Papers

We consider the motion of a charged test particle confined to the round unit sphere in a uniform ambient magnetic field. We use an extension of Noether's theorem for systems with magnetic forces to reduce the problem to a quadrature. The rotation number of a trajectory, the azimuth it gains over one oscillation in latitude, is a complete elliptic integral of the third kind in Legendre form. A trajectory closes if and only if that advance is a rational multiple of a full turn. The criterion holds on every level set of the two first integrals. Then, we differentiate the rotation number with respect to the half-cyclotron frequency and sign that derivative on each of the two branches into which a level set divides. The rotation number sees the charge, the mass, the field and the speed through a single dimensionless ratio, that of the half-cyclotron frequency to the speed. A closed trajectory therefore fixes a value of that ratio. We say how many values carry a given rational winding. The poles are attainable on a single level set, where the motion reduces to a pendulum. The field has no flux through the sphere, so it is globally exact. We compute the Mañé strict critical value of the resulting exact magnetic flow. Above that value the closed trajectories are closed Reeb orbits of an explicit contact form.

Mixed-mode oscillations (MMOs), characterized by the alternation of small-amplitude oscillations (SAOs) and large-amplitude oscillations (LAOs), and bursting oscillations are common forms of complex oscillatory dynamics observed in systems with multiple timescales and have been widely studied across scientific disciplines. Mixed-mode bursting oscillations (MMBOs) combine features of MMOs and bursting, with LAOs organized into burst events. Most existing studies treat MMBOs as two-timescale phenomena, identifying distinct geometric mechanisms depending on how the timescales are grouped. In this work, we use a three-timescale biophysical cortical neuronal oscillator model to demonstrate that a three-timescale implementation of geometric singular perturbation theory (GSPT) provides stronger predictive insight into MMBO dynamics. This perspective unifies mechanisms previously identified from fast-slow analysis, while revealing the canard-delayed-Hopf (CDH) singularity as an organizing center for MMBOs near the singular limit. We perform a detailed bifurcation analysis of the full eight-dimensional model to determine how MMBOs are organized along families of isolas. We then combine GSPT with full-system bifurcation analysis to show how tuning the relative timescales induces transitions among MMOs, MMBOs, and bursting dynamics. Our results highlight the importance of studying MMBOs from a three-timescale perspective and demonstrate that combining GSPT with full-system bifurcation analysis can reveal organizing structures and mechanisms that are not apparent from either approach alone.

We show a birthday paradox for random non-backtracking walk on regular graphs of degree at least $3$: such a walk of length $k$ has high probability of self-intersecting when $k$ is significantly greater than $\sqrt n$, where $n$ is the number of vertices of the graph. This resolves a conjecture of Noga Alon and Yuval Peres for the fixed degree case.

We develop a general framework for instantaneous arithmetic computation using chemical reaction networks. Numerical values are encoded by ratios of species concentrations: an extended nonnegative value $a\in[0,+\infty]$ is represented by a computational pair $(A_0,A_1)$ through the ratio $a=a_1/a_0$. For this ratio encoding, our main theoretical result is a feedforward compositionality theorem: if a collection of modules satisfies two easily checked structural conditions, then any admissible finite feedforward composition computes the corresponding arithmetic expression instantaneously, meaning that every intermediate and output ratio is correct at every positive time. We make this framework concrete by constructing reaction network modules for four elementary operations—identification, inversion, multiplication, and addition—and verifying that each satisfies these structural conditions. As applications, we show that truncated power series and matrix products can be computed instantaneously using reaction networks whose size reflects the number of arithmetic operations in the underlying computation. We also extend the nonnegative ratio encoding to signed quantities by representing a real number as the difference of two nonnegative ratio-encoded values; this allows addition and multiplication over real-valued inputs. We discuss the practical implications of instantaneous computation, including its relationship to other resource constraints such as the number of chemical species required and the cost of reading out the final answer, which requires division and, in the signed case, rectified subtraction.

