# Discovery of a Hypervelocity L Subdwarf at the Star/Brown Dwarf Mass Limit

Adam J. Burgasser Department of Astronomy & Astrophysics, UC San Diego, La
Jolla, CA, USA  Roman Gerasimov Department of Physics & Astronomy, University
of Notre Dame, Notre Dame, IN 46556, USA  Kyle Kremer TAPIR, California
Institute of Technology, Pasadena, CA 91125, USA  Hunter Brooks Department of
Astronomy and Planetary Science, Northern Arizona University, Flagstaff, AZ
86011, USA  Efrain Alvarado III Department of Astronomy, University of
California, Berkeley, CA 94720-3411, USA  Adam C. Schneider US Naval
Observatory, Flagstaff Station, Flagstaff, AZ, USA  Aaron M. Meisner NSF
National Optical-Infrared Astronomy Research Laboratory, 950 North Cherry
Avenue, Tucson, AZ 85719, USA  Christopher A. Theissen Department of Astronomy
& Astrophysics, UC San Diego, La Jolla, CA, USA  Emma Softich Department of
Astronomy & Astrophysics, UC San Diego, La Jolla, CA, USA  Preethi Karpoor
Department of Astronomy & Astrophysics, UC San Diego, La Jolla, CA, USA
Thomas P. Bickle School of Physical Sciences, The Open University, Milton
Keynes, MK7 6AA, UK  Backyard Worlds: Planet 9  Martin Kabatnik Backyard
Worlds: Planet 9  Austin Rothermich Department of Astrophysics, American
Museum of Natural History, Central Park West at 79th Street, NY 10024, USA
Department of Physics, Graduate Center, City University of New York, 365 5th
Ave., New York, NY 10016, USA  Department of Physics and Astronomy, Hunter
College, City University of New York, 695 Park Avenue, New York, NY, 10065,
USA  Dan Caselden Department of Astrophysics, American Museum of Natural
History, Central Park West at 79th Street, NY 10024, USA  J. Davy Kirkpatrick
IPAC, Mail Code 100-22, Caltech, 1200 East California Boulevard, Pasadena, CA
91125, USA  Jacqueline K. Faherty Department of Astrophysics, American Museum
of Natural History, Central Park West at 79th Street, NY 10024, USA  Sarah L.
Casewell Centre for Exoplanet Research, School of Physics and Astronomy
University of Leicester, University Road, Leicester, LE1 7RH, UK  Marc J.
Kuchner Exoplanets and Stellar Astrophysics Laboratory, NASA Goddard Space
Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA

(Received May 28, 2024; Accepted July 8, 2024)

###### Abstract

We report the discovery of a high velocity, very low-mass star or brown dwarf
whose kinematics suggest it is unbound to the Milky Way. CWISE
J124909.08+362116.0 was identified by citizen scientists in the Backyard
Worlds: Planet 9 program as a high proper motion ($\mu$ $=$ 0$\farcs$9/yr)
faint red source. Moderate resolution spectroscopy with Keck/NIRES reveals it
to be a metal-poor early L subdwarf with a large radial velocity
($-$103$\pm$10 km/s), and its estimated distance of 125$\pm$8 pc yields a
speed of 456$\pm$27 km/s in the Galactic rest frame, near the local escape
velocity for the Milky Way. We explore several potential scenarios for the
origin of this source, including ejection from the Galactic center $\gtrsim$3
Gyr in the past, survival as the mass donor companion to an exploded white
dwarf. acceleration through a three-body interaction with a black hole binary
in a globular cluster, and accretion from a Milky Way satellite system. CWISE
J1249+3621 is the first hypervelocity very low mass star or brown dwarf to be
found, and the nearest of all such systems. It may represent a broader
population of very high velocity, low-mass objects that have undergone extreme
accelerations.

Globular star clusters (656) Hypervelocity stars (776), L subdwarfs (896),
Metallicity (1031), Type Ia supernova (1728), Low mass stars (2050), Galactic
archaeology (2178)

††journal: ApJL††facilities: Keck Observatory(NIRES)††software:  astropy
(Astropy Collaboration et al., 2013, 2018, 2022), galpy (Bovy, 2015),
Matplotlib (Hunter, 2007), NumPy (van der Walt et al., 2011), pandas (Wes
McKinney, 2010), SciPy (Virtanen et al., 2020), SpeXTool (Cushing et al.,
2004), SPLAT (Burgasser & Splat Development Team, 2017)

##  1 Introduction

The majority of stars in the neighborhood of the Sun have low relative
velocities ($v$ $\approx$ 10-30 km/s) reflecting their common origin in star
forming clusters concentrated in the plane of the Milky Way. A rare subset of
nearby stars have much higher velocities ($v_{\rm tan}$ $\gtrsim$ 400 km/s;
$<$0.3% of stars within 1 kpc; Favia et al. 2015). These stars may originate
from the Milky Way’s ancient halo population, or underwent strong dynamical
interactions with compact objects such as the Milky Way’s central supermassive
black hole (Hills, 1988) of compact binaries in dense clusters (Yu & Tremaine,
2003; Fragione & Gualandris, 2019), or may be the survivors of the supernova
explosion of a binary companion (Blaauw, 1961; Portegies Zwart, 2000). The
fastest “hypervelocity” stars are unbound to the Milky Way’s gravitational
potential and may even have extragalactic origins (Abadi et al., 2009; Piffl
et al., 2011). These rare objects trace extreme interactions that may be
explored through their trajectories, velocity distributions, and atmospheric
properties (Brown, 2015).

The Gaia mission (Gaia Collaboration et al., 2021) has greatly expanded our
sample of high-velocity stars by providing 5D (position, parallax, and proper
motion) or 6D (plus radial velocity or RV) coordinates for billions of stars
out to kiloparsec distances. These measurements, combined with detailed
chemical abundances from RAVE (Steinmetz et al., 2006), LAMOST (Cui et al.,
2012), APOGEE (Majewski et al., 2017), and other spectral surveys have enabled
the discovery and characterization of over a dozen hypervelocity stars (e.g.,
Du et al. 2019; Quispe-Huaynasi et al. 2022; Liao et al. 2023; Scholz 2024)
that originate from environments as diverse as the Galactic center, globular
clusters, or satellite systems. Current studies focus on deep optical
measurements of rare and distant stars, and primarily sample main sequence and
red giant stars over a limited range of mass (0.7 M⊙ $\lesssim$ M $\lesssim$ 2
M⊙) and age ($\lesssim$10 Gyr for high-velocity OBAFG stars) which may limit
our ability to probe compositions and origins.

The citizen science project Backyard Worlds: Planet 9 (BYW; Kuchner et al.
2017) takes advantage of multi-epoch infrared photometry and astrometry from
the Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010) and its
extended NEOWISE mission (Mainzer et al., 2014) to search for faint, infrared
moving sources identified by a community of citizen scientists. BYW is ideally
designed to find low-mass, high velocity stars and brown dwarfs, including
local low-temperature metal-poor subdwarfs—the L, T, and Y subdwarfs—from the
thick disk and halo populations (Schneider et al., 2020; Meisner et al., 2020,
2021; Kirkpatrick et al., 2021a; Brooks et al., 2022; Burgasser et al., 2024).
In this Letter, we report the discovery of a nearby, metal-poor L subdwarf,
CWISE J124909.08+362116.0 (hereafter J1249+3621) whose speed may exceed the
local escape velocity of the Milky Way, making it the first low-mass
hypervelocity star and the nearest such system to the Sun.

##  2 Identification and Spectral Observations

J1249+3621 was identified by citizen scientists Tom Bickle, Martin Kabatnik,
and Austin Rothermich in multi-epoch unWISE images (Lang, 2014; Meisner et
al., 2018; Schlafly et al., 2019) on the BYW citizen science
portal.111http://www.backyardworlds.org. Its $W2$ magnitude and $J-W2$ color
(Table 2) suggest an early-type L dwarf at an estimated distance of
$\approx$100 pc (Kirkpatrick et al., 2021b). Combining astrometry from
PanSTARRS (Chambers et al., 2016) and the UKIDSS Hemisphere Survey (UHS; Dye
et al. 2018) yields a proper motion of $\mu$ = 884$\pm$5 mas yr-1, suggesting
a tangential velocity of $v_{\rm tan}$ $\approx$ 420 km s-1 and making it a
high priority target for spectroscopic followup.

Table 1: Properties of J1249+3621

Property | Value | Reference
---|---|---
$\alpha_{J2000}$ | 12h49m09$\fs$08 | 1
$\delta_{J2000}$ | +36$\arcdeg$21$\arcmin$16$\arcsec$ | 1
$\mu_{\alpha}\cos\delta$ | 344$\pm$5 mas/yr | 2,3
$\mu_{\delta}$ |  $-$814$\pm$5 mas/yr | 2,3
$i$ (AB) | 21.48$\pm$0.15 mag | 2
$z$ (AB) | 20.01$\pm$0.06 mag | 2
$y$ (AB) | 19.13$\pm$0.05 mag | 2
$J$ (Vega) | 17.10$\pm$0.03 mag | 3
$K$ (Vega) | 16.46$\pm$0.04 mag | 3
$W1$ (Vega) | 15.92$\pm$0.04 mag | 1
$W2$ (Vega) | 15.59$\pm$0.07 mag | 1
SpT | sdL1 | 4
$d_{est}$aa Estimated from the spectral classification, $JKW1W2$ photometry, and the spectral type/absolute magnitude relations of Gonzales et al. (2018) and Zhang et al. (2019). | 125$\pm$8 pc | 4
$v_{\rm tan}$ | 524$\pm$33 km/s | 4
$T_{\rm eff}$bb Based on the $\pm$1$\sigma$ range of Elf Owl and SAND model fits. | 1715 K to 2320 K | 4,5,6
$\log{g}$bb Based on the $\pm$1$\sigma$ range of Elf Owl and SAND model fits. | 4.4 to 5.1 (cm/s2) | 4,5,6
[M/H]bb Based on the $\pm$1$\sigma$ range of Elf Owl and SAND model fits. |  $-$1.4 to $-$0.5 | 4,5,6
[$\alpha$/Fe] | +0.25$\pm$0.07 | 4,5
Est. Mass | 0.082${}^{+0.002}_{-0.003}$ M⊙ | 4,7
$RV$ |  $-$103$\pm$10 km/s | 4
$U_{LSR}$cc Local Standard of Rest (LSR) velocities assuming a solar motion from Schönrich et al. (2010). | 449$\pm$28 km/s | 4
$V_{LSR}$cc Local Standard of Rest (LSR) velocities assuming a solar motion from Schönrich et al. (2010). |  $-$292$\pm$19 km/s | 4
$W_{LSR}$cc Local Standard of Rest (LSR) velocities assuming a solar motion from Schönrich et al. (2010). |  $-$15$\pm$11 km/s | 4
$v_{GRF}$dd Galactic rest frame (GRF) speed assuming $v_{circ}$ = 220 km/s at the Solar radius. | 456$\pm$27 km/s | 4

References. — (1) CatWISE2020 (Marocco et al., 2021) at astrometric epoch 2015
May 28 (UT); (2) PanSTARRS (Chambers et al., 2016); (3) UKIRT Hemisphere
Survey (Dye et al., 2018); (4) This paper; (5) Alvarado et al. (2024); (6)
Mukherjee et al. (2024); (7) Gerasimov et al. (2024)

J1249+3621 was observed on 30 January 2024 (UT) in clear and windy conditions
with the Near-Infrared Echellette Spectrometer (NIRES; Wilson et al. 2004) on
the Keck II 10m telescope, a cross-dispersed spectrograph that provides
$\lambda/\Delta\lambda$ $\approx$ 2700 spectra over 0.9–2.45 $\mu$m. We
obtained six exposures of 300 s each at an average airmass of 1.06 with the
slit aligned with the parallactic angle. Exposures were made in an ABBA
pattern, nodding 10$\arcsec$ along the slit for background subtraction. We
also observed the A0 V star HD 108140 ($V$ = 9.35) at a similar airmass, and
dome flat lamp exposures at the start of the night for pixel response
calibration. Data were reduced using a modified version of the Spextool
package (Cushing et al., 2004), following the procedure of Vacca et al. (2003)
for flux calibration and telluric absorption correction.

