Papers

The goal of this paper is to generalize recent results about the topological classification of the dynamics of a real planar polynomial vector field near infinity. Given a polynomial vector field $X$, using Newton polyhedra one can define its generalized upper principal part $X_{\Gamma}^{U}$. By dropping some monomials of $X_{\Gamma}^{U}$, we define its minimal generalized upper principal part $X_{G}^{U}$. We prove that there exist an open and dense set $\widetilde{\mathfrak{U}}_{1}$ in the set of polynomial vector fields with Newton degenerate upper principal part satisfying the following property: if $X\in\widetilde{\mathfrak{U}}_{1}$, then $X$ and $X_{G}^{U}$ are topologically equivalent near infinity, provided that $X_{G}^{U}$ satisfies some additional non-degeneracy assumptions. Our techniques rely on toric compactification and the Normal Form Theorem.

We determine the Boltzmann–Grad limit of the Ehrenfest “wind-tree” model with a periodic configuration of polyhedral scatterers. The central result is a loss-of-memory effect in dimension $d\geq 3$ for a generic choice of polyhedron. This yields a Markovian limiting transport process on an extended state space and a generalised linear Boltzmann equation for the time evolution of the particle density.

We prove that for any cocompact Fuchsian group the class of Medvedev degrees attained by subshifts of finite type is the class of $\Pi_1^0$ degrees. This result relies on the construction of a rigid hierarchical structure in a graph model of the hyperbolic plane, and on the fact that every such group admits a representation in $\operatorname{PSL}_2(\mathbb{R})$ such that the matrices in the image have coefficients that are computable real numbers.

We construct a nonempty $C^k$-open family of smooth, autonomous, divergence-free velocity fields on $\mathbb{T}^3$ that generate fast dynamos. The proof proceeds by first establishing fast dynamo action for a large class of smooth, time-periodic velocity fields exhibiting the classical stretch–fold–shear mechanism. We then realize the associated period map within a smooth autonomous flow using carefully tuned return dynamics to a global Poincaré section. In both settings, the proof first isolates unstable distributional eigenmodes for the ideal dynamo operator on an anisotropic Banach space of distributions, and then shows that this spectral instability persists under the singular perturbation $\varepsilon \Delta$. The resulting estimates are uniform as $\varepsilon\to0$ and stable under $C^k$ perturbations of the velocity field, yielding an open set of smooth fast dynamo vector fields. This result resolves the Fast Dynamo Conjecture of Zeldovich and Sakharov, as recorded in Arnold's book of problems.

We consider an alternative notion of minimality for C*-dynamical systems, which we refer to as strong minimality. This notion coincides with the usual definition of minimality in the commutative setting, but diverges sharply from it in the noncommutative setting in a way which we argue addresses several fundamental defects in the usual definition. We demonstrate that strong minimality is deeply connected with Rørdam's notion of a C*-irreducible inclusion, giving a new perspective on the study of such inclusions. We provide various characterizations of this property, including for reduced crossed products, tensor products, and subhomogeneous C*-algebras.

We prove that the electric field generated in three-dimensional space by finitely many point charges of one sign has only finitely many equilibrium points, thereby answering a 1969 question of Morse and Cairns (restated by Eremenko in 2008 and, as a conjecture, by Shapiro in 2015). More generally, for nonzero charges $q_i$ of possibly mixed signs at distinct sites $\mathbf a_i\in\mathbf{R}^3$, we show that the Coulomb field has at most $2^{N-4}(N-1)(9N^2+9N+10)$ equilibria in the region where $S(\mathbf x):=\sum_iq_i|\mathbf x-\mathbf a_i|^{-3}$ does not vanish. Of course, for charges of one sign, $S$ is nonzero everywhere. The proof rules out curves of equilibria using algebraic geometry and complex analysis on an associated complex curve, and then applies a Bézout count to obtain a quantitative bound. Well-known examples show that, in the mixed-sign case, the Coulomb field can vanish on curves contained in the zero set of $S$.