This note supplies genuinely non-fibred examples for the manuscript: Rigidity on the Two-Torus and Sarnak's Conjecture. For every $0<\delta<\tfrac12$, we construct $C^\infty$ Lebesgue-area-preserving pseudo-rotations of $\mathbb{T}^2$ which satisfy the $(C,\delta)$-deviation condition but do not have bounded mean motion. We give both semi-irrational and totally irrational rotation vectors and two realizations: a controlled weakly mixing Anosov–Katok construction and an explicit weakly mixing special flow construction. Weak mixing is used as a conjugacy-invariant obstruction to every continuous circle-rotation factor. Consequently, none of the resulting maps is topologically conjugate, by a linear or nonlinear change of coordinates, to a skew product over a circle rotation. In the special-flow realization, the same lacunary Fourier series simultaneously gives weak mixing, the sharp upper bound $O(n^\delta)$, and unbounded deviations; in fact no smaller deviation exponent is possible. The semi-irrational examples meet the assumptions of Theorems~1 and~2 of the cited manuscript, whereas the totally irrational examples meet those of Theorem~1. Each construction produces continuum many maps and continuum many topological conjugacy classes of each rotation type.

Let $\alpha$ be irrational and let $q_j$ be the denominators of its continued-fraction convergents. We study the function \[ h(x)=\sum_{j\geq1}\frac{\cos(2\pi q_jx)}{q_j} \] and the skew product \[f(x,y)=(x+\alpha,y+h(x))\quad\mathrm{mod}\quad\mathbb{Z}^2.\] The function $h$ is Hölder continuous of every exponent below one. A Fourier argument shows that $h$ is not Lipschitz. The map $f$ is a toral pseudo-rotation with rotation vector $(\alpha,0)$, but it has neither bounded mean motion nor $C^0$-rigidity. Suppose $\alpha$ satisfies the Diophantine condition $\mathrm{DC}(\tau)$. Then, $f$ has $(C,1-1/\tau)$-deviation when $\tau>1$; and it has $(C_\delta,\delta)$-deviation for every $0<\delta<1$, but not for $\delta=0$ when $\tau=1$.

We study weighted periodic-orbit zeta functions for an infinitely cusped tree lattice $Y_q$, where $q\ge2$ is even and the quotient is a one-sided comb. The global Euler product fails coefficientwise because infinitely many primitive cycles have length four. A first-return determinant at a finite directed-edge set nevertheless exists, and stationary Schur elimination gives an algebraic formula for the root local zeta and its dominant poles. For the two-step multiplicity potential we compute the Gurevich pressure $P_G=2\log(q+1)$ and pressure at infinity $P_\infty=\log(4q)$, yielding strong positive recurrence and exponential local-orbit asymptotics. The height-damped transition operator is trace class. After subtraction of an explicit integrated-pressure counterterm, the inverse Fredholm determinant has a locally uniform finite part, expressed by a convergent dilogarithmic product and covariant under changes of height.

We study a class of discrete-time interacting particle systems arising from mean-field coupled maps. We establish a Dobrushin-type estimate, a relative entropy estimate and Central Limit Theorem (CLT) type results for this class of systems. First we control the growth of the 1-Wasserstein distance between the empirical measures and the limiting distributions. Then we use the relative entropy method to show the propagation of chaos. Finally we consider the asymptotic behavior of the fluctuations for the empirical measures and prove that the sequence of fluctuation processes converges in distribution to some Gaussian process, where we establish both qualitative and quantitative results.

The Stochastic Simulation Algorithm (SSA), widely considered an exact algorithm for stochastic chemical reaction networks, suffers from high computational cost. In this work, we propose a data-driven effective model that operates on a user-defined coarse time step independent of the underlying microscopic reaction-event scale. This is accomplished by directly approximating the finite-time transition kernel of the continuous-time Markov chain induced by SSA, using a generative machine learning model trained on short bursts of SSA simulation data. The trained model constructs a stochastic propagator that recursively generates statistically consistent trajectories at the constant coarse time step, with significantly reduced computational cost. In this paper, we employ conditional normalizing flow as the stochastic propagator. A comprehensive set of numerical examples is presented to demonstrate the accuracy and efficiency of the proposed method.