##  3 Analysis

###  3.1 Classification and Atmosphere Parameters

Figure 1 compares a smoothed version of our spectrum to near-infrared spectral
standards from the SpeX Prism Library Analysis Toolkit (SPLAT; Burgasser &
Splat Development Team 2017). J1249+3621 exhibits the characteristic features
of L-type dwarfs, with strong H2O absorption at 1.4 $\mu$m and 1.9 $\mu$m, and
FeH, Na I, and K I absorption in the 1.0–1.3 $\mu$m region. Its NIR spectral
slope is distinctly bluer than normal L dwarf spectra, and its 2.3 $\mu$m CO
band is highly suppressed indicating enhanced H2 collision-induced absorption
(CIA) in a low-metallicity, low-temperature atmosphere (Linsky, 1969;
Burgasser et al., 2003). Indeed, the spectrum of J1249+3621 best matches that
of the L subdwarf 2MASS J17561080+2815238 (sdL1; Kirkpatrick et al. 2010) but
is somewhat bluer, while not as blue as the extreme L subdwarf WISE
J043535.80+211509.2 (esdL1; Luhman & Sheppard 2014; Zhang et al. 2017). We
classify J1249+3621 as sdL1 based on comparison to a broad range of dwarf and
subdwarf spectra. This classification, WISE $W1W2$ and UHS $JK$ Vega
magnitudes, and the spectral type/absolute magnitude relations for L sudwarfs
from Gonzales et al. (2018) and Zhang et al. (2019) allow us to estimate a
spectrophotometric distance222This estimate takes into account photometric
uncertainties, uncertainties in the absolute magnitude/spectral type
relations, and a $\pm$1 subtype uncertainty on the classification. We used the
uncertainty-weighted mean across all bands and both relations, which are in
formal agreement within the uncertainties. of 125$\pm$8 pc for J1249+3621,
implying $v_{\rm tan}$ = 524$\pm$33 km/s.

To further evaluate its physical properties, we compared the smoothed spectrum
of J1249+3621 to the Sonora Elf Owl (Mukherjee et al., 2024) and Spectral
ANalog of Dwarfs (SAND; Alvarado et al. 2024) atmosphere models. These models
encompass the temperatures (1500 K $\lesssim$ $T_{\rm eff}$ $\lesssim$ 2400 K)
and subsolar metallicities ([M/H] $\lesssim$ $-$0.5) of L subdwarfs, and
contain up-to-date opacities and treatments for condensation and
disequilibrium chemistry. We used a Metropolis-Hastings Markov Chain Monte
Carlo (MCMC) fitting algorithm (Metropolis et al., 1953; Hastings, 1970) to
fit the models to the apparent spectral flux densities of J1249+3621,
following the procedure described in Burgasser et al. (2024). Our best fit
models and parameters are shown in Figure 1. The Elf Owl and SAND grids yield
distinctly different effective temperatures ($T_{\rm eff}$ = 2260$\pm$60 K
versus 1785$\pm$70 K) and metallicities ([M/H] = $-$0.63$\pm$0.10 versus
$-$1.28$\pm$0.10), but similar surface gravities (logg = 4.66$\pm$0.11 versus
4.88$\pm$0.24). We note that the Elf Owl grid does not extend to [M/H] $<$
$-$1 and does not include a prescription for condensate cloud formation,
although the latter may be less important in metal-poor L subdwarf atmospheres
(Burgasser et al., 2007; Gonzales et al., 2021). The best-fit Elf Owl model is
a marginally better fit to the observed spectrum,333The best fits yield
$\chi^{2}_{r}$ = 9.4 for Elf Owl and $\chi^{2}_{r}$ = 14.4 for SAND, where
$\chi^{2}_{r}$ = $\chi^{2}$/DOF, with DOF = 263. These values indicate a
relative probability $\ln\mathcal{P}$ =
$-\chi^{2}_{r,SAND}/\chi^{2}_{r,ElfOwl}$ = $-$1.5 or $\mathcal{P}$ = 0.22 that
the models equally represent the data. and its scaled surface fluxes are in
good agreement with the spectrophotometric distance estimate ($d$ = 102 pc)
assuming a radius of 0.08 R⊙. The model fit discrepancies could be resolved by
a direct distance measurement. Nevertheless, this analysis confirms our
interpretation of J1249+3621 as a low-temperature, metal-poor object. We note
that the SAND models suggest significant alpha enrichment ([$\alpha$/Fe] =
+0.25$\pm$0.07) and the Elf Owl models marginal C/O enrichment (C/O =
0.71$\pm$0.16), potential clues to the origin of this source.

Adopting generous parameter ranges of 1715 K $\lesssim$ $T_{\rm eff}$
$\lesssim$ 2320 K and $-$1.4 $\leq$ [M/H] $\leq$ $-$0.5, and assuming an age
$\tau$ $\geq$ 5 Gyr, the SAND evolutionary models (SANDee; Gerasimov et al.
2024) predict a mass of 0.082${}^{+0.002}_{-0.003}$ M⊙, placing this source
marginally above the metallicity-dependent Hydrogen Burning Minimum Mass (HBMM
$\approx$ 0.080 M⊙ for [M/H] = $-$1). The relatively narrow uncertainty range
indicated by the evolutionary models is due to the steep decline in
temperatures below the HBMM for old low-temperature sources, and does not
account for potential systematic biases (e.g., non-solar abundance patterns).
We conclude that J1249+3621 is likely a low-mass, metal-poor star, with a 10%
probability of being a high-mass brown dwarf.

Figure 1:  (Left): Keck/NIRES spectrum of J1249+3621 smoothed to a resolution
of $\lambda/\Delta\lambda$ = 150 (black lines), compared to the L dwarf
templates 2MASSW J2130446-084520 (L1, blue dot-dashed line; data from Bardalez
Gagliuffi et al. 2014), 2MASS J17561080+2815238 (sdL1, magenta line; data from
Kirkpatrick et al. 2010), and WISE J043535.80+211509.2 (esdL1, red dashed
line; data from Luhman & Sheppard 2014). The spectrum of J1249+3621 is
normalized at 1.3 $\mu$m and the comparison spectra normalized to maximize
agreement in the 1.0–1.3 $\mu$m range. The dwarf and subdwarf comparisons are
offset for ease of comparison. Major spectral features are labeled, as are
regions of strong telluric absorption at 1.35–1.5 $\mu$m and 1.75–1.85 $\mu$m
(vertical grey bands). (Right) Comparison of the smoothed spectrum of
J1249+3621 (black lines) to best fit models (magenta lines) from the Sonora
Elf Owl (top; Mukherjee et al. 2024) and SAND (bottom; Alvarado et al. 2024)
atmosphere grids. Spectra are scaled to apparent fluxes using the UHS $J$-band
magnitude of J1249+3621, and the models scaled to minimize $\chi^{2}_{r}$.
Model parameters are listed in the figure captions. Difference spectra
(observed minus computed; black line) are compared to the $\pm$1$\sigma$
uncertainty of the observed flux densities (grey band) in the bottom panels.

###  3.2 Radial Velocity and Kinematics

At full resolution, the Keck/NIRES data permit assessment of the radial
velocity of J1249+3621, which we approached using a forward modeling technique
described in Burgasser et al. (2024). In brief, we fit the extracted NIRES
spectrum without telluric correction to a high resolution stellar atmosphere
model, $M(\lambda)$, from Allard et al. (2012, BT-Settl) and an empirical
telluric absorption template, $T(\lambda)$, from Livingston & Wallace (1991)
using the parameterized data model

$D(\lambda+\delta_{\lambda})=C(\lambda)\times\left[\left(M(\lambda^{*})\otimes\kappa_{R}(v\sin{i})\right){\times}T^{\alpha}(\lambda)\right]\otimes\kappa_{G}(v_{b})+\delta_{f}.$
(1)

Here, $C(\lambda)$ is a fifth-order polynomial continuum correction,
$\kappa_{R}(v\sin{i})$ is a rotational broadening profile for projected
velocity $v\sin{i}$, $\alpha$ is a scaling exponent for the strength of the
telluric absorption, $\kappa_{G}({v_{b}})$ is a Gaussian instrumental
broadening profile parameterized by velocity width $v_{b}$, and
$\delta_{\lambda}$ and $\delta_{f}$ represent small offsets to the wavelength
scale and normalized flux density to account for residual calibration errors.
We used a $T_{\rm eff}$ = 2000 K, $\log{g}$ = 5.0, solar metallicity model
evaluated at a shifted wavelength
$\lambda^{*}=\lambda\left(1+\frac{RV+V_{bary}}{c}\right)$, accounting for the
unknown RV and known barycentric motion $v_{bary}$ = 16.82 km s-1 at the time
of observation. After identifying an optimized set of parameters using the
Nelder-Mead algorithm (Nelder, 1965) with a fixed $v\sin{i}$ = 50 km s-1, we
used a Metropolis-Hastings MCMC algorithm to map the parameter uncertainty
space for the five remaining free parameters: RV, $\alpha$, $v_{b}$,
$\delta_{\lambda}$ and $\delta_{f}$. Fits were conducted in two wavelength
regions that contain both stellar and telluric absorption features: 1.10–1.19
$\mu$m which contains Na I (1.138, 1.140 $\mu$m) and K I (1.169, 1.177 $\mu$m)
stellar lines and a telluric complex over 1.11–1.15 $\mu$m, and 1.235–1.28
$\mu$m which contains K I stellar lines (1.243, 1.252 $\mu$m) and a telluric
feature at 1.269 $\mu$m. We avoided the CO band at 2.3 $\mu$m, commonly used
for RV forward modeling of L dwarfs (cf. Blake et al. 2010; Konopacky et al.
2010; Burgasser et al. 2015), due to the suppression of this feature by H2
absorption. Figure 2 displays the best fit models for these regions, which
yield consistent values for RV ($-$92${}^{+13}_{-14}$ km/s and
$-$114${}^{+13}_{-14}$ km/s). We adopt a mean RV = $-$103$\pm$10 km/s for
J1249+3621.