The way numbers are represented strongly influences which arithmetic structures are easy to see. The prime clockwork is a recursively growing discrete dynamical system: a list of autonomous two-hand clocks driven by one common $+1$ signal. No primes or primality labels are supplied. Starting empty, the process appends a clock of period $n$ whenever none already present rings; the primes are generated internally as its growth times. For each installed prime $p$, the seconds reading $R_p$ advances through $0,\ldots,p-1$, and each return to zero increments the minutes reading $M_p$, which counts completed $p$-cycles. The hands use only increment, comparison, reset, and carry, without explicit mod or div operations. At time $n$, $n=pM_p(n)+R_p(n)$. The valuation readout $V_p(n)=\nu_p(n)$ is generated locally: it is zero when the seconds counter is non-zero (silent state) and otherwise (when the p-clock rings) one plus the earlier valuation addressed by the current minutes reading. The valuation vector gives the integer in unique prime-factorized form. Its coordinates add and subtract under multiplication and division, representing every positive rational uniquely; divisibility becomes weak componentwise order, and unique factorization is natural in this representation. Finite seconds arrays form Cartesian-product state spaces whose common orbit visits every joint state once before repeating; this grand cycle is the order-sensitive dynamical counterpart of the Chinese remainder theorem. The same coordinates expose gcd, lcm, perfect powers, Bézout's identity, and Euler's totient. Rational valuation levels reach certain positive algebraic irrationalities, but not algebraic numbers in general.

The topological mating of two postcritically finite polynomials with dendritic Julia sets is encoded by a pair of circle laminations $\Lambda^{\pm}$, whose collapse produces a sphere-filling curve. When a leaf of $\Lambda^{+}$ and a leaf of $\Lambda^{-}$ share an endpoint they form a perfect fit. An unpublished proposition of Epstein, recorded by Petersen and Meyer, shows that for matings of honest degree-$d$ polynomials an infinite-diameter ray equivalence class, i.e. an infinite chain of perfect fits is impossible. We show that this finiteness is a genuinely holomorphic phenomenon. Allowing the dynamics to carry a periodic critical orbit, we construct combinatorially expanding Thurston maps-realized by no expanding rational map-that admit invariant sphere-filling curves yet whose laminations $\Lambda^{\pm}$ contain infinite chains of perfect fits, in fact infinitely many of them.

We study the long-term distributional dynamics and small-noise asymptotics of two-dimensional nonautonomous stochastic primitive equations driven by Gaussian and multiplicative Lévy noise, together with moderate deviations for the purely jump model. For sufficiently small noise, uniform moment bounds and an exponentially weighted terminal estimate yield a pullback absorbing family and tightness, while a lower semicontinuous vertical-moment functional preserves admissibility under weak limits. We prove the existence and uniqueness of a pullback measure attractor in the weak topology of probability measures and establish its upper semicontinuity as both noise components vanish. For the jump-driven equation, we establish a moderate deviation principle in $\mathcal D([0,T];H)\cap L^2(0,T;V)$ with speed $a^2(\epsilon)/\epsilon$. The proof combines continuity of the skeleton map with controlled stochastic convergence based on entropy bounds, truncation, martingale estimates, and direct vertical estimates, avoiding Lipschitz continuity of the vertical derivative of the jump coefficient.

We prove that any real-analytic Hamiltonian in five or more degrees-of-freedom, with a locally integrable non-degenerate elliptic equilibrium with indefinite quadratic part, can be perturbed within the real analytic category, while preserving the Birkhoff normal form at the equilibrium up to any arbitrary order, so that the equilibrium becomes Lyapunov unstable.