The set of $\tau$-approximable numbers, $\mathcal W(\tau)$, has genuinely fractional Hausdorff dimension, whereas the set of inhomogeneously badly approximable numbers, $\Bad^\gamma$, has full Hausdorff dimension. We determine the Hausdorff dimension of their intersection by introducing the $\Psi$-rapid game, a scale-sensitive refinement of the rapid game of Hatefi and Simmons (preprint 2024). For every approximation function $\psi$, we prove that $\mathcal W(\psi)\cap\Bad^\gamma$ is strong $\Psi$-rapid winning for a natural gauge $\Psi$ determined by $\psi$. Unlike Schmidt-type games, whose winning property always implies full Hausdorff dimension, the $\Psi$-rapid game is calibrated to a prescribed Diophantine scale, so that the resulting dimension bound depends explicitly on the decay of $\Psi$. In particular, for $\psi(q)=q^{-\tau}, \tau\ge1,$ we recover the exact Jarník–Besicovitch dimension, that is, $$ \HD\bigl(\mathcal W(\tau)\cap\Bad^\gamma\bigr)=\frac{2}{\tau+1}.$$

Collective dynamics arise in a wide range of physical, biological, and engineering applications. Examples include cell migration, swarm robotics, social dynamics, and animal behavior. A defining characteristic of these systems is the emergence of large-scale coordination from local interactions among agents; a fundamental question is thus to understand the local interactions that give rise to the observed emergent dynamics. We are interested in methods for learning interactions generally, which can describe a wide class of physical systems exhibiting collective dynamics defined by an interaction kernel, without a priori assumptions on the analytical form of this kernel (i.e. it is nonparametric). The advantage of this kernel-based approach is that it incorporates the underlying physics of the model (i.e. collective dynamics), which more general equation-learning approaches may ignore, potentially limiting their effectiveness for model accuracy and predictions. In this work, we extend existing variational learning approaches to collective systems with both interaction kernels and environmental/intra-agent forces. The proposed framework simultaneously infers the interaction kernel non-parametrically while learning the environmental force using either semi-parametric or fully nonparametric representations. The methodology is validated on several benchmark models exhibiting synchronization, alignment, attraction-repulsion, and external environmental forces. We also introduce a model-selection procedure based on our nonparametric learning framework to identify models that optimally explain a given set of trajectory observations. By exploiting the feature-identification capability of the learned models, the proposed procedure can distinguish among different collective dynamics frameworks and recover mechanistic interaction mechanisms directly from trajectory data.

Large language model multi-agent systems (LLM-MAS) rely on inter-agent communication to solve complex reasoning tasks, yet rigorous guarantees relating consensus performance to computational resources remain limited. Here, we present a novel way to characterize collective belief dynamics as a discrete-time switched system in which communication topologies have distinct consensus-contraction rates and token costs. By augmenting the belief dynamics with the remaining computational budget, we define a consensus safe set that jointly captures agreement and resource feasibility. We derive explicit bounds on consensus time and token expenditure and construct a consensus-budget certificate region guaranteeing finite-time convergence without resource exhaustion. We further establish conditions under which adaptive topology switching achieves a trade-off between convergence speed and communication cost relative to fixed-topology strategies. Numerical experiments and live LLM-MAS deployments show the predicted consensus-cost trade-offs, demonstrating how control-theoretic certificates can enable resource-aware coordination in AI systems.

We study pointwise convergence of polynomial ergodic averages over short intervals whose left endpoints tend to infinity. For a polynomial orbit of degree $d\geq2$ and doubly lacunary starting times, we prove $L^p$ variational estimates, and hence almost-everywhere convergence, for $1<p<\infty$ in the range $c>(d-1)/d$. This gives the first pointwise ergodic theorem for polynomial orbits along short intervals. We also show that the endpoint $L^1$ fails along every infinite subsequence. In a different direction, we prove that substantially denser sequences of starting times exhibit the strong sweeping-out property.