Figure 2:  Forward modeling of the Keck/NIRES spectrum of J1249+3621 in the
1.10–1.19 $\mu$m (left), and 1.235–1.28 $\mu$m (right) spectral regions, both
of which contain stellar and telluric absorption features. Each panel displays
from top to bottom: the telluric spectrum in green, the stellar model in blue,
the combined model in magenta overlaid on the observed spectrum in black, and
the difference spectrum (data-model) in black overlaid on the $\pm$1$\sigma$
uncertainty band in grey. The reduced $\chi^{2}$ of each fit is indicated in
the text to the right of the plot.

Combining the measured RV, proper motion, and position of J1249+3621 with its
estimated distance, we computed $UVW$ velocities in the Local Standard of Rest
(LSR).444LSR velocity components assume a right-handed coordinate system
centered on the Sun with $U$ pointed radially inward, $V$ pointed in the
direction of Galactic rotation, and $W$ pointed toward the north Galactic
pole. We assumed solar velocity components of ($U_{\odot}$, $V_{\odot}$,
$W_{\odot}$) = (11.1 km/s, 12.24 km/s, 7.25 km/s) (Schönrich et al., 2010).
The velocities (Table 2) indicate a slightly retrograde motion relative to
Galactic rotation ($V_{LSR}$ = $-$292$\pm$19 km/s), with a trajectory directed
radially inward ($U_{LSR}$ = 449$\pm$28 km/s) and constrained to the Galactic
disk ($W_{LSR}$ = $-$15$\pm$11 km s-1). Assuming a local Galactic circular
velocity of $v_{circ}$ = 220 km/s, the velocity of J1249+3621 translates into
a Galactic rest frame speed of $v_{GRF}$ = 456$\pm$27 km/s, or 0.47$\pm$0.03
kpc/Myr. The median speed is just below the Galactic escape velocity at the
Solar radius, with current estimates ranging from 521${}^{+46}_{-30}$ km/s
(Williams et al., 2017, 1.6$\sigma$ above) to 580$\pm$63 km/s (Monari et al.,
2018, 1.8$\sigma$ above). Given the uncertainties in the inferred velocities
and potential models, we find that J1249+3621 has a significant probability of
being unbound to the Milky Way.

Figure 3:  Projected orbit of J1249+3621 from galpy (Bovy, 2015) for 100
initial conditions sampling uncertainties in distance, proper motion, and RV
(black lines). The top panels display the forward orbit over 10 Gyr projected
onto the Galactic plane on wide (left) and narrow (right) distance scales. The
bottom panels display the backward orbit over 150 Myr projected onto the
Galactic plane (left) and XZ coordinates (right). The top left panel shows the
present-day positions (magenta crosses) and forward-projected orbits (magenta
lines) of 50 satellite galaxies from Fritz et al. (2018), as well as the Milky
Way’s virial radius of 180 kpc (dashed blue circle; Sylos Labini et al. 2023).
The top right panel shows the present-day positions of 161 globular clusters
from Vasiliev (2019, magenta circles) and the closest-approach radius of
$\sim$1 kpc from the Galactic center (yellow dashed circle). Globular clusters
and their backward-projected orbits (magenta lines) are also shown in the
bottom panels.

The Galactic orbit of J1249+3621 was generated using galpy (Bovy, 2015). We
used the axisymmetric MWPotential2014 potential to integrate the trajectory of
J1249+3621 forward and backward in time by up to 10 Gyr, with a finer sampling
of the backward orbit up to 150 Myr. We drew 100 random initial conditions
sampling the uncertainties on distance, proper motion, and radial velocity
assuming independent Gaussian distributions. Figure 3 displays the forward and
backward trajectories projected onto the disk plane and in cylindrical
coordinates. The forward motion of J1249+3621 shows a close approach to the
inner region of the Milky Way, coming within 0.94${}^{+0.28}_{-0.19}$ kpc of
the Galactic center, then extending beyond the Milky Way’s virial radius of
180 kpc (Sylos Labini et al., 2023). The median model remains bound to the
Milky Way on a $\sim$3 Gyr, highly eccentric orbit, but 17% of our simulated
orbits are unbound over 10 Gyr. The backward orbit is approximately radial and
tightly confined to the Galactic plane, converging to within 2$\arcdeg$ of
05h22m25s +38$\arcdeg$37$\arcmin$00$\arcsec$ (galactic coordinate 38$\fd$6
+1$\fd$2) by 50 Myr in the past.

##  4 Assessing the Origins of J1249+3621

###  4.1 Was J1249+3621 Ejected from the Galactic Center?

J1249+3621 has a unique trajectory and speed; less than 0.002% of stars in
Gaia within 200 pc of the Sun have comparable tangential velocities.555This
statistic is based on a search of Gaia DR3 for sources with $\pi$ $>$ 5 mas,
$\pi/\sigma_{\pi}$ $>$ 10, and $v_{\rm tan}$ $>$ 500 km/s, which comprises 34
sources out of 2,234,316 without a tangential velocity constraint. While this
could nevertheless represent the extreme tail of the halo velocity
distribution Hawkins & Wyse (2018), we explored potential origins of
J1249+3621 in the context of currently known hypervelocity stars. Its small
but nonzero orbital angular momentum in the Galactic rest frame ($L_{z}$ =
572${}^{+147}_{-146}$ kpc km s-1) and inward trajectory would seem to argue
against ejection from the Galactic center through the traditional Hill’s
mechanism (Hills, 1988). However, if J1249+3621 is bound to the Milky Way,
which is the case for 83% of our orbit simulations, we could be observing it
on a return pass after intervals of roughly 3 Gyr, with torques imparted by
asymmetries in the Galactic potential; i.e., spiral structure or the inner bar
(Sellwood & Binney, 2002; Daniel & Wyse, 2018). Moreover, while the majority
of Galactic center stars are metal-rich (Carr et al., 2000; Ramírez et al.,
2000; Cunha et al., 2007), recent studies have identified metal-poor, alpha-
enhanced M giants in the nuclear region similar in nature to J1249+3621
(Schultheis et al., 2015, 2020). A small number of metal-poor hypervelocity
stars have also been associated with ejection from the Galactic center (Li et
al., 2012, 2023). Thus, despite its radially inward trajectory, ejection via
the Hill’s mechanism is a possible origin for J1249+3621.

###  4.2 Is J1249+3621 the Surviving Companion of a Type Ia Supernova?

An alternative disk origin for J1249+3621 is as the tight binary companion to
an accreting white dwarf that exceeded the 1.4 M⊙ Chandrasekhar limit
(Chandrasekhar, 1931, 1935), underwent a thermonuclear explosion, and released
J1249+3621 at high speed (Blaauw, 1961; Portegies Zwart, 2000; Shen et al.,
2018). Several accreting short-period white dwarf-brown dwarf pairs (polars
and cataclysmic variables) are known with white dwarf masses extending up to
0.94 M⊙ (Longstaff et al. 2019 and references therein). If J1249+3621 was the
initially more massive donor in such a system, its subsequent ejection speed
would exceed 690$\left(\frac{P}{1~{}hr}\right)^{-1/3}$ km s-1 based on orbital
motion alone, where $P$ is the orbit period at detonation. Periods of 1–1.5 hr
are sufficient for Roche lobe overflow666Following Eggleton (1983), $a$/R∗
$\approx$ 3–10 for a star-white dwarf system that evolves from 0.6 M⊙+0.9 M⊙
to 0.1 M⊙+1.4 M⊙, where $a$ is the orbit semimajor axis and R∗ the radius of
the mass donor. In the latter configuration, P $\approx$ 1–1.5 hr for average
densities of 50–80 g cm-3. for donors near the HBMM, based on theory and as
observed for low-mass cataclysmic variables (e.g., Kolb & Baraffe 1999;
Littlefair et al. 2006). This scaling law yields ejection speeds of 550–700 km
s-1, on par with J1249+3621’s LSR speed of 534 km/s. Contributions from
supernova shockwaves and mass stripping could drive ejection velocities even
higher (Pan et al., 2012; Rau & Pan, 2022).

As the ejection direction in this scenario is isotropic, the probability of an
individual source such as J1249+3621 passing by the Sun is very low. However,
the overall higher density of stars in the Galactic plane makes it more likely
that we would see a closely-passing ejectee with a trajectory confined to the
plane. We note that there are no known supernova remnants in the projected
past position of J1249+3621, but as remnants dissipate and merge with the
interstellar medium within $\lesssim$1 Myr (Leahy & Williams, 2017) this a
relatively weak constraint on the time since this source could have been
ejected.

###  4.3 Was J1249+3621 Ejected from a Globular Cluster?

Another possible origin for J1249+3621 is dynamical ejection from a globular
cluster (GC). The top-heavy present-day mass functions of these clusters is
evidence of efficient tidal dispersion of low-mass objects (Fregeau et al.,
2002), and large velocity kicks exceeding the 10–100 km/s escape velocities of
GCs are a natural outcome of strong three- and four-body dynamical
interactions that are commonplace in these systems (Leonard, 1991). Such kicks
are amplified by interactions with black hole binaries (Cabrera & Rodriguez,
2023), which are now understood to dominate the dynamics of the centers of
most GCs (Kremer et al., 2020a). The characteristic kick velocity imparted to
a star of mass $m$ during a strong encounter with a binary with components of
equal mass $M$ and semi-major axis $a$ is roughly:

$v_{\rm kick}\approx\sqrt{\frac{GM^{2}}{ma}}\approx
600\Bigg{(}\frac{M}{20\,M_{\odot}}\Bigg{)}\Bigg{(}\frac{m}{0.1\,M_{\odot}}\Bigg{)}^{-1/2}\Bigg{(}\frac{a}{10\,\rm{au}}\Bigg{)}^{-1/2}\,\rm{km/s}.$
(2)

Thus, for $a\lesssim 10\,$au and/or compact object masses $M\gtrsim
20\,M_{\odot}$, sufficiently high velocities can be achieved.

To test this mechanism, we searched for low-mass high velocity stars in the
CMC Cluster Catalog (Kremer et al., 2020b), a public suite of $N$-body
simulations intended to serve as a proxy for the Galactic GC population
(Harris, 1996). These simulations adopt a Kroupa (2001) initial mass function
ranging from $0.08-150\,M_{\odot}$, and therefore include stars of mass
comparable to J1249+3621. At times $t>8\,$Gyr, typical of the ages of Milky
Way GCs, we identify roughly $4,000$ low-mass stars (M $<$ 0.2 M⊙) ejected
with velocities of at least $100\,$km/s, corresponding to a rate of roughly 1
Myr-1 across the Milky Way. Of these, six stars are ejected with
$V>500\,$km/s, corresponding to a rate of roughly 2 Gyr-1. all six
hypervelocity subdwarfs awere ejected via dynamical encounters with stellar-
mass black hole binaries with properties similar to the scales in Eqn. 2.
Hence, simulations support the scenario of GC ejection as a means of
generating very high velocity, very low mass objects, albeit as exceedingly
rare events. The isotropic distribution of ejections further reduces the
probability of this scenario, with the same caveat that the higher
concentration of GCs in the Galactic plane makes detection of closely-passing
ejectees on planar orbits more likely.