In dimension 3, following Eliashberg-Thurston and Mitsumatsu, one can associate a transverse pair of oppositely oriented contact structures to any Anosov flow. We show that when the ambient manifold is hyperbolic, the isotopy classes of these contact structures completely determine the flow up to isotopy orbit equivalence. This then implies that on any closed hyperbolic 3-manifold, the number of Anosov flows modulo isotopy orbit equivalence is finite. This approach also yields an explicit bound on the number of orbit equivalence classes of Anosov flows in terms of the number of universally tight contact structures. In addition, we show finiteness of pseudo-Anosov flows for which the complement of the singular orbits is atoroidal and, in the non- hyperbolic case, we obtain a control on the dynamics of Anosov flows on the hyperbolic pieces of the JSJ-decomposition.

We show that exponential maps could have complicated measurable behaviours by giving two examples in the exponential family both with singular value escaping to infinity “slowly", one of which is recurrent (in the sense that every positive measure set will return to itself infinitely many times), while the other is non-recurrent. This also complements earlier results of Lyubich, Rees, Urbański-Zdunik, and Hemke. We also give an example of exponential maps with singular value escaping to infinity arbitrarily slow, which is a parametric analog of a result of Rempe.

Let $M$ be a smooth manifold equipped with a para-Kähler structure. We define an action of the diffeomorphism group $\operatorname{Diff}(M)$ on the space of para-Kähler structures and prove that this action preserves the associated affine structures. Although this result is well known, we provide a proof based on this action, showing in particular that an isometry commutes with the Levi–Civita connection. We then construct several natural liftings of this action to the tangent bundle $TM$, the cotangent bundle $T^*M$, and the Whitney sum $E=TM\oplus T^*M$. In certain cases, these lifted actions induce natural dynamical systems on a distinguished class of $3$-webs.

The most important open problem of symbolic dynamics consists in finding a procedure to decide whether two subshifts of finite type are conjugate or, equivalently, whether two nonnegative integer matrices are strong shift equivalent. While the problem remains hard in general, the literature also contains specific examples of matrices that are not known to be strong shift equivalent: the Baker matrices, and Ashley's eight-by-eight. These specific examples have been open for 30 years. Using computer experiments, we show that the Baker matrices are indeed strong shift equivalent for small values of k, and that Ashley's eight-by-eight matrix is indeed strong shift equivalent to the one-by-one matrix 2

Let $K_{a,m}$ and $K_{b,n}$ be missing-digit Cantor sets with initial consecutive digit sets, and write $c_a=m/(a-1)$ and $c_b=n/(b-1)$. We prove that if $c_a+c_b\geq 1$, then the radial projection of $K_{a,m}\times K_{b,n}$ from every observer has nonempty interior; no multiplicative-independence assumption is needed for this implication. Conversely, when the bases are multiplicatively independent and $c_a+c_b<1$, we exhibit explicit unbounded open sets of observers for which the radial image is compact and nowhere dense. Rational observers with the same property are dense in each exterior corner region and occur arbitrarily close to the four corners of the product. Thus $c_a+c_b=1$ is the exact threshold for the all-observers interior property within the multiplicatively independent initial-block family. In particular, this supplies an explicit two-set counterexample to the nonempty-interior conclusions of two conjectures of Yu. We also establish a sufficient modular phase obstruction for affine translates of such sets. Combining it with a fixed-pin positive-measure theorem of Banaji and Yu yields consecutive-block division sets that are compact, perfect, of positive Lebesgue measure, and nowhere dense, with both factor dimensions tending to one. For the same family, Fourier $l^1$-dimension estimates and the incidence argument used in Yu's product theorem imply that the two self-products contain intervals for all sufficiently large $r$ and $s$, while the cross-division set remains nowhere dense.

Let $g$ be a $C^\infty$ diffeomorphism in the isotopy class of a pseudo-Anosov homeomorphism $f$ such that $g$ and $f$ have the same topological entropy. In 1988, Handel proved that this implies the existence of a semiconjugacy $\pi$ from $g$ to $f$. He stated that, in general, there is at least one point $x$ such that $\pi^{-1}(x)$ is disconnected. We show that this is not the case: for every $x$, the set $\pi^{-1}(x)$ is the intersection of a nested sequence of closed topological disks, and hence is connected. We also prove that there is a unique $g$-invariant probability measure projecting to the measure of maximal entropy of $f$. This measure is entropy-maximizing, hyperbolic, and Bernoulli, and the semiconjugacy induces a metric isomorphism between the corresponding measure-preserving systems.