Suppose that $G$ is a finitely generated group, $M$ is a compact Riemannian manifold and $G\curvearrowright M$ is a free and isometric action. We prove that the associated discretized warped cone does not have geometric property (T). In contrast, we show that the discretized warped cone associated to the natural action $SL_m(\mathbb Z)\curvearrowright \mathbb T^m$ has geometric property (T) for $m\geq 3$. We also prove that, for free probability-measure-preserving Lipschitz actions on compact Riemannian manifolds, geometric property (T) of the discretized warped cone implies Kazhdan's property (T) of the acting group.

We consider the Hamiltonian system with one degree of freedom $$\dot x = y, \quad \dot y = -\partial V(x) / \partial x, \qquad x,y\in{\mathbb R}$$ with the smooth potential $V:{\mathbb R}\to{\mathbb R}$. We assume that the origin is an elliptic singular point: $V'(0)=0$ and $V''(0)>0$. Two potentials $V_0$ and $V_1$ are referred to be isonormal if in a neighborhood of the origin there exists a canonical change of coordinates which transforms the Hamiltonian function $y^2/2 + V_0$ to $y^2/2 + V_1$. In particular, any potential isonormal to $x^2/2$ is isochronous. We obtain a criterium of isonormality for analytic potentials.

This paper studies the longest increasing subsequence (LIS) problem for sequences generated by dyadic-type chaotic interval maps. Starting from a single point $x\in[0,1)$ chosen uniformly at random, we form the order pattern of the first $N$ points of its orbit, with the doubling map as the basic model. Let $\lambda_1^{(N)}$ be the LIS length, equivalently the length of the first row of the Young diagram obtained by Schensted's insertion. We show that $\mathbb E[\lambda_1^{(N)}]/\sqrt N\to 2$, matching the leading asymptotics in the classical Ulam–Hammersley problem for uniform random permutations.

Let $T_n(x)=nx\pmod 1$ on $\T=\mathbb R/\mathbb Z$, and let $\Sigma\subset\mathbb N$ be a finitely generated nonlacunary multiplicative semigroup. We prove that, for every $1\le s\le\infty$, there is an open dense set of potentials $f\in W^{1,s}(\T)$ such that every measure that maximizes or minimizes $\int f\,d\mu$ over the $\Sigma$-invariant probability measures satisfies $h_\mu(T_r)=0$ for all $r\in\Sigma\setminus\{1\}$. For each fixed $s$, a single dense $G_\delta$ subset of $W^{1,s}(\T)$ works simultaneously for all finitely generated nonlacunary multiplicative semigroups. The proof combines Rudolph–Johnson entropy rigidity with periodic-grid perturbations that generically exclude Haar measure from the optimizing faces. In particular, without any uniqueness assumption, the result applies to simultaneous $T_p,T_q$-invariance whenever $p,q\ge2$ are multiplicatively independent.

In this paper, we study the finite-dimensional, homogeneous, all-to-all coupled Kuramoto model. We begin by performing a complete spectral analysis of all equilibria of the system. Motivated by this analysis, we then derive an explicit description of the unstable manifolds associated with the family of incoherent equilibria. Subsequently, we establish the convergence of this family of unstable manifolds to the Ott-Antonsen manifold $\mathcal{M}_{\mathrm{OA}}$, with respect to the Hausdorff distance induced by the $p$-Wasserstein metric. We further carry out an analogous analysis for the corresponding counterpart of $\mathcal{M}_{\mathrm{OA}}$ in the continuum limit. Moreover, we establish the uniform-in-time convergence of trajectories of the finite-dimensional Kuramoto model on these invariant manifolds towards their corresponding mean-field limit trajectories. Our results provide a direct geometric link between finite-dimensional particle systems and their mean-field, or continuum, limits.