There are no known globular clusters within 5$\arcdeg$ of the projected past
position of J1249+3621. However, given the long timescales involved it is
necessary to account for cluster motion. We used our galpy orbits to assess
whether the trajectory of J1249+3621 intersected with any of the 161 GCs in
the kinematic catalog of Vasiliev (2019), projecting all orbits back 150 Myr.
The closest approaches by NGC 3201 and Palomar 1 are more than 4 kpc in
separation, making these improbable origin sites. The low Galactic latitude of
the projected past position of J1249+3621 ($b$ = +1$\fd$2) could argue for an
origin from an as yet undiscovered GC hidden in the Galactic plane.
Alternately, the higher concentration of GCs near the Galactic center implies
that J1249+3621 could have been ejected from one of these systems, and is now
making a return pass $\gtrsim$3 Gyr later. Despite these considerations, we
find the GC ejection scenario less likely than the prior two scenarios.

For completeness, we note that four open clusters lie within 1$\arcdeg$ of the
projected past position of J1249+3621, including the well-studied NGC 1857
system at 3.1 kpc (Herschel, 1864). However, the lower stellar densities of
these systems and lack of compact binaries from massive progenitors makes
dynamic ejection unlikely, and none of these clusters have significantly
subsolar metallicities characteristic of J1249+3621.

###  4.4 Is J1249+3621 an Extragalactic Star?

As the bound orbits of J1249+3621 extend beyond the Milky Way’s virial radius,
it is possible that this source could have an extragalactic origin,
specifically accretion from one of the Milky Way satellites (Abadi et al.,
2009; Piffl et al., 2011). Its present trajectory does not align with distant
extragalactic systems such as M31 or the Magellanic Clouds. We examined the
intersection of J1249+3621’s orbit with 50 Milky Way satellites from Fritz et
al. (2018) using the same procedure as our GCs, integrating back to 10 Gyr.
Only one system, Tucana III (Drlica-Wagner et al., 2015), comes within 5 kpc
of the median orbit of J1249+3621 about 6 Gyr ago. The substantial
uncertainties of the trajectories of J1249+3621 and the known dwarf satellites
over Gyr timescales means we cannot strictly rule out this scenario; however,
we find it the least likely of the scenarios considered here, given the
trajectory of J1249+3621 being confined to the Galactic plane.

Inferring the true origin of J1249+3621 will require further investigation
into its physical and atmospheric properties. A Galactic center origin
requires closer examination of its orbital trajectory through refinement of
its distance (by direct parallax measurement) and velocity components, as well
as a more realistic, non-axisymmetric Galactic potential model. A more
detailed compositional analysis would also help clarify its origin. For
example, if J1249+3621 is the surviving companion of a Type Ia supernova, its
atmosphere may be enriched with heavy elements, particularly nickel, depending
on the degree of mass stripping by the supernova blast wave (Rau & Pan, 2022).
Similarly, if J1249+3621 was ejected from the Galactic center, a globular
cluster, or a satellite system, its detailed abundances may provide the
chemical fingerprint of its origin. Better assessment of composition of
through additional optical and infrared spectra, and improved atmosphere
models exploring specific abundances (e.g., Gerasimov et al. 2022) are needed
to infer a chemical-based origin.

Finally, we note that at least one other metal-poor L subdwarf, the esdL1 ULAS
J231949.36+044559.5 (Zhang et al., 2018), has a high enough estimated
tangential velocity (513${}^{+50}_{-46}$ km/s) to make it a promising
hypervelocity candidate, although no RV has yet been reported for this source.
The existence of at least one and possibly two L subdwarfs within $\sim$200 pc
of the Sun with hypervelocity speeds suggests a considerably larger population
of such sources could exist in the Milky Way system.

We thank Keck Observatory staff Randy Campbell and Max Service for their
support during the Keck/NIRES observations. Figure 3 was made with the MWplot
package created by Henry Leung (https://github.com/henrysky/milkyway_plot) and
graphics created by Robert Hurt
(https://www.spitzer.caltech.edu/image/ssc2008-10a-a-roadmap-to-the-milky-
way). The authors acknowledge helpful discussions with Boris Gaensickle and
Keith Hawkins in the preparation of this manuscript, and a prompt and helpful
review by our anonymous referee. This material is based upon work supported by
the National Science Foundation under grant No. 2009136. Support for KK was
provided by NASA through the NASA Hubble Fellowship grant HST-HF2-51510
awarded by the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., for NASA, under
contract NAS5-26555. This publication makes use of data products from the
Wide-field Infrared Survey Explorer, which is a joint project of the
University of California, Los Angeles, and the Jet Propulsion
Laboratory/California Institute of Technology, and NEOWISE, which is a project
of the Jet Propulsion Laboratory/California Institute of Technology. WISE and
NEOWISE are funded by the National Aeronautics and Space Administration. This
research has benefited from the SpeX Prism Libraries Analysis Toolkit,
maintained by Adam Burgasser at https://github.com/aburgasser/splat. This
research has made use of the SIMBAD database (Wenger et al., 2000), the
“Aladin sky atlas” (Bonnarel et al., 2000), and the VizieR catalogue access
tool developed and operated at CDS, Strasbourg, France Data presented herein
were obtained at Keck Observatory, which is a private 501(c)3 nonprofit
organization operated as a scientific partnership among the California
Institute of Technology, the University of California, and the National
Aeronautics and Space Administration. The Observatory was made possible by the
generous financial support of the W. M. Keck Foundation. The authors recognize
and acknowledge the significant cultural role and reverence that the summit of
Maunakea has within the indigenous Hawaiian community, and that the W. M. Keck
Observatory stands on Crown and Government Lands that the State of Hawai’i is
obligated to protect and preserve for future generations of indigenous
Hawaiians. Portions of this work were conducted at the University of
California, San Diego, which was built on the unceded territory of the
Kumeyaay Nation, whose people continue to maintain their political sovereignty
and cultural traditions as vital members of the San Diego community.

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# Continuity of measure-dimension mappings

Liangang Ma  Dept. of Mathematical Sciences, Binzhou University, Huanghe 5th
Road No. 391, Binzhou 256600, Shandong, P. R. China<EMAIL_ADDRESS>
###### Abstract.

We study continuity and discontinuity of the upper and lower (modified) box-
counting, Hausdorff, packing, (modified) correlation measure-dimension
mappings under the weak, setwise and TV topology on the space of Borel
measures respectively in this work. We give various examples to show that no
continuity can be guaranteed under the weak, setwise or TV topology on the
space of Borel measures for any of these measure-dimension mappings. However,
in some particular circumstances or by assuming some restrictions on the
measures, we do have some (semi-)continuity results for some of these measure-
dimension mappings under the setwise topology. In the end we point out some
applications of our continuity results on deciding the dimensions of measures
on ambient space with some dynamical structures.

The work is supported by ZR2019QA003 from SPNSF and 12001056 from NSFC

##  1\. Introduction

The dimension of a measure on a metrizable space indicates how well the mass
is distributed on the space. Its definition usually depends on the dimensions
of sets in the space. P. Mattila, M. Morán and J. M. Rey gave some criterions
on evaluating the well-posedness of different concepts of dimensions of
measures, see [MMR, p220]. In their work they treat the dimensions of measures
as as a function (functional) from the space of Borel measures to
$\mathbb{R}^{+}=[0,\infty)$: the measure-dimension mappings. From this point
of view it is a natural question to ask the continuity of various measure-
dimension mappings, with suitable topology endowed on the space of Borel
measures. It would be interesting if the measure-dimension mappings could
admit some continuity under some topology in case one is trying to deal with
the corresponding dimensions of Borel measures.

Considering the topological description of the space of Borel measures, the
vague, weak, setwise and TV topology are well-known among probabilists, with
their fineness in increasing order. One can refer to [Kal2] and [Kle] for the
vague and weak topology, while to [FKZ1] and [Las] for the setwise and TV
topology on the space of Borel measures. These topologies have lots of
applications in various circumstances, for example, see the applications of
the vague or weak topology in particle systems by Cox-Klenke-Perkins and
Kallenberg [CKP, Kal1], the applications of the weak, setwise or TV topology
in Markov decision processes by Feinberg-Kasyanov-Zgurovsky, O. Hernández-
Lerma and J. Lasserre in [FKZ2, HL]. The choice of topology is subtle in
different circumstances, with better properties possible under finer topology
at the cost of more difficulty in guaranteeing convergence of concerning
sequences of measures under the finer topology.

Luckily, we find that lots of measure-dimension mappings admit some semi-
continuity properties under the setwise topology on the space of Borel
measures. Especially, the upper measure-dimension mapping induced from some
dimensional mapping satisfying the _countable stability_ admits lower semi-
continuity under the setwise topology, in words of Falconer [Fal1, p40]. Upper
semi-continuity always holds for any induced measure-dimension mappings under
the setwise topology. However, some measure-dimension mappings do not admit
any semi-continuity under any of these topologies, for example, the (modified)
correlation measure-dimension mapping. We will also give examples to show that
none of our concerning measure-dimension mappings admits any continuity if the
topology is relaxed from the setwise level to weak level. There are some
applications of our continuity results to Hausdorff dimensions of ergodic
measures on some Riemannian manifold following Young [You] at the end of the
work.

##  2\. Notations, definitions and the main results

In this section we first introduce some popular measure-dimension mappings,
then we give definitions of the vague, weak, setwise and TV topology on the
space of Borel measures. At the end of the section we present our results on
semi-continuity of some measure-dimension mappings under the setwise topology.

Let $(X,\rho)$ be a metric space with its Borel $\sigma$-algebra
$\mathcal{B}$. Denote by $\mathcal{M}(X)$ to be the collection of all the
Borel measures (note that it is possible that some measures are infinite in
$\mathcal{M}(X)$). The collection of all the finite Borel measures on
$(X,\mathcal{B})$ is denoted by $\hat{\mathcal{M}}(X)$. A _dimensional
mapping_ is a real non-negative function

$dim:\mathcal{B}\rightarrow\mathbb{R}^{+}$.

Of course the concept is interesting if and only if the function satisfies
some reasonable properties, for example, the ones in [Fal1, p40]. We are
particularly interested in those dimensional mappings satisfying the
_countable stability_ , that is, for any collection of countable sets
$\\{A_{i}\\}_{i=1}^{\infty}$,

$dim(\cup_{i=1}^{\infty}A_{i})=\sup\\{dim(A_{i})\\}_{i=1}^{\infty}$.

