arXiv subjects

Christopher Mouron

Publications and source records attributed to Christopher Mouron.

7 recordsLinked to original sources

arXiv2026

Results on Cartesian $1$-capacity of graphs

The concept of graph capacity extends graph span by considering the maximum number of agents that can simultaneously traverse a graph while preserving a prescribed minimum distance. We study the Cartesian $1$-capacity, corresponding to the movement in which exactly one agent moves at each step. We establish general lower and upper bounds based on structural graph properties and derive exact results for trees. Our work highlights the role of branching and bridge structures in determining the Cartesian $1$-capacity and provides new insights into collision-free multi-agent motion on graphs.

math.CO↗

arXiv2026

Span capacities of graphs

The $d$-capacity of a graph $G$ is introduced as the maximum number of players that can simultaneously traverse $G$ such that each player visits all vertices while maintaining a distance of at least $d$ under various movement rules. We determine their values for paths and cycles and provide bounds for bipartite graphs. Furthermore, we characterize topfull graphs, where the 1-capacities reach their theoretical maximum, establishing a connection to graph factorizations and connectivity.

math.CO↗

arXiv2026

The Specification Property on the Lelek Fan

Recent work of Piotr Oprocha and his collaborators has provided a number of delicate examples of dynamical systems separating specification, shadowing, and periodic-point density, primarily in symbolic or totally disconnected spaces. The goal of the present paper is to demonstrate that similar - and in some cases sharper - separations occur on the Lelek fan, a smooth one-dimensional continuum. Our constructions rely on Mahavier products of closed relations. By carefully choosing relations on the unit interval, we obtain Mahavier products that are homeomorphic to the Lelek fan whose associated shift maps display diverse dynamical behavior. This approach yields a unified framework for producing and analyzing examples on a familiar continuum.

math.DS↗

arXiv2025

Euclidean-type algorithm for functions with same or similar origin

Maps $f,g\colon X\to X$ are called kin if they are forward iterates of the same map $\varphi\colon X\to X$, up to a composition with a commuting homeomorphism. Kin form an important class of commuting maps on $X$. In this paper, we characterize kin, and give an Euclidean-type algorithm which tests when two maps $f,g\colon X\to X$ are kin. Furthermore, we compute the topological entropy of diagonal maps induced by commuting diagonal kin diagrams.

math.DS↗

arXiv2025

Turbulent Closed Relations

This paper generalizes the classical notion of turbulence from dynamical systems generated by continuous functions to those defined by closed relations on compact metric spaces. Using the Mahavier product and the associated shift map, we introduce and explore CR-turbulence and reverse CR-turbulence, analyzing their relationship to topological entropy. A key focus is understanding when turbulence implies entropy and vice versa, with results showing that for finite closed relations, these properties are equivalent. However, examples are provided to demonstrate that this equivalence can fail for more general relations. We also construct a large class of explicit turbulent closed relations on the unit interval that are dynamically rich yet structurally simple. Additionally, since homeomorphisms cannot admit turbulence, we investigate a weakened notion of turbulence: separated, continuum-wise semi-turbulence for homeomorphisms, and then prove that smooth fans cannot support even this weaker form of turbulent dynamics. The paper includes new examples, counterexamples, and open questions that deepen the understanding of turbulence in non-classical settings.

math.DS↗

arXiv2020

Strongly commuting interval maps

Maps $f,g\colon I\to I$ are called strongly commuting if $f\circ g^{-1}=g^{-1}\circ f$. We show that strongly commuting, piecewise monotone maps $f,g$ can be decomposed into a finite number of invariant intervals (or period 2 intervals) on which $f,g$ are either both open maps, or at least one of them is monotone. As a consequence, we show that strongly commuting piecewise monotone interval maps have a common fixed point. Results of the paper also have implications in understanding dynamical properties of certain maps on inverse limit spaces.

math.DS↗

arXiv2020

Topological entropy of diagonal maps on inverse limit spaces

We give an upper bound for the topological entropy of maps on inverse limit spaces in terms of their set-valued components. In a special case of a diagonal map on the inverse limit space $\underleftarrow{\lim}(I,f)$, where every diagonal component is the same map $g\colon I\to I$ which strongly commutes with $f$ (i.e. $f^{-1}\circ g=g\circ f^{-1}$), we show that the entropy equals $\max\{\textrm{Ent}(f),\textrm{Ent}(g)\}$. As a side product, we develop some techniques for computing topological entropy of set-valued maps.

math.DS↗