We provide a proof of the (well-known) result that the Poincare exponent of a non-elementary Kleinian group is a lower bound for the upper box dimension of the limit set. Our proof only uses elementary hyperbolic and fractal geometry.
We present the spectroscopic and photometric study of five intermediate-luminosity red transients (ILRTs), namely AT 2010dn, AT 2012jc, AT 2013la, AT 2013lb, and AT 2018aes. They share common observational properties and belong to a family of objects
similar to the prototypical ILRT SN~2008S. These events have a rise time that is less than 15 days and absolute peak magnitudes of between $-11.5$ and $-14.5$ mag. Their pseudo-bolometric light curves peak in the range $0.5$ - $9.0 times10^{40}~mathrm{erg~s}^{-1}$ and their total radiated energies are on the order of (0.3 - 3) $times$~10$^{47}$~erg. After maximum brightness, the light curves show a monotonic decline or a plateau, resembling those of faint supernovae IIL or IIP, respectively. At late phases, the light curves flatten, roughly following the slope of the $^{56}$Co decay. If the late-time power source is indeed radioactive decay, these transients produce $^{56}$Ni masses on the order of $10^{-4}$ to $10^{-3}$~msun. The spectral energy distribution of our ILRT sample, extending from the optical to the mid-infrared (MIR) domain, reveals a clear IR excess soon after explosion and non-negligible MIR emission at very late phases. The spectra show prominent H lines in emission with a typical velocity of a few hundred km~s$^{-1}$, along with Ca~II features. In particular, the [Ca~II] $lambda$7291,7324 doublet is visible at all times, which is a characteristic feature for this family of transients. The identified progenitor of SN~2008S, which is luminous in archival Spitzer MIR images, suggests an intermediate-mass precursor star embedded in a dusty cocoon. We propose the explosion of a super-asymptotic giant branch star forming an electron-capture supernova as a plausible explanation for these events.
A $(d,k)$-set is a subset of $mathbb{R}^d$ containing a $k$-dimensional unit ball of all possible orientations. Using an approach of D.~Oberlin we prove various Fourier dimension estimates for compact $(d,k)$-sets. Our main interest is in restricted
$(d,k)$-sets, where the set only contains unit balls with a restricted set of possible orientations $Gamma$. In this setting our estimates depend on the Hausdorff dimension of $Gamma$ and can sometimes be improved if additional geometric properties of $Gamma$ are assumed.
Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the orbital set is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of Euclidean spac
e. We prove that the upper box dimension of an orbital set is given by the maximum of three quantities: the upper box dimension of the given set; the Poincare exponent of the Kleinian group; and the upper box dimension of the limit set of the Kleinian group. Since we do not make any assumptions about the Kleinian group, none of the terms in the maximum can be removed in general. We show by constructing an explicit example that the (hyperbolic) boundedness assumption on $C$ cannot be removed in general.
Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are replaced with
a countable hierarchy of `well-spread points, which we refer to as abstract rationals. We prove various Jarnik-Besicovitch type dimension bounds and investigate their sharpness.
We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of contractions. Our primary focus is on the intermediate dimensions: a family of dimensions depending on a parameter $theta in [0,1]$
which interpolate between the Hausdorff and box dimensions. Our main results are in the case when all the contractions are conformal. Under a natural separation condition we prove that the intermediate dimensions of the limit set are the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This builds on work of Mauldin and Urbanski concerning the Hausdorff and upper box dimension. We give several (often counter-intuitive) applications of our work to dimensions of projections, fractional Brownian images, and general Holder images. These applications apply to well-studied examples such as sets of numbers which have real or complex continued fraction expansions with restricted entries. We also obtain several results without assuming conformality or any separation conditions. We prove general upper bounds for the Hausdorff, box and intermediate dimensions of infinitely generated attractors in terms of a topological pressure function. We also show that the limit set of a generic infinite iterated function system has box and intermediate dimensions equal to the ambient spatial dimension, where generic can refer to any one of (i) full measure; (ii) prevalent; or (iii) comeagre.
In the second of two papers on the peculiar interacting transient AT 2016jbu, we present the bolometric lightcurve, identification and analysis of the progenitor candidate, as well as preliminary modelling to help elucidate the nature of this event.
We identify the progenitor candidate for AT 2016jbu in quiescence, and find it to be consistent with a $sim$20 M$_{odot}$ yellow hypergiant surrounded by a dusty circumstellar shell. We see evidence for significant photometric variability in the progenitor, as well as strong H$alpha$ emission consistent with pre-existing circumstellar material. The age of the resolved stellar population surrounding AT 2016jbu, as well as integral-field unit spectra of the region support a progenitor age of >16 Myr, again consistent with a progenitor mass of $sim$20 M$_{odot}$. Through a joint analysis of the velocity evolution of AT 2016jbu, and the photospheric radius inferred from the bolometric lightcurve, we find that the transient is consistent with two successive outbursts or explosions. The first outburst ejected a shell of material with velocity 650 km $s^{-1}$, while the second more energetic event ejected material at 4500 km $s^{-1}$. Whether the latter is the core-collapse of the progenitor remains uncertain, as the required ejecta mass is relatively low (few tenths of M$_{odot}$). We also place a restrictive upper limit on the ejected $^{56}$Ni mass of <0.016 M$_{odot}$. Using the BPASS code, we explore a wide range of possible progenitor systems, and find that the majority of these are in binaries, some of which are undergoing mass transfer or common envelope evolution immediately prior to explosion. Finally, we use the SNEC code to demonstrate that the low-energy explosion of some of these systems together with sufficient CSM can reproduce the overall morphology of the lightcurve of AT 2016jbu.