Predicting storm surge induced by landfalling tropical cyclones is crucial for flood mitigation and coastal risk management. Traditionally, physics-based numerical models simulate storm surge by solving the Navier–Stokes equations using numerical methods, but these simulations are computationally expensive. Generative models are promising for storm surge emulation because they can generate diverse realizations rather than producing a single deterministic prediction. However, their use for storm surge emulation remains largely unexplored. In this paper, we leverage diffusion models for storm surge surrogate modeling, combining a baseline prediction stage with conditional generation to provide a more interpretable modeling framework. We develop SurgeGen, a two-stage generative framework for generating storm surge scenarios conditioned on hypothetical storms with parameters defined in a continuous space. First, a baseline model produces a coarse estimate of the storm surge height. This estimate then conditions a diffusion model, which generates refined storm surge scenarios that better capture spatial patterns and variability. We demonstrate that our approach can generate realistic and diverse storm surge scenarios under conditions both within and outside the training distribution.

We settle two questions of Serge Tabachnikov regarding the trapping of light rays by mirrors. We show (i) that the largest dimension of a trapped family of rays in $\mathbb{R}^n$ is $2n-3$, and (ii) that in $\mathbb{R}^3$ one can trap a 2-parameter family of rays that is not locally normal to any smooth surface. We first establish that no $2n-2$-parameter family can be trapped by showing that no open subset of $L_n$ can be trapped due to Poincaré's recurrence theorem, yielding an upper bound of $2n-3$. We then show that a system of mirrors traps a $2n-3$ parameter family of rays by noting that a compact subset of the set of states of rays in the central "waist" of the trap is a normally hyperbolic invariant manifold, making it amenable to a stable manifold theorem for NHIMs, which implies that the set of rays asymptotic to the waist of the trap has dimension $2n-3$. We then truncate our mirror system, which permits the entrance of rays from arbitrarily far outside of the trap, attaining the upper bound $2n-3$ and resolving the first question. We then show that in $\mathbb{R}^3$, there exists within the trapped family of rays a 2-parameter subfamily on which the canonical symplectic form on the space of oriented lines is nowhere vanishing, implying that the subfamily is not locally normal to any smooth surface, which settles the second question.

We introduce a family of recursively generated finite probability distributions obtained from left and right embeddings with overlaps. The construction interpolates between the classical binomial distribution and the non-overlapping Bernoulli product distribution. We derive explicit formulas for the expectation, variance, and the generating function of higher moments, and formulate a conjecture asserting that the Shannon entropy is concave. The conjecture is proved in the two extremal cases and supported by symbolic computations for numerous overlap sequences.

In this paper, we consider one-dimensional discrete Schrödinger operators \begin{align} Hu(n)=(\Delta+V)u(n)\nonumber \end{align} on $\ell^2(\mathbb{N})$ with a self-adjoint boundary condition at $n=0$, where $\Delta$ denotes the discrete Laplacian and $V(n)$ is a real-valued perturbation satisfying $$V(n)=\frac{O(1)}{1+n}.$$ We determine the sharp transition for the asymptotic coefficient of \(V\) governing the existence and nonexistence of embedded eigenvalues.

