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We find sufficient conditions for a self-map of the unit ball to converge uniformly under iteration to a fixed point or idempotent on the entire ball. Using these tools, we establish spectral containments for weighted composition operators on Hardy and Bergman spaces of the ball. When the compositional symbol is in the Schur–Agler class, we establish the spectral radii of these weighted composition operators. 

Abstract Previously, spectra of certain weighted composition operators W ѱ, φ on H 2 were determined under one of two hypotheses: either φ converges under iteration to the Denjoy-Wolff point uniformly on all of 𝔻 rather than simply on compact subsets, or φ is “essentially linear fractional.” We show that if φ is a quadratic self-map of 𝔻 of parabolic type, then the spectrum of W ѱ, φ can be found when these maps exhibit both of the aforementioned properties, and we determine which symbols do so. 

We investigate the origin of the elliptical ring structure observed in the images of the supermassive black hole M87^*, aiming to disentangle contributions from gravitational, astrophysical, and imaging effects. Leveraging the enhanced capabilities of the Event Horizon Telescope (EHT)'s 2018 array, including improved (u,v)-coverage from the Greenland Telescope, we measured the ring's ellipticity using five independent imaging methods, obtaining a consistent average value ofτ = 0.08_−0.02^+0.03with a position angle ofξ = 50.1_−7.6^+6.2 degrees. To interpret this measurement, we compared it to general relativistic magnetohydrodynamic (GRMHD) simulations spanning a wide range of physical parameters including the thermal or nonthermal electron distribution function, spins, and ion-to-electron temperature ratios in both low- and high-density regions. We find no statistically significant correlation between spin and ellipticity in GRMHD images. Instead, we identify a correlation between ellipticity and the fraction of non-ring emission, particularly in nonthermal models and models with higher jet emission. These results indicate that the ellipticity measured from the M87^*emission structure is consistent with that expected from simulations of turbulent accretion flows around black holes, where it is dominated by astrophysical effects rather than gravitational ones. Future high-resolution imaging, including space very long baseline interferometry and long-term monitoring, will be essential to isolate gravitational signatures from astrophysical effects. 

Aims.We investigated the polarization and Faraday properties of Messier 87 (M87) and seven other radio-loud active galactic nuclei (AGNs) atλ0.87 mm (345 GHz) using the Atacama Large Millimeter/submillimeter Array (ALMA). Our goal was to characterize the linear polarization (LP) fractions, measure Faraday rotation measures (RMs), and examine the magnetic field structures in the emission regions of these AGNs. Methods.We conducted full-polarization observations as part of the ALMA Band 7 very long baseline interferometry (VLBI) commissioning during the April 2021 Event Horizon Telescope (EHT) campaign. We analyzed the LP fractions and RMs to assess the nature of Faraday screens and magnetic fields in the submillimeter emission regions. Results.We find LP fractions between 1% and 17% and RMs exceeding 10⁵ rad m⁻², which are 1–2 orders of magnitude higher than typically observed at longer wavelengths (λ>3 mm). This suggests denser Faraday screens or stronger magnetic fields. Additionally, we present the first submillimeter polarized images of the M87 jet and the observed AGNs, revealing RM gradients and sign reversals in the M87 jet indicative of a kiloparsec-scale helical magnetic field structure. Conclusions.Our results provide essential constraints for calibrating, analyzing, and interpreting VLBI data from the EHT at 345 GHz, representing a critical step toward submillimeter VLBI imaging. 

ABSTRACT We present ALMA Band 7 polarization observations of the OMC-1 region of the Orion molecular cloud. We find that the polarization pattern observed in the region is likely to have been significantly altered by the radiation field of the >104 L⊙ high-mass protostar Orion Source I. In the protostar’s optically thick disc, polarization is likely to arise from dust self-scattering. In material to the south of Source I – previously identified as a region of ‘anomalous’ polarization emission – we observe a polarization geometry concentric around Source I. We demonstrate that Source I’s extreme luminosity may be sufficient to make the radiative precession time-scale shorter than the Larmor time-scale for moderately large grains ($$\gt 0.005\!-\!0.1\, \mu$$m), causing them to precess around the radiation anisotropy vector (k-RATs) rather than the magnetic field direction (B-RATs). This requires relatively unobscured emission from Source I, supporting the hypothesis that emission in this region arises from the cavity wall of the Source I outflow. This is one of the first times that evidence for k-RAT alignment has been found outside of a protostellar disc or AGB star envelope. Alternatively, the grains may remain aligned by B-RATs and trace gas infall on to the Main Ridge. Elsewhere, we largely find the magnetic field geometry to be radial around the BN/KL explosion centre, consistent with previous observations. However, in the Main Ridge, the magnetic field geometry appears to remain consistent with the larger-scale magnetic field, perhaps indicative of the ability of the dense Ridge to resist disruption by the BN/KL explosion.