We present comprehensive, multi-wavelength observations of AT 2016jbu, an interacting transient. High cadence photometric coverage reveals that AT 2016jbu underwent significant photometric variability followed by two luminous events, the latter of wh
ich reached an absolute magnitude of M$_Vsim$-18.5 mag. This is similar to the transient SN 2009ip whose nature is still debated. Spectra are dominated by narrow emission lines and show a blue continuum during the peak of the second event. AT 2016jbu shows signatures of a complex, non-homogeneous circumstellar material (CSM). We see slowly evolving asymmetric hydrogen line profiles, with velocities of 500 km $s^{-1}$ seen in narrow emission features from a slow moving CSM, and up to 10,000 km $s^{-1}$ seen in broad absorption from some high velocity material. Similar velocities are seen in other SN 2009ip-like transients. Late-time spectra ($sim$+1 year) show a lack of forbidden emission lines expected from a core-collapse supernova during the nebular phase and are dominated by strong emission from H, He I and Ca II. Strong asymmetric emission features, a bumpy lightcurve, and continually evolving spectra suggest late time CSM interaction is inhibiting the emergence of any nebular features. We compare the evolution of H$alpha$ among SN 2009ip-like transients and find possible evidence for orientation angle effects. The light-curve evolution of AT 2016jbu suggests similar, but not identical, circumstellar environments to other SN 2009ip-like transients. In Paper II we continue the discussion of AT 2016jbu and SN 2009ip-like transients and using the data presented here, we focus on the local environment, the progenitor, and on modelling the transient itself.
We present a study of optical and near-infrared (NIR) spectra along with the light curves of SN 2013ai. These data range from discovery until 380 days after explosion. SN 2013ai is a fast declining type II supernova (SN II) with an unusually long ris
e time; $18.9pm2.7$d in $V$ band and a bright $V$ band peak absolute magnitude of $-18.7pm0.06$ mag. The spectra are dominated by hydrogen features in the optical and NIR. The spectral features of SN 2013ai are unique in their expansion velocities, which when compared to large samples of SNe II are more than 1,000 kms faster at 50 days past explosion. In addition, the long rise time of the light curve more closely resembles SNe IIb rather than SNe II. If SN 2013ai is coeval with a nearby compact cluster we infer a progenitor ZAMS mass of $sim$17 M$_odot$. After performing light curve modeling we find that SN 2013ai could be the result of the explosion of a star with little hydrogen mass, a large amount of synthesized $^{56}$Ni, 0.3-0.4 M$_odot$, and an explosion energy of $2.5-3.0times10^{51}$ ergs. The density structure and expansion velocities of SN 2013ai are similar to that of the prototypical SN IIb, SN 1993J. However, SN 2013ai shows no strong helium features in the optical, likely due to the presence of a dense core that prevents the majority of $gamma$-rays from escaping to excite helium. Our analysis suggests that SN 2013ai could be a link between SNe II and stripped envelope SNe.
We present the results of our monitoring campaigns of the luminous red novae (LRNe) AT 2020hat in NGC 5068 and AT 2020kog in NGC 6106. The two objects were imaged (and detected) before their discovery by routine survey operations. They show a general
trend of slow luminosity rise, lasting at least a few months. The subsequent major LRN outbursts were extensively followed in photometry and spectroscopy. The light curves present an initial short-duration peak, followed by a redder plateau phase. AT 2020kog is a moderately luminous event peaking at ~7 x 10^40 erg/s, while AT 2020hat is almost one order of magnitude fainter than AT 2020kog, although it is still more luminous than V838 Mon. In analogy with other LRNe, the spectra of AT 2020kog change significantly with time. They resemble those of type IIn supernovae at early phases, then they become similar to those of K-type stars during the plateau, and to M-type stars at very late phases. In contrast, AT 2020hat already shows a redder continuum at early epochs, and its spectrum shows the late appearance of molecular bands. A moderate-resolution spectrum of AT 2020hat taken at +37 d after maximum shows a forest of narrow P Cygni lines of metals with velocities of 180 km/s, along with an Halpha emission with a full-width at half-maximum velocity of 250 km/s. For AT 2020hat, a robust constraint on its quiescent progenitor is provided by archival images of the Hubble Space Telescope. The progenitor is clearly detected as a mid-K type star, with an absolute magnitude of MF606W = -3.33+-0.09 mag and a colour of F606W-F814W = 1.14+-0.05 mag, which are inconsistent with the expectations from a massive star that could later produce a core-collapse supernova. Although quite peculiar, the two objects nicely match the progenitor versus light curve absolute magnitude correlations discussed in the literature.