Notable dimensional mappings in this work are the upper and lower (modified)
box-counting, Hausdorff and packing dimensional mappings, which are denoted by

$\overline{dim}_{B},\underline{dim}_{B}(\overline{dim}_{MB},\underline{dim}_{MB}),dim_{H},dim_{P}$

respectively, see [Fal1]. Denote by $dim_{B}$ ($dim_{MB}$) in case
$\overline{dim}_{B}=\underline{dim}_{B}$
($\overline{dim}_{MB}=\underline{dim}_{MB}$). A _measure-dimension mapping_ is
a real non-negative function

$dim:\mathcal{M}(X)\rightarrow\mathbb{R}^{+}$.

We abuse notations as one can easily distinguish its meaning of dimensional
mapping or measure-dimension mapping from the contexts. Refer to [MMR, p220]
on natural properties of reasonable measure-dimension mappings. The following
two dual measure-dimension mappings are highlighted in various circumstances.

######  2.1 Definition.

Let $dim$ be a dimensional mapping on $(X,\rho)$. The induced _lower_ and
_upper measure-dimension mappings_ from $\mathcal{M}(X)$ to $[0,\infty)$ are
defined respectively to be:

$dim^{L}(\nu)=\inf\\{dim(A):\nu(A)>0,A\in\mathcal{B}\\},$

and

$dim^{U}(\nu)=\inf\\{dim(A):\nu(X\setminus A)=0,A\in\mathcal{B}\\}$

for a measure $\nu\in\mathcal{M}(X)$.

It is easy to see that

$dim^{L}(\nu)\leq dim^{U}(\nu)$

for any $\nu\in\mathcal{M}(X)$. When taking the dimensional mappings to be

$dim_{B},\overline{dim}_{B},\underline{dim}_{B}(\overline{dim}_{MB},\underline{dim}_{MB}),dim_{H},dim_{P}$,

the corresponding lower and upper measure-dimension mappings are denoted
respectively by

$dim_{B}^{L},\overline{dim}_{B}^{L},\underline{dim}_{B}^{L}(\overline{dim}_{MB}^{L},\underline{dim}_{MB}^{L}),dim_{H}^{L},dim_{P}^{L}$

and

$dim_{B}^{U},\overline{dim}_{B}^{U},\underline{dim}_{B}^{U}(\overline{dim}_{MB}^{U},\underline{dim}_{MB}^{U}),dim_{H}^{U},dim_{P}^{U}$.

In case the lower and upper measure-dimension mappings coincide with each
other at some measure $\nu\in\mathcal{M}(X)$, denote the values simply by

$dim_{B}(\nu),\overline{dim}_{B}(\nu),\underline{dim}_{B}(\nu)\big{(}\overline{dim}_{MB}(\nu),\underline{dim}_{MB}(\nu)\big{)},dim_{H}(\nu),dim_{P}(\nu)$.

Following Mattila-Morán-Rey, we also pay attention to the correlation measure-
dimension mapping $\dim_{C}$ and its modified version $\dim_{MC}$. The
correlation measure-dimension mapping is introduced by P. Grassberger and I.
Procaccia [GP1, GP2].

######  2.2 Definition.

The _correlation dimension_ of $\nu$ (see [Cut]) is defined to be

$dim_{C}(\nu)=\lim_{r\rightarrow 0}\cfrac{\log\int\nu(B(x,r))d\nu}{\log r}$

for a measure $\nu\in\mathcal{M}(X)$, in case the above limit exists.

The modified one $\dim_{MC}$ is given by Pesin [Pes].

######  2.3 Definition.

The _modified correlation dimension_ of $\nu$ is defined to be

$dim_{MC}(\nu)=\lim_{\delta\rightarrow 0}\sup_{\\{A\in\mathcal{B}:\nu(A)\geq
1-\delta\\}}\lim_{r\rightarrow 0}\cfrac{\log\int_{A}\nu(B(x,r))d\nu}{\log r}$

for a measure $\nu\in\mathcal{M}(X)$, in case the above limit exists.

######  2.4 Remark.

Regardless of the existence of the above two limits, one can define the
_upper_ and _lower correlation dimension_ as well as the _upper_ and _lower
modified correlation dimension_ as [MMR, (2),(4)]. We adopt these definitions
directly because all the measures in our concerns in this work have the same
upper and lower (modified) correlation dimensions.

By [Fal2, Proposition 10.2] and [MMR, Lemma 2.8], we have

$dim_{H}^{L}(\nu)\leq dim_{C}(\nu)\leq dim_{MC}(\nu)$

for any $\nu\in\mathcal{M}(X)$.

In order to discuss the continuity of the measure-dimension mappings, some
topology on $\mathcal{M}(X)$ is necessary. The following concepts apply to
$\mathcal{M}(X)$ with topological ambient space $X$.

######  2.5 Definition.

The _vague topology_ is the topology with basis

$\\{\varrho\in\mathcal{M}(X):|\int_{X}f(x)d\varrho-\int_{X}f(x)d\nu|<\epsilon\\}$

on $\mathcal{M}(X)$ for any continuous function $f:X\rightarrow\mathbb{R}$
with compact support and any real $\epsilon>0$.

Refer to [Kal2, Kle]. As the coarsest topology, no continuity can be
guaranteed for any measure-dimension mappings in our concerns under it.
However, we still would like to pose it here in case of particular interests
for some readers. A finer topology is the weak topology.

######  2.6 Definition.

The _weak topology_ on $\mathcal{M}(X)$ is the topology with basis

$\\{\varrho\in\mathcal{M}(X):|\int_{X}f(x)d\varrho-\int_{X}f(x)d\nu|<\epsilon\\}$

for any bounded continuous function $f:X\rightarrow\mathbb{R}$ and any real
$\epsilon>0$.

Refer to [Bil1, Bil2, Kle, Mat]. Unfortunately, still no (semi-)continuity can
be guaranteed for any measure-dimension mappings in our concerns under the
weak topology in general, see Theorem 3.1.

######  2.7 Definition.

The _setwise topology_ on $\mathcal{M}(X)$ is the topology with basis

$\\{\varrho\in\mathcal{M}(X):|\int_{X}f(x)d\varrho-\int_{X}f(x)d\nu|<\epsilon\\}$

for any bounded measurable function $f:X\rightarrow\mathbb{R}$ and any real
$\epsilon>0$.

Refer to [Doo, FKZ1, GR, HL, Las, LY]. After escalating to the setwise
topology, we are excited to find that some measure-dimension mappings admit
some semi-continuity under it. Denote by

$\nu_{n}\stackrel{{\scriptstyle
v}}{{\rightarrow}}\nu,\nu_{n}\stackrel{{\scriptstyle
w}}{{\rightarrow}}\nu,\nu_{n}\stackrel{{\scriptstyle
s}}{{\rightarrow}}\nu,\nu_{n}\stackrel{{\scriptstyle TV}}{{\rightarrow}}\nu$

as $n\rightarrow\infty$ for sequences of measures in $\mathcal{M}(X)$
converging under the vague, weak, setwise or TV topology (to be defined later)
respectively.

######  2.8 Theorem.

Let $dim$ be a dimensional mapping satisfying the countable stability
property. The induced upper measure-dimension mapping $dim^{U}$ is lower semi-
continuous under the setwise topology on $\mathcal{M}(X)$, that is, if
$\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$ in $\mathcal{M}(X)$ as
$n\rightarrow\infty$, then

(2.1) $\liminf_{n\rightarrow\infty}dim^{U}(\nu_{n})\geq dim^{U}(\nu).$

Alternatively, we have upper semi-continuity for the lower measure-dimension
mapping $dim^{L}$, which shows the two measure-dimension mappings $dim^{U}$
and $dim^{L}$ are dual to each other in some sense.

######  2.9 Theorem.

Let $dim$ be a dimensional mapping. The induced lower measure-dimension
mapping $dim^{L}$ is upper semi-continuous under the setwise topology on
$\mathcal{M}(X)$, that is, if $\nu_{n}\stackrel{{\scriptstyle
s}}{{\rightarrow}}\nu$ in $\mathcal{M}(X)$ as $n\rightarrow\infty$, then

(2.2) $\limsup_{n\rightarrow\infty}dim^{L}(\nu_{n})\leq dim^{L}(\nu).$

As the finest topology in this work, the TV topology is induced from an
explicit metric on $\mathcal{M}(X)$.

######  2.10 Definition.

The _TV (total-variation) metric_ is defined to be

$\|\nu-\varrho\|_{TV}=\sup_{A\in\mathcal{B}}\\{|\nu(A)-\varrho(A)|\\}$

for two measures $\nu,\varrho\in\mathcal{M}(X)$.

Refer to [Doo, FKZ1, HL, Las, PS]). Considering the two measure-dimension
mappings $dim_{C}$ and $dim_{MC}$, they do not admit any semi-continuity under
the finest topology on $\mathcal{M}(X)$ in general, see Theorem 3.13.

##  3\. Proofs of the semi-continuity results and examples on discontinuity of
the measure-dimension mappings under various topology

First, to provide a general intuition on the continuity of the measure-
dimension mappings in our consideration, we present the readers with the
following result.

######  3.1 Theorem.

The measure-dimension mappings

$dim_{B}^{L},\overline{dim}_{B}^{L},\underline{dim}_{B}^{L}(\overline{dim}_{MB}^{L},\underline{dim}_{MB}^{L}),dim_{H}^{L},dim_{P}^{L},$

$dim_{B}^{U},\overline{dim}_{B}^{U},\underline{dim}_{B}^{U}(\overline{dim}_{MB}^{U},\underline{dim}_{MB}^{U}),dim_{H}^{U},dim_{P}^{U}$

and

$dim_{C},dim_{MC}$

from $\mathcal{M}(X)$ to $[0,+\infty)$ are not continuous under the weak,
setwise or TV topology in general.

###### Proof.

See for instance our Example 3.6 and 3.11. ∎

######  3.2 Remark.

It would be interesting to ask whether some measure-dimension mappings can be
defined to be equipped with continuity under some topology, possibly at the
cost of losing some known properties of some popular measure-dimension
mappings.

