Mathematics

This article shortly provides related proofs of the ergodic theorems of von Neumann, Birkhoff, Wiener, and Rokhlin's lemma for $Z^d$-actions with an invariant measure. It is shown how some deviations of ergodic averages can be structured.…

Given a finite abelian group $G$ and a Sylow $p$-subgroup $N_p$, we prove that the $KU_G/p$-local sphere spectrum is equivalent to the homotopy fixed points of a $p$-complete $KO_{N_p}$-module spectrum. Then we compute the…

We investigate transient clustering dynamics in nonlocal aggregation-diffusion systems from an energetic perspective. Starting from a stochastic interacting particle system, we study the associated macroscopic McKean-Vlasov equation on the…

Recent literature shows that hypocoercivity properties of linear evolution equations (in particular their exponential decay and the sharp short time decay of their propagator norm) carry over to their discretization via the midpoint rule.…

Motivated by Berg's notion of quasi-disjointness for ergodic systems, we introduce and investigate the concept of quasi-disjointness for minimal systems. Several equivalent characterizations are provided. We prove that quasi-disjointness is…

We study the action of the Hecke triangle groups $G_q$ on $\lambda_q \mathbb{Q}(\lambda_q^2) \cup \{\infty\}$ with $\lambda_q = 2 \cos (\pi / q)$. When $q = 18$, we show the existence of infinitely many distinct orbits of fixed points of…

Algebraic representations of time series are symbolic representations whose symbols belong to a finite group. Precisely, the framework of the present paper is the analysis of coupled time series in algebraic representations and, more…

Convection-dominated problems are known for their slow Kolmogorov $n$-width decays and are challenging for model order reduction (MOR). In this work, we propose a hybrid surrogate modeling approach and a non-intrusive variant that overcome…

Lloyd algorithm is the standard iterative method for computing quantizers and codebooks in source coding and vector quantization. In this article, we study the dynamical and stability properties of the Lloyd map on the unit circle $\mathbb…

We establish a generalization of Anush Tserunyan and Jenna Zomback's 2024 Backward Ergodic Theorem. We remove the countable-to-one assumption and thus provide a backward ergodic theorem for arbitrary measure-preserving transformations.…

This paper studies the $\alpha$-stability property of differentially flat nonlinear dynamical systems. The results build off the recently introduced notion of $\alpha$-stability, which is particularly amenable to characterize the ability of…

Coupled oscillator networks underlie many biological systems, from cardiac cycles to circadian rhythms. Phase-reduced models such as the Kuramoto model have been widely used to study synchronization, but they typically assume that…

The Ellis semigroup of a topological dynamical system contains algebraic, topological and dynamical information. It is invariant under conjugacy. Despite this wealth of structure, two non-conjugate dynamical systems can have the same Ellis…

We propose a minimal dynamical model of adaptive softmax routing for a two-expert Mixture-of-Experts (MoE) layer. The model is obtained as a mean-field limit of a discrete reinforcement rule: the selected expert receives a small score…

Let $\Gamma_{2n}^\omega(p)$ be the level-$p$ principal congruence subgroup of $\text{Sp}_{2n}(\mathbb{Z})$ for all prime $p$. Borel--Serre demonstrated that the cohomology of $\Gamma_{2n}^\omega(p)$ vanishes above degree $n^2$. We prove…

We prove a function-field analogue of Bourgain's $L^2$ pointwise ergodic theorem. Let $q$ be a power of a prime $p$, let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$, and let $\mathbb{F}_q[t][u]$ be the…

We give an equivalent condition for the existence of invariant Gibbs measures for sequences of continuous functions on one-sided subshifts and, more generally, for the existence of Gibbs measures. These extend the results of Kim [6] and…

In this work, we extend results of Kakutani; Adler and Flatto; Smilansky; Pollicott and Sewell on the equidistribution of endpoints generated by interval-splitting procedures. We study a nonlinear version of the problem generated by a…

For a compact metric space $X$ with a group $G$ acting on it continuously, an invariant random compact is a Borel probability measure on the space of nonempty compact subsets of $X$ that is invariant under the action of $G$. The action is…

We study a circular opinion dynamics model with local midpoint interactions, extended to allow parallel updates of multiple sites. On a ring, the dynamics admits twisted states associated with integer winding numbers. We investigate how…