In 1981, Arnoux and Yoccoz gave the first examples of pseudo-Anosov maps with odd degree stretch factors. In 1985, D. Fried deduced the existence of a pseudo-Anosov map in genus three with the same stretch factor as the Arnoux-Yoccoz example in that genus, and asked if these were the same. We show that they are distinct. We do this by, in a sense, reversing Fried's construction; we show that the mapping torus of the pseudo-Anosov map induced by the Arnoux-Yoccoz map on the surface obtained by blowing-up its two singularities has no cross section which is a torus with two points blown-up.
We introduce tropical skeletons for Berkovich spaces based on results of Ducros. Then we study harmonic functions on good strictly analytic spaces over a non-trivially valued non-Archimedean field. Chambert-Loir and Ducros introduced bigraded sheaves of smooth real-valued differential forms on Berkovich spaces by pulling back Lagerberg forms with respect to tropicalization maps. We give a new approach in which we allow pullback by more general harmonic tropicalizations to get a larger sheaf of differential forms with essentially the same properties, but with a better cohomological behavior. A crucial ingredient is that tropical varieties arising from harmonic tropicalization maps are balanced.
Zariski decomposition plays an important role in the theory of algebraic surfaces due to many applications. For irreducible symplectic manifolds Boucksom provided a characterization of his divisorial Zariski decomposition in terms of the Beauville-Bogomolov-Fujiki quadratic form. Different variants of singular holomorphic symplectic varieties have been extensively studied in recent years. In this note we first show that the Boucksom-Zariski decomposition holds for effective divisors in the largest possible framework of varieties with symplectic singularities. On the other hand in the case of projective surfaces, it was recently shown that there is a strict relation between the boundedness of coefficients of Zariski decompositions of pseudoeffective integral divisors and the bounded negativity conjecture. In the present note, we show that an analogous phenomenon can be observed in the case of projective irreducible symplectic manifolds. We furthermore prove an effective analog of the bounded negativity conjecture in the smooth case. Combining these results we obtain information on the denominators of Boucksom-Zariski decompositions for holomorphic symplectic manifolds. From such a bound we easily deduce a result of effective birationality for big line bundles on projective holomorphic symplectic manifolds, answering to a question asked by F. Charles.
We consider the motion of several solids in a bounded cavity filled with a perfect incompressible fluid, in two dimensions. The solids move according to Newton's law, under the influence of the fluid's pressure, and the fluid dynamics is driven by the 2D incompressible Euler equations, which are set on the time-dependent domain corresponding to the cavity deprived of the sets occupied by the solids. We assume that the fluid vorticity is initially bounded and that the circulations around the solids may be non-zero. The existence of a unique corresponding solution, \`a la Yudovich, to this system, up to a possible collision, is known. In this paper we identify the limit dynamics of the system when the radius of some of the solids converge to zero depending on how, for each body, the inertia is scaled with the radius. We obtain in the limit some point vortex systems for the solids converging to particles and a form of Newton's law for the solids that have a fixed radius; for the fluid we obtain an Euler-type system. This extends earlier works to the case of several moving rigid bodies. A crucial point is to understand the interaction, through the fluid, between small moving solids, and for that we use some normal forms of the ODEs driving the motion of the solids in two steps: first we use a normal form for the system coupling the time-evolution of all the solids to obtain a rough estimate of the acceleration of the bodies, then we turn to some normal forms that are specific to each small solid, with an appropriate modulation related to the influence of the other solids and of the fluid vorticity, to obtain some precise uniform a priori estimates of the velocities of the bodies, and then pass to the limit.
We find the decomposition matrices of the Category O for rational Cherednik algebras of Coxeter groups of types H4, F4, E6, E7 and E8. The decomposition matrix is an important numerical invariant of the Category O: the matrix entries are the multiplicities of irreducible modules in standard modules (also called Verma modules). In the case of F4, E6, E7 and E8, the parameter is not a half-integer and for E8 we only consider the blocks of defect 2. In the case of F4, we consider the case of equal parameters only. As a consequence, we obtain character formulas for irreducible representations in the Category O, as well as a classification of the finite-dimensional representations of these rational Cherednik algebras.