Using monthly Niño-3.4 anomalies through July 2026, we investigate how much predictive information is contained in delayed observations of the index. Ridge regression identifies informative delays, while multilayer perceptron and sparse identification of nonlinear dynamics (SINDy) models test whether nonlinear complexity provides additional direct forecast skill; gated recurrent unit (GRU) and long short-term memory (LSTM) networks provide a complementary test in which the temporal representation is learned internally. Delayed observations substantially improve forecasts over persistence and climatology at leads of up to six months, but increasing model complexity provides no systematic improvement. Historical recursive experiments favor a simple explicit SINDy recurrence and select shallow recurrent architectures, with no appreciable gain from learning the temporal representation internally. These results support a compact predictive representation of Niño-3.4 evolution in which the representation of past information is more consequential than model complexity. As a prospective application, the selected models are used to forecast the developing 2026 event beyond the last available observation and to compare its predicted evolution with completed historical El Niño events.

At an $O(2)$-equivariant Hopf bifurcation a spatially periodic system selects between traveling waves and standing waves, and which branch appears and whether it is stable is decided by two cubic normal form coefficients. Obtaining these coefficients for a given PDE has required a derivation carried out afresh for each model. We remove that step for two-component systems with conservative polynomial differential nonlinearities on a one-dimensional periodic domain, deriving closed form formulas for both coefficients. They are expressed directly in terms of the array of nonlinear PDE coefficients and the spectral data of the linearization, its critical eigenvectors, adjoint eigenvectors, and non-critical resolvents. We give verifiable conditions under which the underlying center manifold reduction is valid: a structural condition on the principal part of the linearization that yields the required resolvent estimate, and a condition on the Fourier symbol that we show is equivalent to the required spectral gap. In contrast to the model specific computations available previously, the resulting formulas apply to any system in the class without further derivation. A companion implementation evaluates the two normal form coefficients from the coefficient array and verifies the assumptions for a given system. We specialize the formulas to two applications: a strain-gradient regularization of the nonlinear $p$-system, whose quadratic case contains a previously studied model as a special case, and a first order conservative bilinear family, not previously analyzed, in which every bifurcation scenario allowed by the general classification is realized.

In this paper, for quadratic polynomials with Cremer points or Siegel disks whose boundaries don't contain critical points, we give an arithmetic condition for their Julia sets to have positive area. Our proof relies on the Siegel Return Machinery as well as the pseudo-Siegel disk theory developed by Dudko and Lyubich.

We consider the box dimension of the graphs of the generalized Riemann-type functions $G_\delta(x)=\sum_{n=1}^{\infty}g(n^{2}x)n^{-1-\delta}$ with 1-periodic real-valued continuous functions $g$ and $0<\delta\le 1$. Firstly, we establish a rational-level non-vanishing criterion for the lower bound of lower box dimension of the graph of $G_\delta$. More precisely, We prove that the lower bound $\dim_B(\mathrm{graph}\,G_\delta)\ge\frac74-\frac\delta2$ under a mild decay condition of the Fourier coefficients of $g$ and non-vanishing of the square-class chirp functional $S_{d}(a;q)$ at a single rational $a/q$. A resolution theorem then asserts that for any nonconstant real trigonometric polynomial $g$, the chirp functional $S(a;q)$ cannot vanish at every rational simultaneously; consequently, $\dim_B(\mathrm{graph}\,G_\delta)=\frac74-\frac\delta2$ for all such $g$ with $0<\delta\le1$ which gives a negative answer to \cite[problem 2]{Wu-Zhan2026}. Finally, two guiding examples distinguish structural vanishing from genuinely arithmetic vanishing related to modular elliptic curve and governed by the Prime Number Theorem.