We first give an example to show that for a weakly convergent sequence
$\nu_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}\nu$ as
$n\rightarrow\infty$, any of the sequences

(3.1)
$\begin{array}[]{cc}\\{dim_{B}^{L}(\nu_{n})\\}_{n=1}^{\infty},\\{\overline{dim}_{B}^{L}(\nu_{n})\\}_{n=1}^{\infty},\\{\underline{dim}_{B}^{L}(\nu_{n})\\}_{n=1}^{\infty},\\{\overline{dim}_{MB}^{L}(\nu_{n})\\}_{n=1}^{\infty},\\\
\\{\underline{dim}_{MB}^{L}(\nu_{n})\\}_{n=1}^{\infty},\\{dim_{H}^{L}(\nu_{n})\\}_{n=1}^{\infty},\\{dim_{P}^{L}(\nu_{n})\\}_{n=1}^{\infty}\end{array}$
(3.2)
$\begin{array}[]{cc}\\{dim_{B}^{U}(\nu_{n})\\}_{n=1}^{\infty},\\{\overline{dim}_{B}^{U}(\nu_{n})\\}_{n=1}^{\infty},\\{\underline{dim}_{B}^{U}(\nu_{n})\\}_{n=1}^{\infty},\\{\overline{dim}_{MB}^{U}(\nu_{n})\\}_{n=1}^{\infty},\\\
\\{\underline{dim}_{MB}^{U}(\nu_{n})\\}_{n=1}^{\infty},\\{dim_{H}^{U}(\nu_{n})\\}_{n=1}^{\infty},\\{dim_{P}^{U}(\nu_{n})\\}_{n=1}^{\infty}\end{array}$

and

(3.3)
$\\{dim_{C}(\nu_{n})\\}_{n=1}^{\infty},\\{dim_{MC}(\nu_{n})\\}_{n=1}^{\infty}$

may not converge. In this and all the examples below, we set
$X\subset\mathbb{R}^{d}$ endowed with the Euclidean metric $\rho_{u}$ for some
positive integer $d$. Let $\mathfrak{L}^{d}$ be the $d$-dimensional Lebesgue
measure on $\mathbb{R}^{d}$.

######  3.3 Example.

Define a sequence of probability measures $\\{\nu_{n}\\}_{n\in\mathbb{N}}$ on
$[0,1]$ to be

$\nu_{n}=\left\\{\begin{array}[]{ll}n\mathfrak{L}^{1}|_{[0,\frac{1}{n}]}&\mbox{
if }n\mbox{ is }odd,\\\ \delta_{\frac{1}{n}}&\mbox{ if }n\mbox{ is }even.\\\
\end{array}\right.$

in which $\delta_{\frac{1}{n}}$ is the Dirac measure at the point
$\frac{1}{n}$. Let $\nu=\delta_{0}$ be the Dirac measure at $0$.

Obviously we have $\nu_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}\nu$ as
$n\rightarrow\infty$ (but not setwisely). By [You, Proposition 2.1] and [MMR,
Lemma 2.8], direct computations give that

$\begin{array}[]{ll}&dim_{B}(\nu_{n})=dim_{MB}(\nu_{n})=dim_{H}(\nu_{n})=dim_{P}(\nu_{n})=dim_{C}(\nu_{n})=dim_{MC}(\nu_{n})\\\
=&\left\\{\begin{array}[]{ll}1&\mbox{ if }n\mbox{ is }odd,\\\ 0&\mbox{ if
}n\mbox{ is }even.\\\ \end{array}\right.\end{array}$

So any of the sequences in (3.1)(3.2)(3.3) does not converge in Example 3.3.

We then give an example to show that in case of
$\nu_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}\nu$, even if all the
sequences

$\\{dim_{B}(\nu_{n})\\}_{n=1}^{\infty},\\{dim_{MB}(\nu_{n})\\}_{n=1}^{\infty},\\{dim_{H}(\nu_{n})\\}_{n=1}^{\infty}$,
$\\{dim_{P}(\nu_{n})\\}_{n=1}^{\infty},\\{dim_{C}(\nu_{n})\\}_{n=1}^{\infty},\\{dim_{MC}(\nu_{n})\\}_{n=1}^{\infty}$

converge, their limits may not equal

$dim_{B}(\nu),dim_{MB}(\nu),dim_{H}(\nu),dim_{P}(\nu),dim_{C}(\nu)$ and
$dim_{MC}(\nu)$

respectively. The example is borrowed from [Bil1, Example 2.2] essentially.

######  3.4 Example (Billingsley).

Define a sequence of probability measures $\nu_{n}$ on $[0,1]$ to be

$\nu_{n}=\frac{1}{n}\Sigma_{i=1}^{n}\delta_{\frac{i}{n}}$

for $n\in\mathbb{N}$.

Obviously $\nu_{n}\stackrel{{\scriptstyle
w}}{{\rightarrow}}\mathfrak{L}^{1}|_{[0,1]}$ as $n\rightarrow\infty$ (but not
setwisely). However, we have

$\lim_{n\rightarrow\infty}dim_{B}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{MB}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{H}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{P}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{C}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{MC}(\nu_{n})=0$,

while

$dim_{B}(\mathfrak{L}^{1}|_{[0,1]})=dim_{MB}(\mathfrak{L}^{1}|_{[0,1]})=dim_{H}(\mathfrak{L}^{1}|_{[0,1]})=dim_{P}(\mathfrak{L}^{1}|_{[0,1]})=dim_{C}(\mathfrak{L}^{1}|_{[0,1]})=dim_{MC}(\mathfrak{L}^{1}|_{[0,1]})=1$.

One might think that things will be better if the atomic measures are
excluded, however, the following example shows that the discontinuity still
exists among sequences of non-atomic measures.

######  3.5 Example.

Suppose $k\geq 2$ is an integer. Let

$S=\\{s_{i}:[0,1]\rightarrow[0,1]\\}_{i=1}^{k}$

be an affine IFS with the contraction ratio $0<|s_{i}|:=|s_{i}^{\prime}(x)|<1$
being a fixed number for any $1\leq i\leq k$ on $X=[0,1]$. We require it
satisfies the _strong separation condition_ :

$s_{i}([0,1])\cap s_{j}([0,1])=\emptyset$

for any $1\leq i\neq j\leq k$. We do not specify the terminals of the the
intervals $\\{s_{i}([0,1])\\}_{i=1}^{k}$ as they have no effect on the
properties we would like to demonstrate through the example. Let $0<h<1$ be
the unique solution of the _Bowen equation_

$\sum_{i=1}^{k}|s_{i}|^{h}=1$.

There is an unique ergodic measure (with respect to the push-forward of the
shift map under the projection) $\nu$ on the attractor $J$ in this case.

Let

$\mathbb{N}_{k}:=\\{1,2,\cdots,k\\}$

be the $k$-truncation of $\mathbb{N}$. For every fixed $n\in\mathbb{N}$, let

$\mathbb{N}_{k}^{n}=\\{\omega:\omega=\omega_{1}\omega_{2}\cdots\omega_{n},\omega_{i}\in\mathbb{N}_{k}\mbox{\
for any }1\leq i\leq n\\}$

be the collection of length-$n$ concatenation-words of $\mathbb{N}_{k}$. Let

$X_{\omega}=s_{\omega}([0,1])=s_{\omega_{1}}\circ s_{\omega_{2}}\cdots\circ
s_{\omega_{n}}([0,1])$

for any $\omega\in\mathbb{N}_{k}^{n}$ and any $n\in\mathbb{N}$. Define a
sequence of probability measures $\\{\nu_{n}\\}_{n\in\mathbb{N}}$ to be

$\nu_{n}=\sum_{\omega\in\mathbb{N}_{k}^{n}}|s_{\omega}|^{h}\mathfrak{L}^{1}|_{X_{\omega}}=\sum_{\omega\in\mathbb{N}_{k}^{n}}(|s_{\omega_{1}}||s_{\omega_{2}}|\cdots|s_{\omega_{n}}|)^{h}\mathfrak{L}^{1}|_{X_{\omega}}$

on $[0,1]$ for any $n\in\mathbb{N}$. It is supported on
$\cup_{\omega\in\mathbb{N}_{k}^{n}}X_{\omega}$ for each $n\in\mathbb{N}$
obviously. In fact $\nu_{n}$ is the $n$-th canonical mass distribution of the
linear cookie-cutter map in each cutting step (see for example, the _self-
similar_ measures [MR]).

Note that none of the measures above is atomic. It is easy to show that for
any open set $L\subset[0,1]$, we have $\nu_{n}(L)\geq\nu(L)$. So by [Bil1,
Theorem 2.1], we have

$\nu_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}\nu$

as $n\rightarrow\infty$ (but not setwisely). One can show that (left to the
readers)

$\lim_{n\rightarrow\infty}dim_{B}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{MB}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{H}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{P}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{C}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{MC}(\nu_{n})=1$,

while

$dim_{B}(\nu)=dim_{MB}(\nu)=dim_{H}(\nu)=dim_{P}(\nu)=dim_{C}(\nu)=dim_{MC}(\nu)=h$.

One might also hope that escalating the strength of convergence of measures
may save the continuity, but we will give an example to show that convergence
of dimensions is not true even for TV convergent sequences of measures (see
also Example 3.11).

######  3.6 Example.

Define a sequence of probability measures $\nu_{n}$ on $[0,2]$ to be

$\nu_{n}=\frac{n-1}{n}\delta_{0}+\frac{1}{n}\mathfrak{L}^{1}|_{[1,2]}.$

It is easy to see that $\|\nu_{n}-\delta_{0}\|_{TV}=\frac{1}{n}\rightarrow 0$
as $n\rightarrow\infty$, so $\nu_{n}\stackrel{{\scriptstyle
TV}}{{\rightarrow}}\delta_{0}$. However,

$\begin{array}[]{ll}&\lim_{n\rightarrow\infty}dim_{B}^{U}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{MB}^{U}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{H}^{U}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{P}^{U}(\nu_{n})=1\\\
>&0=dim_{B}(\delta_{0})=dim_{MB}(\delta_{0})=dim_{H}(\delta_{0})=dim_{P}(\delta_{0}).\end{array}$

In fact, continuity of these measure-dimension mappings has no chance to be
true if there is no dimensional restrictions on the sequences in
(3.1)(3.2)(3.3). Luckily we do have some semi-continuity property for some
measure-dimension mappings under the setwise topology. Now we prove Theorem
2.8.

Proof of Theorem 2.8:

###### Proof.

Assume that (2.1) does not hold, that is,
$\liminf_{n\rightarrow\infty}dim^{U}(\nu_{n})<dim^{U}(\nu)$. Then we can find
a sequence of positive integers $\\{n_{k}\\}_{k=1}^{\infty}$ and a real number
$a<dim^{U}(\nu)$, such that

$\lim_{k\rightarrow\infty}dim^{U}(\nu_{n_{k}})=a$.

So there exists a sequence of measurable sets $\\{A_{k}\\}_{k=1}^{\infty}$,
such that

$\nu_{n_{k}}(X\setminus A_{k})=0$ and $dim(A_{k})<a+\varepsilon<dim^{U}(\nu)$

for some small $\varepsilon>0$. Now let $A=\cup_{k=1}^{\infty}A_{k}$. Since
$\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$, we have

$\nu(X\setminus A)=\lim_{k\rightarrow\infty}\nu_{n_{k}}(X\setminus A)=0.$

However, as $dim$ satisfies the countable stability property, then

$dim(A)=\sup_{k}\\{dim(A_{k})\\}<a+\varepsilon<dim^{U}(\nu)$.

This contradicts the definition of $dim^{U}(\nu)$, which justifies our
theorem. ∎

######  3.7 Remark.

Theorem 2.8 does not hold for weakly convergent sequences
$\nu_{n}\stackrel{{\scriptstyle w}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$
in $\mathcal{M}(X)$, as one can see from our Example 3.4. Together with
Example 3.5, one can see that both lower semi-continuity and upper semi-
continuity are not true for general measure-dimension mappings under the weak
topology.

