arXiv subjects

Pietro Majer

Publications and source records attributed to Pietro Majer.

14 recordsLinked to original sources

arXiv2026

On the independence number of de Bruijn graphs

We derive the asymptotic formula $\alpha(k,q)=\lambda_{k-1}q^k+o(q^k)$, where $\alpha(k,q)$ is the independence number of the de Bruijn graph $B(k,q)$, and $\lambda_{k-1}$ is a constant arising from a variational problem on the unit $(k-1)$-dimensional cube. When $k=4$, we show the bounds $91/240\le \lambda_3\le 11/28$. For odd prime $k$, we analyse the binary case $q=2$ via a phase reduction on rotation orbits. For $k=11,13,17$ this yields compact orbit-marker certificates for optimal constructions. Combined with a lifting theorem by Lichiardopol, these certificates give exact formulas for $\alpha(11,q)$, $\alpha(13,q)$, and $\alpha(17,q)$ for all $q\ge2$, extending the known cases $k=3,5,7$.

math.CO↗

arXiv2023

Game extensions of floppy graph metrics

A $graph$ $metric$ on a set $X$ is any function $d: E_d \to\mathbb R_+:=\{x\in\mathbb R:x>0\}$ defined on a connected graph $ E_d \subseteq[X]^2:=\{A\subseteq X:|A|=2\}$ and such that for every $\{x,y\}\in E_d$ we have $d(\{x,y\})\le\hat d(x,y):=\inf\big\{\sum_{i=1}^nd(\{x_{i-1},x_i\}):\{x,y\}=\{x_0,x_n\}\;\wedge\;\{\{x_{i-1},x_i\}:0 \check d(x,y:= \sup\{d(\{a,b\})-\hat d(a,u)-\hat d(b,y):\{a,b\}\in E_d \}$ for every $x,y\in X$ with $\{x,y\}\notin E_d $. We prove that for every floppy graph metric $d: E_d \to\mathbb R_+$ on a set $X$, every points $x,y\in X$ with $\{x,y\}\notin E_d $, and every real number $r$ with $\frac 13\check d(x,y)+\frac23\hat d(x,y)\le r<\hat d(x,y)$ the function $d\cup\{\langle\{x,y\},r\rangle\}$ is a floppy graph metric. This implies that for every floppy graph metric $d: E_d \to\mathbb R_+$ with countable set $[X]^2\setminus E_d $ and for every indexed family $(F_e)_{e\in[X]^2\setminus E_d }$ of dense subsets of $\mathbb R_+$, there exists an injective function $r\in\prod_{e\in[X]^2\setminus E_d}F_e$ such that $d\cup r$ is a full metric. Also, we prove that the latter result does not extend to partial metrics defined on uncountable sets.

math.CO↗

arXiv2023

Infinite-volume states with irreducible localization sets for gradient models on trees

We consider general classes of gradient models on regular trees with values in a countable Abelian group $S$ such as $\mathbb{Z}$ or $\mathbb{Z}_q$, in regimes of strong coupling (or low temperature). This includes unbounded spin models like the p-SOS model and finite-alphabet clock models. We prove the existence of families of distinct homogeneous tree-indexed Markov chain Gibbs states $μ_A$ whose single-site marginals concentrate on a given finite subset $A \subset S$ of spin values, under a strong coupling condition for the interaction, depending only on the cardinality $\vert A \vert$ of $A$. The existence of such states is a new and robust phenomenon which is of particular relevance for infinite spin models. These states are not convex combinations of each other, and in particular the states with $\vert A \vert \geq 2$ can not be decomposed into homogeneous Markov-chain Gibbs states with a single-valued concentration center. As a further application of the method we obtain moreover the existence of new types of $\mathbb{Z}$-valued gradient Gibbs states, whose single-site marginals do not localize, but whose correlation structure depends on the finite set $A$.

math.PR↗

arXiv2020

Stable foliations and CW-structure induced by a Morse-Smale gradient-like flow

We prove that a Morse-Smale gradient-like flow on a closed manifold has a "system of compatible invariant stable foliations" that is analogous to the object introduced by Palis and Smale in their proof of the structural stability of Morse-Smale diffeomorphisms and flows, but with finer regularity and geometric properties. We show how these invariant foliations can be used in order to give a self-contained proof of the well-known but quite delicate theorem stating that the unstable manifolds of a Morse-Smale gradient-like flow on a closed manifold $M$ are the open cells of a $CW$-decomposition of $M$.

math.DS↗

arXiv2014

A non-squeezing theorem for convex symplectic images of the Hilbert ball

We prove that the non-squeezing theorem of Gromov holds for symplectomorphisms on an infinite-dimensional symplectic Hilbert space, under the assumption that the image of the ball is convex. The proof is based on the construction by duality methods of a symplectic capacity for bounded convex neighbourhoods of the origin. We also discuss some examples of symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which would be impossible in finite dimensions.