We study higher Hochschild homology evaluated on wedges of circles, viewed as a functor on the category of free groups. The principal results use coefficients arising from square-zero extensions; this is motivated in part by work of Turchin and Willwacher in relation to hairy graph cohomology. Working over a field of characteristic zero, achieve effective calculations of higher Hochschild homology, generalizing initial results of Turchin and Willwacher. This exploits exponential functors, which arise both in calculating higher Hochschild homology and in relation to polynomial functors on free groups. It is a crucial observation that these are related. For the main cases of interest, the action of automorphisms of free groups on Hochschild homology factors across outer automorphisms. As the appropriate underlying framework, we introduce and make essential use of the category of outer functors, the full subcategory of functors on free groups on which inner automorphisms act trivially.
In this paper we produce unconditionally new instances of Galois number field extensions exhibiting strong discrepancies in the distribution of Frobenius elements among conjugacy classes of the Galois group. We first prove an inverse Galois theoretic statement showing a dichotomy between “extreme Chebyshev biases” and “equal prime ideal counting”. We further introduce a group theoretic property that implies extreme biases. In the case of abelian extensions this leads to a complete characterization of Galois groups enabling extreme biases. In the case where the Galois group is a p-group, a simple criterion is deduced for the existence of extreme biases, and associated effective statements of Linnik type are obtained.
Fresnel and de Mathan proved that the p-adic Fourier transform is surjective. We reinterpret their result in terms of analytic boundaries, and extend it beyond the cyclotomic case. We also give some applications of their result to Schneider and Teitelbaum's p-adic Fourier theory, in particular to generalized Mahler expansions and to the geometry of the character variety.
One of the versions of the wind-tree model of Boltzmann gas, suggested by Paul and Tatiana Ehrenfest more than a century ago, can be seen as a billiard in the plane endowed with $\mathbb{Z}\oplus\mathbb{Z}$-periodic rectangular obstacles. In the breakthrough paper by V. Delecroix, P. Hubert and S. Lelievre the authors proved, that the diffusion rate of trajectories in such a billiard is equal to $\frac{2}{3}$, that is the \textit{maximal} distance from the origin achieved by a point of a typical trajectory on a segment of time $[0,t]$ grows roughly as $t^\frac{2}{3}$ for large $t$. Here $\frac{2}{3}$ is the Lyapunov exponent of the associated renormalizing dynamical system. This pioneering result does not tell, however, whether trajectories spend most of the time close or far from the initial point. In the current paper, we prove that the \textit{average} distance from the origin grows with the same rate $t^\frac{2}{3}$. In plain terms, it means that trajectories mostly stay as far as possible from the initial point (though, it is known that the wind-tree billiard is recurrent, so trajectories occasionally pass close to the initial point). More generally, fundamental rigidity results by A.Eskin and M.Mirzakhani completed by certain genericity results by J.Chaika and A.Eskin imply that the diffusion rate of almost all flat geodesic rays on any $\mathbb{Z}^d$-cover of a closed translation surface $S$ is given by certain Lyapunov exponent of the Kontsevich--Zorich cocycle on the $\text{SL}_2(\mathbb{R})$-orbit closure of $S$. In this paper we prove that in this most general setting, the \textit{average} and \textit{maximum} diffusion rates coincide.
Over any non-Archimedean local field of characteristic not equal to $2$, Takeda and Wood constructed types for the two blocks containing the even and odd Weil representations of the metaplectic group $\tilde{G}$, and identified the resulting Hecke algebras $H_\psi^{\pm}$ with the Iwahori-Hecke algebras of odd orthogonal groups $G^{\pm}$ of the same rank. We describe normalized parabolic induction and Jacquet modules in terms of Hecke modules using a suitable variant of Bushnell-Kutzko theory. Furthermore, we match the standard intertwining operators of $\tilde{G}$ and $G^{\pm}$ by proving a variant of Gindikin-Karpelevich formula for $\tilde{G}$. As an application, we describe the behavior of normalized intertwining operators of $\tilde{G}$ in these blocks under Aubert involution, reducing everything to the $G^{\pm}$ side. This is mainly motivated by Arthur's local intertwining relations.
Let (M, g) be a compact manifold endowed with a possibly singular Riemannian metric. The metric induces a norm on the homology of M , called the stable norm. We provide explicit computations of the stable norm of flat slit tori using the Farey sequence. We then glue several slit tori together to produce half-translation surfaces whose unit ball of the stable norm has faces of maximal dimension. Furthermore, we give a sub-quadratic estimate for the asymptotic counting of simple homology classes on these surfaces.