We develop a general theory for the dynamics of the geodesic flow of locally CAT(k) branched coverings and we show that it does not depend on the specific covering but only on the base space, the branching set and the degree of the covering. We find an explicit formula for the entropy: it equals the topological pressure of the associated natural dynamical system on the space of broken geodesics with a natural geometric potential. We find the exact asymptotic of the entropy as the number of sheets of the branched covering goes to infinity, as well as asymptotic properties of the measures of maximal entropy.

Spatial patterns, such as those in dryland vegetation models, have historically been studied in systems of reaction-diffusion systems with pattern onset via a Turing bifurcation from a spatially uniform state. More recently, spatial patterns have been considered in models that incorporate spatially extended interactions via nonlocal interaction kernels. It remains largely underexplored if and how the choice of nonlocal interaction kernel contributes to differences in pattern formation and persistence, particularly in models that contain competition and facilitation. Here, we investigate spatial patterns in a reaction-diffusion model for a single species that includes nonlocal competition and facilitation processes; Gaussian, exponential, algebraic, hat, and smooth hat kernels are considered as specific examples. Via a center manifold analysis, and using the relative spatial scale of competition to facilitation and the death rate as bifurcation parameters, we identify that the choice of kernel has impacts on the pattern forming bifurcation. Bifurcations using the Gaussian, exponential, and algebraic kernels largely follow expectations of Turing patterns, but patterns in the hat and smooth hat kernels can form even when the scale of competition is less than that of facilitation. The dynamics of patterns far from onset are investigated via numerical continuation methods. The model produces the so-called "Turing-before-Tipping" phenomenon demonstrating that the arrangement into spatial patterns is an effective resilience mechanism against harsh conditions. Again, there is a kernel-dependent dichotomy in pattern behavior. Early warning signs for population extinction are observed with the Gaussian, exponential, and algebraic kernels, but not under the hat or smooth hat cases.

Although the Heil–Ramanathan–Topiwala conjecture is false in full generality, its failure makes the classification of positive geometric and arithmetic regimes more urgent. We settle the critical mixed-arithmetic regime for four time-frequency shifts. For \(z=(x,\omega)\) and \(w=(y,\eta)\), set \(\sigma(z,w)=x\eta-y\omega\). Let \(u,v\in\R^2\) satisfy \(\lvert\sigma(u,v)\rvert=1\), write \(\nu=\alpha u+\beta v\), and assume that \(0,u,v,\nu\) are distinct. If \(\dim_{\Q}\operatorname{span}_{\Q}\{1,\alpha,\beta\}=2\) then, for every nonzero \(f\in L^2(\R)\), the vectors \(f,\pi(u)f,\pi(v)f,\pi(\nu)f\) are linearly independent. This includes both four-point configurations in Chris Heil's Conjecture~9.2. The proof first converts a putative dependence into a scalar cocycle over an irrational rotation. On a positive-measure family of zero-free fibres, winding and continued-fraction returns force the periodic holonomy to be constant. The return multiplier is Laurent polynomial, however, so its ungauged holonomy is algebraic; the constant-holonomy law simultaneously makes it an irrational character. These conclusions are incompatible. The principal theorem and its complete formal dependency chain have been fully certified end-to-end in Lean~4, including the physical all-nonzero case, the complete three-point HRT theorem, the coefficient-reduction step, and the final four-point conclusion. The accompanying \href{https://www.dropbox.com/scl/fi/jd9xeqqera147wdc82r8k/3f260460-07d5-45f5-8d1c-ee1d2879c0af-aristotle-32-.tar.gz?rlkey=a24njadfds5ty9ez473lk03lv&e=1&dl=0} {Lean~4 certification archive} contains reproducible source, pinned build instructions, and transitive axiom audits.