We develop an efficient and accurate multiscale computational framework for predicting the buckling and post-buckling behaviour of elastic beams containing periodically distributed soft inclusions. The framework extends our previous multiscale, patch, computational homogenisation for linear elasticity by incorporating nonlinearity, and thus enables accurate prediction of both the buckling onset and the subsequent post-buckling response. Microscale computations are performed only within a sparse set of small subdomains (patches), while the macroscale behaviour is recovered through a proven patch-coupling algorithm. The scheme is assessed through quarter- and half-domain patch computations for beams with inclusion-to-matrix Young's modulus ratios ranging from 0.001 to 1. The results show that reducing the inclusion stiffness lowers both the critical buckling strain and the critical buckling stress, indicating an increased susceptibility to instability, while producing a milder post-buckling response with smaller transverse deflections and stress drops. Eigenvalue analysis of the Jacobian matrix accurately predicts the onset of instability and the corresponding critical strain and stress. Bifurcation diagrams of the nonlinear buckled configurations under compressive loading, and a quantitative analysis of the effect of the interpolation order on the predicted buckling and post-buckling responses, are also presented. Comparisons with full-domain simulations demonstrate that the proposed framework accurately predicts both the buckling threshold and the post-buckling behaviour while substantially reducing the computational cost. The methodology is readily extendable to heterogeneous beams, plates, shells, and other engineering structures.

We investigate the dynamics of Chebyshev's method applied to the polynomial family $p_n(z)=z(z^n-1)$ for $n>1$. The resulting map is denoted by $C_n$. It is proved that the immediate basins corresponding to the non-zero roots are unbounded and simply connected. We also show that the Julia set of $C_n$ is connected. It is proved that the immediate basin of the root at the origin exhibits a different behavior: it is unbounded for $n\leq 16$ and bounded for $n\geq 17$. We establish that $C_n$ is convergent whenever $n\leq 16$ or $n$ is odd. Finally, we determine the symmetry group of $C_n$ and prove that it coincides with the symmetry group of the polynomial $p_n$, thereby confirming, for this family, a conjecture proposed by Nayak and Pal.

We review the paper `Models for the Eremenko–Lyubich class' by Chris Bishop, which appeared in the Journal of the London Mathematical Society in 2015, the first of two LMS papers for which Bishop received the LMS Senior Berwick Prize in 2024. This is one of a series of papers where Bishop introduced his new technique of quasiconformal folding, which transformed the range of approaches to constructing examples of transcendental entire functions leading to the resolution of many open problems, in particular in transcendental dynamics.

Let $H\in C^2(T^*M)$ be a Tonelli Hamiltonian on a closed connected manifold and let $u_\lambda$ solve \[ \lambda u_\lambda+H(x,Du_\lambda)=c(H)\qquad\text{in }M. \] We study the convergence rate of $u_\lambda$ to the selected critical solution $u_0$. Assume that the lifted Aubry set is a finite union $\widetilde{A}=\Gamma_1\sqcup\cdots\sqcup\Gamma_N$, where each $\Gamma_i$ is either a hyperbolic equilibrium or a periodic orbit hyperbolic in the critical energy level. We prove \[ -C\lambda\le u_\lambda-u_0\le C\lambda|\log\lambda|. \] Let $\mu_i$ be the projected Mather measure associated with $\Gamma_i$. If \[ \int_M u_0\,\mathrm d\mu_i=0\qquad \text{for every }i, \] then \[ \|u_\lambda-u_0\|_\infty\le C\lambda. \] In particular, if the lifted Aubry set consists of a single hyperbolic equilibrium or a single hyperbolic periodic orbit, the convergence rate is $O(\lambda)$. We give examples showing that both convergence rates $O(\lambda)$ and $O(\lambda|\log\lambda|)$ are optimal. Without hyperbolicity, finite-order degenerate examples give lower bounds of order $\lambda^{1/(2r-1)}$ with $r\ge2$. We also construct infinite-order degenerate examples with arbitrarily slow convergence. Taken together, these results provide, to our knowledge, the first systematic quantitative theory for the vanishing discount problem in the Tonelli setting.