Theorem 2.8 has some interesting extensions in some special cases.

######  3.8 Corollary.

Let $dim$ be a dimensional mapping satisfying the countable stability
property. Let $dim^{U}$ be the induced upper measure-dimension mapping. For
$\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$ as
$n\rightarrow\infty$, if

$\operatorname*{ess\,sup}\\{dim^{U}(\nu_{n})\\}_{n=1}^{\infty}\leq
dim^{U}(\nu)$,

then

$\lim_{n\rightarrow\infty}dim^{U}(\nu_{n})=dim^{U}(\nu)$.

###### Proof.

This follows instantly from Theorem 2.8. ∎

######  3.9 Corollary.

The measure-dimension mappings $dim_{MB}^{U},dim_{H}^{U},dim_{P}^{U}$ are all
lower semi-continuous under the setwise topology on $\mathcal{M}(X)$.

###### Proof.

This is because they are respectively induced from dimensional mappings

$dim_{MB},dim_{H},dim_{P}$

with countable stability property, refer to [Fal1]. ∎

As to $dim_{B}^{U}$ ($\overline{dim}_{B}^{U},\underline{dim}_{B}^{U}$), since
the dimensional mapping $dim_{B}$ ($\overline{dim}_{B},\underline{dim}_{B}$)
inducing it does not satisfy countable stability, Theorem 2.8 does not apply
to it. The following example shows that it is not lower semi-continuous under
the setwise topology.

######  3.10 Example.

Let $X=\\{\frac{1}{i}\\}_{i=1}^{\infty}$. Consider the sequence of measures

$\nu_{n}=\sum_{i=1}^{n}\frac{1}{i^{2}}\delta_{\frac{1}{i}}$

for $n\in\mathbb{N}$ on $X$. Let

$\nu=\sum_{i=1}^{\infty}\frac{1}{i^{2}}\delta_{\frac{1}{i}}$.

Direct computations (left to the readers) show that
$\nu_{n}\stackrel{{\scriptstyle TV}}{{\rightarrow}}\nu$ while

$\lim_{n\rightarrow\infty}dim_{B}^{U}(\nu_{n})=0<\frac{1}{2}=dim_{B}^{U}(\nu)$

in Example 3.10. These measures can be normalized to be probabilities.

Note that the proof of Theorem 2.8 can not be applied to the measure-dimension
mapping $dim^{L}$, because if we choose a sequence of sets
$\\{A_{k}\\}_{k=1}^{\infty}$, such that

$\nu_{n_{k}}(A_{k})>0$ and $dim^{L}(A_{k})<a+\varepsilon<dim^{L}(\nu)$

we can not guarantee $\nu(A)=\lim_{k\rightarrow\infty}\nu_{n_{k}}(A)>0.$ In
fact one can see from the following example that lower semi-continuity of
typical measure-dimension mapping $dim^{L}$ is usually not true under setwise
topology on $\mathcal{M}(X)$.

######  3.11 Example.

Define a sequence of probability measures $\\{\nu_{n}\\}_{n\in\mathbb{N}}$ on
$[0,1]$ to be

$\nu_{n}=\frac{1}{n}\delta_{0}+\mathfrak{L}^{1}|_{[\frac{1}{n},1]}$

for $n\in\mathbb{N}$.

It is easy to see that
$\|\nu_{n}-\mathfrak{L}^{1}|_{[0,1]}\|_{TV}=\frac{1}{n}\rightarrow 0$ as
$n\rightarrow\infty$, so $\nu_{n}\stackrel{{\scriptstyle
TV}}{{\rightarrow}}\delta_{0}$. However,

$\lim_{n\rightarrow\infty}dim_{B}^{L}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{MB}^{L}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{H}^{L}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{P}^{L}(\nu_{n})=0<1=dim_{B}(\mathfrak{L}^{1}|_{[0,1]})=dim_{MB}(\mathfrak{L}^{1}|_{[0,1]})=dim_{H}(\mathfrak{L}^{1}|_{[0,1]})=dim_{P}(\mathfrak{L}^{1}|_{[0,1]})$.

Now we prove the semi-continuity result for the measure-dimension mapping
$dim^{L}$.

Proof of Theorem 2.9:

###### Proof.

Accoriding to the definition of $dim^{L}(\nu)$, for any small $\varepsilon>0$,
we can find a measurable set $A$ with $\nu(A)>0$ and $dim(A)\leq
dim^{L}(\nu)+\varepsilon$. Since $\nu_{n}\stackrel{{\scriptstyle
s}}{{\rightarrow}}\nu$ as $n\rightarrow\infty$, we can guarantee that

$\nu_{n}(A)>0$

for any $n$ large enough. This implies that

$\limsup_{n\rightarrow\infty}dim^{L}(\nu_{n})\leq dim^{L}(\nu)+\varepsilon$.

The proof is finished by letting $\epsilon\rightarrow 0$.

∎

The following result is instant in virtue of Theorem 2.9.

######  3.12 Corollary.

The measure-dimension mappings
$dim_{B}^{L}(\overline{dim}_{B}^{L},\underline{dim}_{B}^{L}),dim_{MB}^{L},dim_{H}^{L},dim_{P}^{L}$
are all upper semi-continuous under the setwise topology on $\mathcal{M}(X)$.

Considering Corollary 3.9 and 3.12, it would be interesting to ask whether the
two measure-dimension mappings $dim_{C}$ or $dim_{MC}$ admits any semi-
continuity. It turns out that they do not have any semi-continuity under the
TV topology in general.

######  3.13 Theorem.

The two measure-dimension mappings $dim_{C}$ and $dim_{MC}$ from
$\mathcal{M}(X)$ to $[0,+\infty)$ are neither upper semi-continuous nor lower
semi-continuous under the TV topology.

###### Proof.

Simple calculations show that, in Example 3.11,

$dim_{C}(\nu_{n})=dim_{MC}(\nu_{n})=\lim_{r\rightarrow
0}\cfrac{\log\int\nu_{n}(B(x,r))d\nu_{n}}{\log
r}=\cfrac{\log\big{(}(\frac{1}{n})^{2}+2r\frac{n-1}{n}\big{)}}{\log r}=0$,

while

$dim_{C}(\mathfrak{L}^{1}|_{[0,1]})=dim_{MC}(\mathfrak{L}^{1}|_{[0,1]})=1$.

Thus Example 3.11 justifies the fact that the two measure-dimension mappings
$dim_{C}$ and $dim_{MC}$ can not be lower semi-continuous under the TV
topology. For an example violating the upper semi-continuity of $dim_{C}$ and
$dim_{MC}$ under the TV topology, see Example 3.16. ∎

However, in virtue of [MMR, Theorem 2.6], if some restrictions are set on the
sequence of the measures $\\{\nu_{n}\\}_{n=1}^{\infty}$, we have some partial
semi-continuity results.

######  3.14 Corollary.

For a sequence of measures $\\{\nu_{n}\in\mathcal{M}(X)\\}_{n=1}^{\infty}$, if
$\nu_{n}$ has an essentially bounded density with respect to $\nu$ (which
implies that $\nu_{n}$ is absolutely continuous with respect to $\nu$) for any
$n\in\mathbb{N}$, then

$\liminf_{n\rightarrow\infty}dim_{C}(\nu_{n})\geq dim_{C}(\nu)$.

The restriction on the sequence of the measures $\\{\nu_{n}\\}_{n=1}^{\infty}$
can be relaxed, by [MMR, Theorem 2.10], to give the following lower semi-
continuity result for $dim_{MC}$ in due course.

######  3.15 Corollary.

For any sequence of measures $\\{\nu_{n}\in\mathcal{M}(X)\\}_{n=1}^{\infty}$,
if $\nu_{n}$ is absolutely continuous with respect to $\nu$ for any
$n\in\mathbb{N}$, then

$\liminf_{n\rightarrow\infty}dim_{MC}(\nu_{n})\geq dim_{MC}(\nu)$.

Considering Theorem 3.13, we still owe the readers an example of a convergent
sequence of measures violating the upper semi-continuity of $dim_{C}$ and
$dim_{MC}$ under the TV topology. The following example is an evolution of
[MMR, Example 2.5].

######  3.16 Example (Mattila-Morán-Rey).

Let $a\in(0,1)$ be a real number. Consider the sequence
$\\{a^{n^{2}}\\}_{n=0}^{\infty}\subset[0,1]$. Define a sequence of probability
measures $\nu_{n}$ on $[0,1]$ to be

$\nu_{n}=\cfrac{1-a}{1-a^{n+1}}\sum_{i=0}^{n}\cfrac{a^{i}}{a^{i^{2}}-a^{{(i+1)}^{2}}}\mathfrak{L}^{1}|_{[a^{i^{2}},a^{{(i+1)}^{2}}]}$

for any $n\in\mathbb{N}$. Let $\nu$ be the probability measure on $[0,1]$ with

$\nu=(1-a)\sum_{i=0}^{\infty}\cfrac{a^{i}}{a^{i^{2}}-a^{{(i+1)}^{2}}}\mathfrak{L}^{1}|_{[a^{i^{2}},a^{{(i+1)}^{2}}]}$.

First we show the sequence of measures converges under the TV topology.

######  3.17 Lemma.

For the sequence of measures $\\{\nu_{n}\\}_{n\in\mathbb{N}}$ in Example 3.16,
we have

$\nu_{n}\stackrel{{\scriptstyle TV}}{{\rightarrow}}\nu$ as
$n\rightarrow\infty$.

###### Proof.

In this case we have the following estimation on the total variation of the
difference between the two measures $\nu_{n}$ and $\nu$,

$\|\nu_{n}-\nu\|_{TV}=\max\\{a^{n+1},\cfrac{a^{n+1}}{1-a^{n+1}}\\}=\cfrac{a^{n+1}}{1-a^{n+1}}$

for any $n\in\mathbb{N}$. So
$\lim_{n\rightarrow\infty}\|\nu_{n}-\nu\|_{TV}=0$, which justifies its
convergence under the TV topology. ∎

Although the correlation dimension and modified correlation dimension appear
as global concepts, they are sometimes dominated by some properties on a
subset of $X$ of relatively small measure, as indicated by the following
result.

######  3.18 Lemma.

Let $\nu\in\mathcal{M}(X)$ be a probability measure. For any $r$ small enough,
if there exists a subset $Y_{r}\subset X$ of positive measure, such that for
_a.e._ $x\in Y_{r}$, the following estimation holds,

(3.4) $\nu(B(x,r))\nu(Y_{r})\geq r^{o(1)},$

in which $o(1)$ is a positive term satisfying $\lim_{r\rightarrow 0}o(1)=0$.
Then we have

$dim_{C}(\nu)=dim_{MC}(\nu)=0$.

###### Proof.

Under the above assumptions, we have

$\int\nu(B(x,r))d\nu(x)\geq\nu(B(x,r))\nu(Y_{r})$.

So

$\log\int\nu(B(x,r))d\nu(x)\geq\log\big{(}\nu(B(x,r))\nu(Y_{r})\big{)}\geq
o(1)\log r$,

which gives that

$\cfrac{\log\int\nu(B(x,r))d\nu(x)}{\log r}\leq o(1)$.

