Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.23992

Mathematics of Herding

Sayomi Kamimoto
View original source ↗Read source document ↗Export metadata ↓

Abstract

A herd of animals often follows a leader who sets the direction of travel. The leader sometimes turns unexpectedly and the followers reshuffle. Yet the herd regroups itself and stays coherent. Our work aims to understand the herding behavior mathematically. We construct a minimal discrete-time nonequilibrium dynamical system with one-way coupling, and the minimal structure allows an exact analysis. Take the location of each follower in the frame of the leader. The recorded locations form a distribution. The limit of the distribution is the asymptotic spatial measure, whose geometric representation we study. The dynamics of the herd have an exact operator representation with two factors, the operator of a fixed leader-follower pair and the operator of the interaction rules, which reshuffle the pairs. We show that the influence of the leader passes through some follower more than once. As a result, two followers can have reciprocal influence, despite the one-way coupling.

Explore related subjects

math.DSmath.FAnlin.AOnlin.CD

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sayomi Kamimoto. 2026-10-14. Mathematics of Herding. https://arxiv.org/abs/2610.23992

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Unstable entropy for Anosov diffeomorphisms on the 3-torus

For Anosov diffeomorphisms on the $3$-torus which are strongly partially hyperbolic with expanding center, we construct systems of strong unstable and center stable Margulis measures which are holonomy-invariant. This allows us to obtain a characterization of the measures of maximal unstable entropy in terms of their conditional measures along the strong unstable leaves. Moreover, we show that the Margulis systems identify with Pollicott-Ruelle (co)-resonant states for the action on 2-forms. This shows that the unstable topological entropy and a measure of maximal unstable entropy can be retrieved from the spectral approach.

math.DS↗
arXiv2026

Thermodynamic formalism and multifractal analysis of Birkhoff averages for non-uniformly expanding Rényi interval maps with countably many branches

In this paper, we study the multifractal spectrum of Birkhoff averages for non-uniformly expanding Rényi interval maps with countably many branches. Our main theorem substantially strengthens conditional variational formulas established by Jaerisch and Takahasi. Furthermore, our results enable a detailed analysis of Khinchin exponents and arithmetic means of backward continued fraction expansions in terms of the Hausdorff dimension. We also give a positive answer to the conjecture of Jaerisch and Takahasi. In addition, we develop the thermodynamic formalism for non-uniformly expanding Rényi interval maps with countably many branches.

math.DS↗
arXiv2026

On the intersection of Cantor sets and products of random matrices

Author's follow-up comment: The line of research pursued in this paper has been developed substantially further in my subsequent paper, Lyapunov exponents for random products of non-negative matrices (arXiv:2602.23317). I will therefore not pursue separate journal publication of the present paper. Original abstract: Kenyon and Peres (1991) showed that the Hausdorff dimension of intersections of randomly translated Cantor sets can be expressed in terms of the top Lyapunov exponent of a product of random matrices, and this exponent can be written as an integral with respect to stationary measures on the projective line. Although explicit computations are available when stationary measures are discrete, the continuous case has remained challenging. In this paper we introduce new combinatorial and analytic tools that allow us to compute the Lyapunov exponent, and hence the Hausdorff dimension, in a broad class of examples where stationary measures are continuous. As an application, we complete the dimension computation in the setting where a single digit is forbidden; for example, we determine the Hausdorff dimension of the intersection of the middle-seventh Cantor set with a random translate of itself.

math.DS↗