We prove measure-rigidity results for regular polynomial endomorphisms of~$\mathbb{C}^k$. For maps of the same degree whose leading homogeneous parts differ by an invertible linear map on the target, equality of equilibrium measures is equivalent to postcomposition by an affine symmetry of the Julia set. We then show that, for $f$ in a nonempty Zariski open subset of the parameter space, $\mu_g=\mu_f$ implies $g=f$, with no hypothesis relating the leading terms; in dimension two this holds across all degrees, with $g$ an iterate of~$f$. We also establish finiteness results for maps with a prescribed equilibrium measure under essentially necessary hypotheses. Our approach rests on the stable manifold structure near the hyperplane at infinity. Equality of equilibrium measures yields a Zariski-dense family of common local stable-manifold germs, whose intrinsic first-order jets lead to a divisibility incidence problem on a Grassmannian. We resolve this problem by analyzing the power-map fiber and applying a generic stabilizer argument. In dimension two, applications include a characterization in terms of preperiodic sets, a Tits-type alternative, results on iterated centralizers, and an arithmetic non-density theorem.

In this paper, we introduce the Inertial Manifold Neural Operator (IMNO) for solving dissipative time-dependent partial differential equations (PDEs). The long-time dynamics of such systems often exhibit an effective low-dimensional structure due to dissipation. Unlike standard neural operator architectures such as the Fourier Neural Operator (FNO), IMNO explicitly leverages the low-dimensional structure to achieve better physical interpretability, accuracy, and stability in long-horizon autoregressive training and prediction for nonlinear dissipative PDEs. For shift-equivariant PDEs, we further introduce a shift-equivariant variant (IMNO-SE) of the proposed neural operator, ensuring that a spatial shift in the input induces the same spatial shift in the output. This symmetry-preserving inductive bias substantially improves its performance in shift-equivariant PDEs. Extensive benchmark experiments are presented to evaluate IMNO's performance numerically.

We study travelling front solutions of a family of degenerate Fisher-KPP equations $u_t=(u^n u_x)_x+u(1-u^n)$, where $n$ is a positive integer. At the minimal wave speed $c_{\rm min}=\tfrac{1}{\sqrt{1+n}}$, these equations admit sharp front solutions, corresponding to an explicit heteroclinic orbit in the travelling wave phase-space. We show how this sharp front is perturbed when the wave speed is increased to $c=\tfrac{1}{\sqrt{1+n}}+\varepsilon$, with $\varepsilon$ sufficiently small. We use a well-motivated geometric desingularisation (also known as blow-up) near the degenerate equilibrium at the leading edge. By analysing the resulting directional and rescaling charts, we construct a singular heteroclinic orbit connecting the relevant asymptotic states, providing a simple geometric proof of the transition from sharp to smooth travelling fronts.

Classical results establish that the Lyapunov exponents of an ergodic measure for linear cocycles over hyperbolic systems can be approximated by the Lyapunov exponents of periodic orbits, provided the cocycle is Hölder continuous. A recent counterexample by Bochi demonstrates that this approximation property fails in general if the Hölder assumption is relaxed to mere continuity. In this paper, we introduce a geometric condition that successfully substitutes this analytical regularity hypothesis. More precisely, we prove that if a cocycle - even a discontinuous one - admits a continuous family of invariant holonomies, the periodic approximation of Lyapunov exponents still holds. Our geometric approach yields a proof that is substantially simpler and more direct than existing arguments in the literature, even when applied to classical settings such as fiber-bunched cocycles for which previous results were already available.

The Kuramoto model provides a paradigmatic framework for studying synchronization of interacting oscillators, and has been generalized to arbitrary dimensions to describe swarms, flocks and multi-dimensional opinion dynamics. Yet, existing formulations neglect inertia, a key mechanism known to enhance information propagation and collective responsiveness. Here, we introduce and analyze an inertial Kuramoto model in arbitrary dimensions. We show that inertia fundamentally alters the nature of the synchronization transition, inducing a crossover from continuous to discontinuous behavior, with the onset of hysteresis depending explicitly on both inertia and dimensionality parity. Our analytical theory is supported by extensive numerical simulations. These results establish inertia as a crucial ingredient of high-dimensional collective dynamics and reveal a novel universal structure in synchronization phenomena.