Undecidable properties of self-affine sets and multi-tape automata

We study the decidability of the topological properties of some objects coming from fractal geometry. We prove that having empty interior is undecidable for the sets defined by two-dimensional graph-directed iterated function systems. These results are obtained by studying a particular class of self-affine sets associated with multi-tape automata. We first establish the undecidability of some language-theoretical properties of such automata, which then translate into undecidability results about their associated self-affine sets.Comment: 10 pages, v2 includes some corrections to match the published versio

Global periodicity conditions for maps and recurrences via Normal Forms

We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two and three dimensional classes of polynomial or rational maps. In particular we find the global periodic cases for several Lyness type recurrences.Comment: 25 page

Small unions of affine subspaces and skeletons via Baire category

Our aim is to find the minimal Hausdorff dimension of the union of scaled and/or rotated copies of the k-skeleton of a fixed polytope centered at the points of a given set. For many of these problems, we show that a typical arrangement in the sense of Baire category gives minimal Hausdorff dimension. In particular, this proves a conjecture of R. Thornton. Our results also show that Nikodym sets are typical among all sets which contain, for every x∈Rn, a punctured hyperplane H∖{x} through x. With similar methods we also construct a Borel subset of Rn of Lebesgue measure zero containing a hyperplane at every positive distance from every point. © 201

On the number of Mather measures of Lagrangian systems

In 1996, Ricardo Ricardo Ma\~n\'e discovered that Mather measures are in fact the minimizers of a "universal" infinite dimensional linear programming problem. This fundamental result has many applications, one of the most important is to the estimates of the generic number of Mather measures. Ma\~n\'e obtained the first estimation of that sort by using finite dimensional approximations. Recently, we were able with Gonzalo Contreras to use this method of finite dimensional approximation in order to solve a conjecture of John Mather concerning the generic number of Mather measures for families of Lagrangian systems. In the present paper we obtain finer results in that direction by applying directly some classical tools of convex analysis to the infinite dimensional problem. We use a notion of countably rectifiable sets of finite codimension in Banach (and Frechet) spaces which may deserve independent interest

Variations, approximation, and low regularity in one dimension

We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general of the form of a standard Lipschitz "variation". Part of this investigation, but of interest in its own right, is an example of a nowhere locally Lipschitz minimizer which serves as a counter-example to any putative Tonelli partial regularity statement. Under these low assumptions we find it nonetheless remains possible to derive necessary conditions for minimizers, in terms of approximate continuity and equality of the one-sided derivatives.Comment: v3, 60 pages. To appear in CoVPDE. Minor cosmetic correction

Discrete integrable systems and Poisson algebras from cluster maps

We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutation-periodic with period 1. Such quivers were completely classified by Fordy and Marsh, who characterised them in terms of the skew-symmetric matrix that defines the quiver. The associated nonlinear recurrences are equivalent to birational maps, and we explain how these maps can be endowed with an invariant Poisson bracket and/or presymplectic structure. Upon applying the algebraic entropy test, we are led to a series of conjectures which imply that the entropy of the cluster maps can be determined from their tropical analogues, which leads to a sharp classification result. Only four special families of these maps should have zero entropy. These families are examined in detail, with many explicit examples given, and we show how they lead to discrete dynamics that is integrable in the Liouville-Arnold sense.Comment: 49 pages, 3 figures. Reduced to satisfy journal page restrictions. Sections 2.4, 4.5, 6.3, 7 and 8 removed. All other results remain, with minor editin

A Multi-epoch, Multiwavelength Study of the Classical FUor V1515 Cyg Approaching Quiescence

Historically, FU Orionis-type stars are low-mass, pre-main-sequence stars. The members of this class experience powerful accretion outbursts and remain in an enhanced accretion state for decades or centuries. V1515 Cyg, a classical FUor, started brightening in the 1940s and reached its peak brightness in the late 1970s. Following a sudden decrease in brightness, it stayed in a minimum state for a few months, then started brightening for several years. We present the results of our ground-based photometric monitoring complemented with optical/near-infrared spectroscopic monitoring. Our light curves show a long-term fading with strong variability on weekly and monthly timescales. The optical spectra show P Cygni profiles and broad blueshifted absorption lines, common properties of FUors. However, V1515 Cyg lacks the P Cygni profile in the Ca II 8498 Å line, a part of the Ca infrared triplet, formed by an outflowing wind, suggesting that the absorbing gas in the wind is optically thin. The newly obtained near-infrared spectrum shows the strengthening of the CO bandhead and the FeH molecular band, indicating that the disk has become cooler since the last spectroscopic observation in 2015. The current luminosity of the accretion disk dropped from the peak value of 138 L ⊙ to about 45 L ⊙, suggesting that the long-term fading is also partly caused by the dropping of the accretion rate

Photometric and Spectroscopic Properties of Type Ia Supernova 2018oh with Early Excess Emission from the Kepler 2 Observations

Supernova (SN) 2018oh (ASASSN-18bt) is the first spectroscopically confirmed Type Ia supernova (SN Ia) observed in the Kepler field. The Kepler data revealed an excess emission in its early light curve, allowing us to place interesting constraints on its progenitor system. Here we present extensive optical, ultraviolet, and near-infrared photometry, as well as dense sampling of optical spectra, for this object. SN 2018oh is relatively normal in its photometric evolution, with a rise time of 18.3 ± 0.3 days and Δm 15(B) = 0.96 ± 0.03 mag, but it seems to have bluer B − V colors. We construct the "UVOIR" bolometric light curve having a peak luminosity of 1.49 × 1043 erg s−1, from which we derive a nickel mass as 0.55 ± 0.04 M ⊙ by fitting radiation diffusion models powered by centrally located 56Ni. Note that the moment when nickel-powered luminosity starts to emerge is +3.85 days after the first light in the Kepler data, suggesting other origins of the early-time emission, e.g., mixing of 56Ni to outer layers of the ejecta or interaction between the ejecta and nearby circumstellar material or a nondegenerate companion star. The spectral evolution of SN 2018oh is similar to that of a normal SN Ia but is characterized by prominent and persistent carbon absorption features. The C ii features can be detected from the early phases to about 3 weeks after the maximum light, representing the latest detection of carbon ever recorded in an SN Ia. This indicates that a considerable amount of unburned carbon exists in the ejecta of SN 2018oh and may mix into deeper layers