The study of evolution in temporally fluctuating environments often relies on the analysis of Lyapunov exponents, which quantify the exponential growth of populations. However, when model parameters depend on a stochastic process, calculating the selection gradient, a key tool for predicting the evolution of phenotypic traits, becomes a mathematical challenge. While the periodic case has been resolved, a general approach for random environments remains to be developed. This article proposes a rigorous method to: Establish the differentiability of the leading Lyapunov exponent with respect to a parameter, providing an explicit integral formula for its derivative. Develop a numerical algorithm to approximate the derivative by solving an extended differential equation. Apply these results to the analysis of mutant invasion in a resident population at equilibrium, identifying the selection gradient as the derivative of the top Lyapunov exponent.

In this paper, we study the emergent collective dynamics of multi-agent systems driven by infinite distributed fading memory of Volterra type. We establish a unified theoretical framework covering both first-order opinion consensus dynamics and second-order velocity alignment flocking kinematics. By introducing Dafermos past-history transformations, the governing integro-differential systems are reformulated into dynamical systems on an extended product Hilbert spaces. For first-order dynamics, we prove that fading memory inherently provides a hidden dissipative mechanism, guaranteeing unconditional global exponential consensus with or without instantaneous communication forces. For second-order dynamics, we obtain unconditional exponential flocking for the pure fading memory system and we give a sufficient condition for flocking when an instantaneous interaction is also present. In particular, this condition is always satisfied, namely the flocking occurs unconditionally, when the influence function has a divergent tail.

We establish a sharp rigidity–chaos dichotomy for planar two-centre billiards, motivated by a natural analogue of the Birkhoff–Poritsky conjecture: the only tables integrable at every energy should be ellipses confocal with the two centres. Let $\Omega$ be a bounded domain with $\mathcal C^1$ boundary containing the segment joining the centres. At every fixed energy $h\geq 0$, if $\partial\Omega$ is not a confocal ellipse, we construct billiard trajectories that shadow the stable and unstable manifolds of the collision–reflection orbit and realise arbitrarily prescribed sequences of sufficiently large winding numbers around the segment. This yields an invariant set semiconjugate to the full shift on a countable alphabet, periodic trajectories with prescribed finite itineraries, and compact invariant subsystems with arbitrarily large topological entropy. If, in addition, $\partial\Omega$ is real-analytic, every real-analytic function on the fixed-energy phase space $M_h$ that is invariant under the billiard map is constant. This establishes the real-analytic form of the two-centre Birkhoff–Poritsky conjecture throughout the non-negative-energy regime.

This paper is about disjoint F-transitive operators on Hilbert C*-modules, where F is a finitely invariant Furstenberg family of subsets of N. We characterize disjoint F-transitivity for generalized bilateral weighted shift operators on the standard Hilbert module over C*-algebra of compact operators on the separable Hilbert space. Then we give a sufficient condition for these operators to be disjoint chaotic and give concrete examples. We also characterize disjoint F-transitivity for generalized translation operators on the non-unitary C*-algebra and provide some applications.

Following B. Yu's work on Lyapunov graphs of non-singular Smale flows on $S^1\times S^2$, we characterize the Lyapunov graphs of non-singular Morse-Smale flows on $S^1 \times S^2$ by using filtrating neighborhoods as the local data attached to the vertices. More precisely, we determine which oriented graphs with vertices labeled by filtrating neighborhoods can be realized as such Lyapunov graphs.

We study how network connectivity and heterogeneous phase-delays shape the spatiotemporal dynamics of finite oscillator networks. Phase-delays can destabilize global synchronization and promote phase-locked patterns, including states with uniform phase gradients and more complex combinations of these modes. Yet, how connectivity and phase-delays jointly determine which states the network selects remains unclear. Here, we show that the spectrum of a composite matrix, which combines connectivity and phase-delays, governs not only the linear stability of the network's collective states but also their basin sizes. This, in turn, enables analytical estimates of basin size of phase-locked states for individual networks from connectivity and phase-delays alone. Applying this framework to nonlocal and global networks, including cases with random phase-delays, we uncover multistability and strong asymmetries in basin sizes, revealing chiral dynamics that conventional stability analysis cannot detect.