math.SG↗

arXiv2014

A survey on non-autonomous basins in several complex variables

Consider a holomorphic automorphism which acts hyperbolically on some invariant compact set. Then for every point in the compact set there exists a stable manifold, which is a complex manifold diffeomorphic to real Euclidean space. If the point is fixed, then the stable manifold is even biholomorphic to complex Euclidean space. In fact, it is known that the stable manifold of a generic point is biholomorphic to Euclidean space, and it has been conjectured that this holds for every point. In this article we survey the history of this problem, addressing both known results and the techniques used to obtain those results. Moreover, we present a list of seemingly simpler open problems and prove several new results, all pointing towards a positive answer to the conjecture discussed above.

math.CV↗

arXiv2012

Monotone paths in random hypergraphs

We determine the probability thresholds for the existence of monotone paths, of finite and infinite length, in random oriented graphs with vertex set $\mathbb N^{[k]}$, the set of all increasing $k$-tuples in $\mathbb N$. These graphs appear as line graph of uniform hypergraphs with vertex set $\mathbb N$.

math.PR↗

arXiv2011

Stable manifolds for holomorphic automorphisms

We give a sufficient condition for the abstract basin of attraction of a sequence of holomorphic self-maps of balls in \mathbb{C}^{d} to be biholomorphic to \mathbb{C}^{d}. As a consequence, we get a sufficient condition for the stable manifold of a point in a compact hyperbolic invariant subset of a complex manifold to be biholomorphic to a complex Euclidean space. Our result immediately implies previous theorems obtained by Jonsson-Varolin and by Peters; in particular, we prove (without using Oseledec's theory) that the stable manifold of any point where the negative Lyapunov exponents are well-defined is biholomorphic to a complex Euclidean space. Our approach is based on the solution of a linear control problem in spaces of subexponential sequences, and on careful estimates of the norm of hte conjugacy operator by a lower triangular matrix on the space of \textit{k}-homogeneous polynomial endomorphisms of \mathbb{C}^{d}.

math.DS↗

arXiv2011

Global stable manifolds in holomorphic dynamics under bunching conditions

We prove that the stable manifold of every point in a compact hyperbolic invariant set of a holomorphic automorphism of a complex manifold is biholomorphic to a complex vector space, provided that a bunching condition, which is weaker than the classical bunching condition for linearizability, holds.

math.DS↗

arXiv2008

A Morse complex for Lorentzian geodesics

We prove the Morse relations for the set of all geodesics connecting two non-conjugate points on a class of globally hyperbolic Lorentzian manifolds. We overcome the difficulties coming from the fact that the Morse index of every geodesic is infinite, and from the lack of the Palais-Smale condition, by using the Morse complex approach.

math.DG↗

arXiv2008

Infinite dimensional Grassmannians

We study the analytic and homotopy properties of some infinite dimensional Grassmannians, useful for developing a Morse theory for infinite dimensional manifolds. We study the space of Fredholm pairs of a Hilbert space, we determine its homotopy type, and we define a determinant bundle over it. We study the space of compact perturbations of a given closed linear subspace, and the related concept of essential Grassmannian.

math.AT↗

arXiv2005

On the global stable manifold

We give an alternative proof of the stable manifold theorem as an application of the (right and left) inverse mapping theorem on a space of sequences. We investigate the diffeomorphism class of the global stable manifold, a problem which in the general Banach setting gives rise to subtle questions about the possibility of extending germs of diffeomorphisms.

math.DS↗

arXiv2005

A Morse complex for infinite dimensional manifolds - Part I

In this paper and in the forthcoming Part II we introduce a Morse complex for a class of functions f defined on an infinite dimensional Hilbert manifold M, possibly having critical points of infinite Morse index and coindex. The idea is to consider an infinite dimensional subbundle - or more generally an essential subbundle - of the tangent bundle of M, suitably related with the gradient flow of f. This Part I deals with the following questions about the intersection W of the unstable manifold of a critical point x and the stable manifold of another critical point y: finite dimensionality of W, possibility that different components of W have different dimension, orientatability of W and coherence in the choice of an orientation, compactness of the closure of W, classification, up to topological conjugacy, of the gradient flow on the closure of W, in the case dim W=2.

math.DS↗

arXiv2004

When the Morse index is infinite

Let f be a smooth Morse function on an infinite dimensional separable Hilbert manifold, all of whose critical points have infinite Morse index and co-index. For any critical point x choose an integer a(x) arbitrarily. Then there exists a Riemannian structure on M such that the corresponding gradient flow of f has the following property: for any pair of critical points x,y, the unstable manifold of x and the stable manifold of y have a transverse intersection of dimension a(x)-a(y).

math.DS↗