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Let $f$, $p$, and $q$ be Laurent polynomials with integer coefficients in one or several variables, and suppose that $f$ divides $p+q$. We establish sufficient conditions to guarantee that $f$ individually divides $p$ and $q$. These conditions involve a bound on coefficients, a separation between the supports of $p$ and $q$, and, surprisingly, a requirement on the complex variety of $f$ called atorality satisfied by many but not all polynomials. Our proof involves a related dynamical system and the fundamental dynamical notion of homoclinic point. Without the atorality assumption our methods fail, and it is unknown whether our results hold without this assumption., Comment: 8 pages

We construct natural symbolic representations of intrinsically ergodic, but not necessarily expansive, principal algebraic actions of countably infinite amenable groups and use these representations to find explicit generating partitions (up to null-sets) for such actions, Comment: 26 pages

Let $f$ be a Laurent polynomial in $d$ commuting variables with integer coefficients. Associated to $f$ is the principal algebraic $\mathbb{Z}^d$-action $\alpha_f$ on a compact subgroup $X_f$ of $\mathbb{T}^{\mathbb{Z}^d}$ determined by $f$. Let $N\ge1$ and restrict points in $X_f$ to coordinates in $N\mathbb{Z}^d$. The resulting algebraic $N\mathbb{Z}^d$-action is again principal, and is associated to a polynomial $g_N$ whose support grows with $N$ and whose coefficients grow exponentially with $N$. We prove that by suitably renormalizing these decimations we can identify a limiting behavior given by a continuous concave function on the Newton polytope of $f$, and show that this decimation limit is the negative of the Legendre dual of the Ronkin function of $f$. In certain cases with two variables, the decimation limit coincides with the surface tension of random surfaces related to dimer models, but the statistical physics methods used to prove this are quite different and depend on special properties of the polynomial., Comment: 33 pages with 8 figures. An informal account, with background and motivation, in contained in a talk by Lind that can be viewed at https://www.youtube.com/watch?v=jT2OQifgldk&t=120s

For a continuous $\mathbb{N}^d$ or $\mathbb{Z}^d$ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic $\mathbb{Z}$ actions of positive entropy are Bohr-chaotic. The same is proved for principal algebraic $\mathbb{Z}^d$ ($d\ge 2$) actions of positive entropy under the condition of existence of summable homoclinic points.

We give examples of principal algebraic actions of the noncommutative free group $F$ of rank two, as well as other groups, by automorphisms of a connected compact abelian group for which there is an explicit measurable isomorphism to a Bernoulli action of the group. The isomorphism is defined using homoclinic points, a method that has been used earlier to construct symbolic covers of algebraic actions. To our knowledge, these are the first nontrivial examples of the Bernoullicity of an algebraic action of $F$., Comment: 13 pages. This update to our previous version generalizes our arguments to more acting groups and a more general class of principal actions. It also includes a different proof that the Bernoulli measure maps to Haar measure using a Fourier coefficient argument due to Ben Hayes

Every copula $ C $ for a random vector $ {\bf X}=(X_1,\dots,X_d) $ with identically distributed coordinates determines a unique copula $ C_{:d} $ for its order statistic $ {\bf X}_{:d}=(X_{1:d},\dots,X_{d:d}) $. In the present paper we study the dependence structure of $ C_{:d} $ via Kendall's tau, denoted by $ \kappa $. As a general result, we show that $ \kappa[C_{:d}] $ is at least as large as $ \kappa[C] $. For the product copula $ \Pi $, which corresponds to the case of independent coordinates of $ {\bf X} $, we provide an explicit formula for $ \kappa[\Pi_{:d}] $ showing that the inequality between $ \kappa[\Pi] $ and $ \kappa[\Pi_{:d}] $ is strict. We also compute Kendall's tau for certain multivariate margins of $ \Pi_{:d} $ corresponding to the lower or upper coordinates of $ {\bf X}_{:d} $.

After Furstenberg had provided a first glimpse of remarkable rigidity phenomena associated with the joint action of several commuting automorphisms (or endomorphisms) of a compact abelian group, further key examples motivated the development of an extensive theory of such actions. Two of Mahler's achievements, the recognition of the significance of Mahler measure of multivariate polynomials in relating the lengths and heights of products of polynomials in terms of the corresponding quantities for the constituent factors, and his work on additive relations in fields, have unexpectedly played important roles in the study of entropy and higher order mixing for these actions. This article briefly surveys these connections between Mahler's work and dynamics. It also sketches some of the dynamical outgrowths of his work that are very active today, including the investigation of the Fuglede-Kadison determinant of a convolution operator in a group von Neumann algebra as a noncommutative generalization of Mahler measure, as well as diophantine questions related to the growth rates of periodic points and their relation to entropy., Comment: 20 pages, to appear in the book "Mahler Selecta" in the book series of Documenta Mathematica. This version corrects a few typos and also Example 2.2

In the present paper we study quantile risk measures and their domain. Our starting point is that, for a probability measure $ Q $ on the open unit interval and a wide class $ \mathcal{L}_Q $ of random variables, we define the quantile risk measure $ \varrho_Q $ as the map which integrates the quantile function of a random variable in $ \mathcal{L}_Q $ with respect to $ Q $. The definition of $ \mathcal{L}_Q $ ensures that $ \varrho_Q $ cannot attain the value $ +\infty $ and cannot be extended beyond $ \mathcal{L}_Q $ without losing this property. The notion of a quantile risk measure is a natural generalization of that of a spectral risk measure and provides another view at the distortion risk measures generated by a distribution function on the unit interval. In this general setting, we prove several results on quantile or spectral risk measures and their domain with special consideration of the expected shortfall.

In the present paper we propose and study estimators for a wide class of bivariate measures of concordance for copulas. These measures of concordance are generated by a copula and generalize Spearman's rho and Gini's gamma. In the case of Spearman's rho and Gini's gamma the estimators turn out to be the usual sample versions of these measures of concordance., Comment: 13 pages

This article contains a Wiener Lemma for the convolution algebra $\ell^1(\mathbb H,\mathbb C)$ and group $C^\ast$-algebra $C^\ast(\mathbb H)$ of the discrete Heisenberg group $\mathbb H$. At first, a short review of Wiener's Lemma in its classical form and general results about invertibility in group algebras of nilpotent groups will be presented. The known literature on this topic suggests that invertibility investigations in the group algebras of $\mathbb H$ rely on the complete knowledge of $\widehat{\mathbb H}$ -- the dual of $\mathbb H$, i.e., the space of unitary equivalence classes of irreducible unitary representations. We will describe the dual of ${\mathbb H}$ explicitly and discuss its structure. Wiener's Lemma provides a convenient condition to verify invertibility in $\ell^1(\mathbb H,\mathbb C)$ and $C^\ast(\mathbb H)$ which bypasses $\widehat{\mathbb H}$. The proof of Wiener's Lemma for $\mathbb H$ relies on local principles and can be generalised to countable nilpotent groups. As our analysis shows, the main representation theoretical objects to study invertibility in group algebras of nilpotent groups are the corresponding primitive ideal spaces. Wiener's Lemma for $\mathbb H$ has interesting applications in algebraic dynamics and Time-Frequency Analysis which will be presented in this article as well., Comment: Monatshefte f\"ur Mathematik 2016