt _Introduction to the Modern Theory of Dynamical Systems._ Encyclopedia of Mathematics and its Applications. 1995. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511809187
  * [Ki90] Y. Kiffer, _Large deviations in dynamical systems and stochastic processes_ , Transactions of the American Mathematical Society, 321(1990),505-524.
  * [Le00] R. Leplaideur, _Local product structure for equilibrium states_ , Trans. Amer. Math. Soc. 352 (2000), no. 4, 1889–1912. MR 1661262 doi:10.1090/S0002-9947-99-02479-4
  * [Le21] G. Leclerc, _Julia sets of hyperbolic rational maps have positive Fourier dimension_
preprint, arXiv:2112.00701.

  * [Li18] J. Li, _Discretized Sum-product and Fourier decay in_ $\mathbb{R}^{n}$, JAMA 143, 763–800 (2021).
doi:10.1007/s11854-021-0169-0, arXiv:1811.06852

  * [LNP19] Li, Naud, Pan, _Kleinian Schottky groups, Patterson-Sullivan measures and Fourier decay_
Duke Math. J. 170 (4) 775 - 825, 15 March 2021.
doi:10.1215/00127094-2020-0058,  arXiv:1902.01103.

  * [LQZ03] P. Liu, M. Qian, Y. Zhao, _Large deviations in Axiom A endomorphisms_ Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 133 , Issue 6 , December 2003 , pp. 1379 - 1388 doi:10.1017/S0308210500002997
  * [Mc34] J. McShane _Extension of range of functions_ Bull. Amer. Math. Soc. 40 (1934), 837–842.
  * [Pe19] Y. Pesin, _Sinai’s Work on Markov Partitions and SRB Measures_ The Abel Prize 2013-2017 (pp.257-285) doi:10.1007/978-3-319-99028-6_11
  * [PP90] W. Parry, M. Pollicott, _Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics_ Asterisque; 187-188, société mathématique de France, 1990.
  * [PR02] A. Pinto, D. Rand, _Smoothness of Holonomies for codimension 1 hyperbolic dynamics_ , Bulletin of the London Mathematical Society , Volume 34 , Issue 3 , May 2002 , pp. 341 - 352
doi:10.1112/S0024609301008670

  * [PSW97] C. Pugh, M. Shub, A. Wilkinson, _Hölder laminations_ , Duke Math. J. 86 (1997), p. 517–546.
  * [PSW00] C. Pugh, M. Shub, A. Wilkinson, _correction to “Hölder laminations”_ , Duke Math. J. 104 (2000) no. 1, p. 105-106
  * [QR03] M. Queffélec and O. Ramaré, _Analyse de Fourier des fractions continues à quotients restreints_ , Enseign. Math. (2), 49(3-4):335–356, 2003
  * [Ru78] D. Ruelle, _Thermodynamic Formalism_ , Second edition, Cambridge University Press 2004
  * [Sa51] R. Salem, _On singular monotonic functions whose spectrum has a given Hausdorff dimension._ Ark. Mat., 1:353–365, 1951.
  * [Sa19] J. Sakhr, _Semi-Poisson nearest-neighbor distance statistics for a 2D dissipative map_ , Results in Physics, Volume 16, 102906 (2020) arXiv:1909.09728.
  * [SS20] T. Sahlsten, C. Stevens, _Fourier transform and expanding maps on Cantor sets_
preprint, 2020.  arXiv:2009.01703

  * [Yo02] L-S. Young, _What Are SRB Measures, and Which Dynamical Systems Have Them ?_ Journal of Statistical Physics. 108. 733-754. doi:10.1023/A:1019762724717.