Related papers: Transition Matrix without Continuation in the Conl…

This article represents a major step in the unification of the theory of algebraic, topological and singular transition matrices by introducing a definition which is a generalization that encompasses all of the previous three. When this…

Connection matrices are one of the central tools in Conley's approach to the study of dynamical systems, as they provide information on the existence of connecting orbits in Morse decompositions. They may be considered a generalisation of…

This paper provides a unified framework connecting dynamical systems with tools from topological data analysis and geometric topology and inspires new interactions among dynamical systems, topology, and nonlinear analysis. To this end, we…

Matrix theory, foundational in diverse fields such as mathematics, physics, and computational sciences, typically categorizes matrices based strictly on their invertibility-determined by a sharply defined singular or nonsingular…

The connection matrix is a powerful algebraic topological tool from Conley index theory that captures relationships between isolated invariant sets. Conley index theory is a topological generalization of Morse theory in which the connection…

There are several approaches to study occurrences of consecutive patterns in permutations such as the inclusion-exclusion method, the tree representations of permutations, the spectral approach and others. We propose yet another approach to…

In this paper we study the Lorenz equations using the perspective of the Conley index theory. More specifically, we examine the evolution of the strange set that these equations posses throughout the different values of the parameter. We…

In this paper we introduce filtration pairs for isolated invariant sets of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an…

We review the passage from the supermembrane to matrix theory via a consistent truncation following a non-commutative deformation in light-cone gauge. Some indications are given that there should exist a generalisation of non-commutativity…

We introduce a novel characterization of phase transitions based on hypothesis testing. In our formulation, a phase transition is defined as the breakdown of statistical indistinguishability under vanishing parameter perturbations in the…

This text is based on a talk by the first named author at the first congress of the SMF (Tours, 2016). We present Bloch's conductor formula, which is a conjectural formula describing the change of topology in a family of algebraic varieties…

By a result of Blokh from 1984, every transitive map of a tree has the relative specification property, and so it has finite decomposition ideal, positive entropy and dense periodic points. In this paper we construct a transitive dendrite…

We give a precise description of spectra of the Maryland model $ (h_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+ \lambda \tan \pi(\theta+n\alpha)u_n$ for all values of parameters. We introduce an arithmetically defined index $\delta…

We give a sufficient geometric condition for a subshift to be measurably isomorphic to a domain exchange and to a translation on a torus. And for an irreducible unit Pisot substitution, we introduce a new topology on the discrete line and…

We investigate the breakdown of normal hyperbolicity of a manifold of equilibria of a flow. In contrast to classical bifurcation theory we assume the absence of any flow-invariant foliation at the singularity transverse to the manifold of…

This is a self-contained tour of the Conley index and connection matrices. The starting point is Conley's fundamental theorem of dynamical systems. There is a short stop at the necessary topological background, before we proceed to the…

We introduce a persistence-type invariant for finite weighted graphs based on combinatorial multivector dynamics. For each threshold parameter, a relation matrix determines a graph multivector field, whose induced directed dynamics admits a…

We present a novel class of real symmetric matrices in arbitrary dimension $d$, linearly dependent on a parameter $x$. The matrix elements satisfy a set of nontrivial constraints that arise from asking for commutation of pairs of such…

Bifurcation characterizes the qualitative changes in parameterized dynamical systems and is one of the major topics in the field. In this work, we study combinatorial bifurcations within the framework of combinatorial dynamical systems -- a…

We give a natural definition for the transitivity of a matrix. Using an endomorphism d of a base ring R and a transitive nxn matrix over the center Z(R), we construct the subalgebra M_{n}(R,d,T) of the full nxn matrix algebra M_{n}(R)…