For a topological space $X$ a topological contraction on $X$ is a closed mapping $f:X\to X$ such that for every open cover of $X$ there is a positive integer $n$ such that the image of the space $X$ via the $n$th iteration of $f$ is a subset of some element of the cover. Every topological contraction in a compact $T_1$ space has a unique fixed point. As in the case of metric spaces and the classical Banach fixed point theorem, this analogue of Banach's theorem is true not only in compact but also in complete (here in the sense of \v{C}ech) $T_1$ spaces. We introduce a notion of weak topological contraction and in Hausdorff spaces we prove the existence of a unique fixed point for such continuous and closed mappings without assuming completeness or compactness of the space considered. These theorems are applied to prove existence of fixed points for mappings on compact subsets of linear spaces with weak topologies and for compact monoids. We also prove some fixed point results for $T_1$ locally Hausdorff spaces and, introduced here, peripherally Hausdorff spaces. An iterated function system on a topological space, IFS, is a finite family of closed mappings from the space into itself. It is contractive if for every open cover of $X$ for some positive integer $n$ the image of $X$ via a composition of $n$ mappings from the IFS is contained in an element of the cover. We show that in $T_1$ compact topological spaces the Hutchinson operator of a contractive IFS may not be closed as the mapping in the hyperspace of closed subsets of the space. Nevertheless, the Hutchinson operator of a contractive IFS has always a unique fixed point.
We study two models of random multiplicative functions: Rademacher random multiplicative functions supported on the squarefree integers $f$, and Rademacher random completely multiplicative functions $f^*$. We prove that the partial sums $\sum_{n\leq x}f^*(n)$ and $\sum_{n\leq x}\frac{f(n)}{\sqrt{n}}$ change sign infinitely often as $x\to\infty$, almost surely. The case $\sum_{n\leq x}\frac{f^*(n)}{\sqrt{n}}$ is left as an open question and we stress the possibility of only a finite number of sign changes, with positive probability.
Let phi = (phi(n))(n <= N )denote a sequence of complex numbers. The main object of this paper is the quadratic form V(phi, Q) =& sum;(Q", Q) are well understood when Q = o( N) and V (p, Q) behaves like root a Riemann sum when N = o(Q). The behavior in the range Q is an element of [root N, 100N] is the object of our queries. In particular, we present a full spectral analysis when Q >= N-3/4 in terms of the eigenvalues of a one-parameter family of nuclear difference operators. Among the consequences, we show that V(p, Q)/(Q(2) & sum;(N) |phi(n)|(2)) stays between two positive constants uniformly in p, when N/Q stays similarly between two positive constants, and that (a smoothed version of) the quadratic form V (", Q) may stay away from the diagonal form p 7 -> (6/pi(2))Q(2) & sum;(N) |phi(n)|(2) when N/Q stays between two positive constants, although only on a vector space of p of positive but small dimension.
In this document, we develop a new model for the category of dg-categories. Following Rezk's example in the case of classic Segal spaces, we define dg-Segal spaces: functors between free dg-categories of finite type and simplicial spaces to which we add certain properties. We define also complete dg-Segal spaces, and make their relationship to classic Segal spaces explicit. With the help of two new hypercover constructions, and up to a certain hypothesis, we prove that there exists an equivalence between the homotopy category of dg-categories and the homotopy category of functors defined above with a model structure making the complete dg-Segal spaces into its fibrant objects.
We introduce a category of filtered sheaves on a circle to describe the Stokes phenomenon of linear difference equations with mild singularity. The main result is a mild difference analog of the Riemann-Hilbert correspondence for germs of meromorphic connections in one complex variable by Deligne-Malgrange.
In this paper we consider the problem of computing the difference Galois groups of order three equations for a large class of difference operators including the shift operator (Case S), the q-difference operator (Case Q), the Mahler operator (Case M) and the elliptic case (Case E). We show that the general problem can be reduced to several ancillary problems. We prove criteria to detect the irreducible and imprimitive Galois groups. Finally, we give a sufficient condition of differential transcendence of solutions of order three difference equations. We also compute the difference Galois group of an equation suggested by Wadim Zudilin.
In this article, we introduce infinitesimal cohomology for rigid analytic spaces that are not necessarily smooth, with coefficients in a p-adic field or Fontaine's de Rham period ring.