The proof ends by letting $r\rightarrow 0$ in the above estimation. ∎

Now we pay attention to the correlation and modified correlation dimensions of
the limit measure of the sequence $\\{\nu_{n}\\}_{n\in\mathbb{N}}$ in Example
3.16.

######  3.19 Proposition (Mattila-Morán-Rey).

For the limit measure $\nu$ in Example 3.16, we have

$dim_{C}(\nu)=dim_{MC}(\nu)=0$.

###### Proof.

We achieve the conclusion by Lemma 3.18. Suppose $r$ is a number small enough
such that

(3.5) $a^{n_{r}^{2}}\leq r<a^{(n_{r}+1)^{2}}$

for some $n_{r}\in\mathbb{N}$. Let $Y_{r}=Y_{n_{r}}=[0,a^{n_{r}^{2}}]$, then
we have

$\nu(Y_{n_{r}})=(1-a)(a^{n_{r}}+a^{n_{r}+1}+\cdots)=(1-a)\cfrac{a^{n_{r}}}{(1-a)}=a^{n_{r}}$.

For any $x\in Y_{n_{r}}$, since $Y_{n_{r}}\subset B(x,r)$, we have

$\nu(B(x,r))\geq\nu(Y_{n_{r}})=a^{n_{r}}$.

So

$\nu(B(x,r))\nu(Y_{r})\geq a^{2n_{r}}$

for any $x\in Y_{r}$. Considering (3.5), the condition (3.4) is satisfied in
this case. So by Lemma 3.18, we have

$dim_{C}(\nu)=dim_{MC}(\nu)=0$.

∎

Obviously for the sequence of measures $\\{\nu_{n}\\}_{n\in\mathbb{N}}$ in
Example 3.16, we have

$dim_{C}(\nu_{n})=dim_{MC}(\nu_{n})=1$.

Combining this with Lemma 3.17 and Proposition 3.19, we conclude that the
measure dimension mappings $dim_{C}$ and $dim_{MC}$ can not be upper semi-
continuous under the TV topology in general.

We end the section by some results on comparing the dimensions between two
comparable measures.

######  3.20 Lemma.

For any two measures $\nu_{1},\nu_{2}\in\mathcal{M}(X)$, if $\nu_{1}$ is
absolutely continuous with respect to $\nu_{2}$, then the induced upper
measure-dimension mapping satisfies

$\dim^{U}(\nu_{1})\leq\dim^{U}(\nu_{2})$.

###### Proof.

For $A\in\mathcal{B}$, if $\nu_{2}(X\setminus A)=0$, since $\nu_{1}$ is
absolutely continuous with respect to $\nu_{2}$, we have $\nu_{1}(X\setminus
A)=0$, so

$dim^{U}(\nu_{2})=\inf\\{dim(A):\nu_{2}(X\setminus
A)=0\\}\geq\inf\\{dim(A):\nu_{1}(X\setminus A)=0\\}=dim^{U}(\nu_{1})$.

∎

One is recommended to compare the lemma with [MMR, p220(a)]. These conclusions
again show duality of the two measure-dimension mappings.

######  3.21 Corollary.

For any two measures $\nu_{1},\nu_{2}\in\mathcal{M}(X)$, if $\nu_{1}$ is
equivalent to $\nu_{2}$, then

$\dim^{U}(\nu_{1})=\dim^{U}(\nu_{2})$

for any induced upper measure-dimension mapping.

See also [HS, Lemma 3.1(3)].

##  4\. Applications of the semi-continuity results to dimensions of measures
originated from dynamical systems

Most of our results until now are set on measures on ambient space $X$ without
dynamics on it. In this section we focus on probability measures carrying some
dynamical structures on $X$.

Let $T:X\rightarrow X$ be a transformation. Denote by
$\mathcal{M}_{\sigma}(X)$ and $\mathcal{M}_{e}(X)$ respectively to be the
collections of all the invariant and ergodic probability measures on
$(X,\mathcal{B})$, with respect to $T$. The invariant and ergodic probability
measures are crucial in many dynamical systems, while dimensions of these
measures usually reflect important properties on these dynamical systems. A
transformation $T:X\rightarrow X$ is said to be _inverse-dimension-expanding_
with respect to some dimensional mapping $dim$ if there exists some
$A\in\mathcal{B}$, such that

$dim(T^{-1}(A))>dim(A)$.

######  4.1 Young’s Lemma.

For a transformation $T$ on $(X,\mathcal{B})$ such that $T$ is not inverse
dimension-expanding with respect to $dim$, then we have

$dim^{L}(\nu)=dim^{U}(\nu)$

for any ergodic measure $\nu\in\mathcal{M}_{e}(X)$ with respect to $T$.

###### Proof.

See [You, P115]. ∎

Combining Young’s Lemma and Corollary 3.21, we have the following result.

######  4.2 Corollary.

For two equivalent measures $\nu_{1},\nu_{2}\in\mathcal{M}(X)$ with one of
them being ergodic with respect to some non-inverse-dimension-expanding
transformation $T$ on $X$, we have

$\dim^{L}(\nu_{1})=\dim^{U}(\nu_{1})=\dim^{L}(\nu_{2})=\dim^{U}(\nu_{2})$.

This corollary suggests a possible way to deal with the dimensions
$\dim^{L}(\nu)$ and $\dim^{U}(\nu)$ of some (non-invariant) measure
$\nu\in\mathcal{M}(X)$ simultaneously. That is, for some concerning measure
$\nu\in\mathcal{M}(X)$, if one can find an ergodic measure
$\nu_{e}\in\mathcal{M}_{e}(X)$ equivalent with $\nu$, then the dimension of
$\nu_{e}$ gives the dimension of $\nu$. As there are dynamical structure with
respect to $\nu_{e}$, the dimension of $\nu_{e}$ may be easier to be
calculated. In case of non-existence of such an ergodic measure equivalent
with $\nu$, one may resort to the ergodic decomposition of $\nu$. The method
still has some meaning for an invariant measure $\nu$, though an invariant
measure $\nu$ absolutely continuous with respect to an ergodic one $\nu_{e}$
necessarily satisfies

$\nu=\nu_{e}$

according to [Wal, P153, Remarks(1)].

In some cases it is difficult to decide the dimensions of ergodic (invariant)
measures admitting some singularity (for example, when the measure has
infinite entropy with respect to the transformation $T$). In these cases our
semi-continuity of measures-dimension mappings under the setwise topology
together with Young’s Lemma provide a systematic way to approximate the
dimensions of these singular ergodic (invariant) measures by sequences of
dimensions of non-singular ergodic measures converging setwisely to the
singular ones. The method usually involves the following steps.

  1. 1.

For ones’ target (singular or non-singular) measure $\nu$ on $X$ with a
transformation $T$ being not inverse-dimension-expanding, find a sequence of
non-singular measures $\\{\nu_{n}\\}_{n\in\mathbb{N}}$ converging to the
target measure $\nu$ under the setwise topology.

  2. 2.

Show the existence of ergodic measures $\\{\nu_{n}^{*}\\}_{n\in\mathbb{N}}$
(with respect to some appropriate dynamical structures on $X$) equivalent with
the ones $\\{\nu_{n}\\}_{n\in\mathbb{N}}$ respectively for any
$n\in\mathbb{N}$ (sometimes they coincide with each other).

  3. 3.

Apply Young’s Lemma and our semi-continuity results-Theorem 2.8 and 2.9 to
show the convergence of the sequence of dimensions of the measures
$\\{\nu_{n}\\}_{n\in\mathbb{N}}$ and $\\{\nu_{n}^{*}\\}_{n\in\mathbb{N}}$ to
the dimension of the concerning measure $\nu$.

For example, as a tool to tackle the obstacle of exploding entropy of some
measure with respect to some partitions on the $X$ space, we have the
following result, as an extension of [You, Main Theorem].

######  4.3 Corollary.

Let $X$ be a (non-compact) two Riemannian manifold, $T:X\rightarrow X$ is a
$C^{2}$ endomorphism. For an ergodic measure $\nu\in\mathcal{M}(X)$ with
respect to $T$, if one can find a sequence of ergodic measures
$\\{\nu_{n}\in\mathcal{M}(X)\\}_{n\in\mathbb{N}}$ with respect to a
corresponding sequence of $C^{2}$ endomorphisms $\\{T_{n}:X\rightarrow
X\\}_{n\in\mathbb{N}}$ such that

$\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$

as $n\rightarrow\infty$ and $0\leq
h_{\nu_{n}}(T_{n}),\lambda_{1}(T_{n}),\lambda_{2}(T_{n})<\infty$ for any
$n\in\mathbb{N}$, then

$\begin{array}[]{ll}&dim_{H}^{L}(\nu)=dim_{H}^{U}(\nu)=\lim_{n\rightarrow\infty}dim_{H}^{L}(\nu_{n})=\lim_{n\rightarrow\infty}dim_{H}^{U}(\nu_{n})\\\
=&\lim_{n\rightarrow\infty}h_{\nu_{n}}(T_{n})\Big{(}\cfrac{1}{\lambda_{1}(T_{n})}-\cfrac{1}{\lambda_{2}(T_{n})}\Big{)},\end{array}$

in which $\lambda_{1}(T_{n})\geq\lambda_{2}(T_{n})$ are the Lyapunov exponents
of the maps $T_{n}$ and $T_{n}^{-1}$ respectively (they are constants for
_a.e._ $x\in X$ due to ergodicity of $T_{n}$).

###### Proof.

First, considering Young’s Lemma, apply [You, Main Theorem] to the measure
$\nu_{n}$ with finite entropy and Lyapunov exponents, we have

(4.1)
$dim_{H}^{L}(\nu_{n})=dim_{H}^{U}(\nu_{n})=h_{\nu_{n}}(T_{n})\Big{(}\cfrac{1}{\lambda_{1}(T_{n})}-\cfrac{1}{\lambda_{2}(T_{n})}\Big{)}.$

Since $\nu_{n}\stackrel{{\scriptstyle s}}{{\rightarrow}}\nu$ as
$n\rightarrow\infty$, apply Theorem 2.8 to the sequence of measures, we get

(4.2) $\liminf_{n\rightarrow\infty}dim^{U}(\nu_{n})\geq dim^{U}(\nu),$

while applying 2.9 to the sequence of measures, we get

(4.3) $\limsup_{n\rightarrow\infty}dim^{L}(\nu_{n})\leq dim^{L}(\nu).$

Due to the ergodicity and Young’s Lemma we have $dim^{U}(\nu)=dim^{L}(\nu)$,
combing with (4.2) and (4.3) together we get

(4.4) $\limsup_{n\rightarrow\infty}dim^{L}(\nu_{n})\leq
dim^{L}(\nu)=dim^{U}(\nu)\leq\liminf_{n\rightarrow\infty}dim^{U}(\nu_{n}).$

(4.1) and (4.4) together justify the corollary. ∎

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