Papers

For more than half a century, iterated Pearson correlation has underpinned methods for network blockmodeling, clustering, seriation, and information visualization. Despite its continued use and the longstanding assumption of convergence, the global convergence problem remained open even in dimension three. We resolve this problem completely for the $3\times3$ correlation map: every admissible orbit is well defined for all forward iterates and converges to one of exactly seven fixed points. In arbitrary dimension, we establish an exact row-wise Gram factorization and the identity $\operatorname{rank}C(A)=\operatorname{rank}(AH_n)$, which gives the precise rank-reduction mechanism and forces every fixed point to be singular. We identify the all-ones matrix as the unique Pearson-degenerate correlation matrix, prove forward invariance of the Pearson-nondegenerate elliptope, and classify the $2^{n-1}-1$ nondegenerate sign-valued fixed points. For $n=3$, we prove a complete analytical fixed-point classification: three rank-one patterned points, three rank-two mixed points, and one rank-two equicorrelation point. We also prove the complete relative Lyapunov stability classification: precisely the patterned points are locally asymptotically stable, while the mixed and equicorrelation points are unstable. The global proof reduces the rank-two dynamics to a one-dimensional projective kernel coordinate; an exact order identity produces monotone projective ratios, excludes nontrivial periodic and recurrent limit sets, and forces convergence to a fixed point. Finally, the initial conditions converging to the four unstable fixed points form a Lebesgue-null set. Hence almost every admissible initial condition converges to a patterned fixed point; the three patterned basins are relatively open and permutation-equivalent, and each has Lebesgue measure exactly one third of that of the elliptope.

Unit excitation (UE) is a common assumption in stochastic saddle avoidance: the stochastic error must have a uniformly positive component along every direction, in expectation. This condition gives a direct way to rule out convergence to strict saddles, but it also oversimplifies the actual noise structure, and does not match many stochastic optimization regimes. In overparameterized or interpolation models, the noise may vanish near stationarity. In finite-sum problems, the stochastic gradient noise may lie in a low-dimensional, data-dependent subspace. In these (common) scenarios, UE is naturally not satisfied. In this paper, we prove an abstract almost sure avoidance theorem for stochastic recursions without UE. The theorem replaces UE-type requirements by verifiable pathwise conditions. In applications, these conditions follow, e.g., from local smoothness and finite-moment assumptions under standard i.i.d. sampling, or from the finite-sum structure under without-replacement sampling. Since the stochastically sampled maps generally do not share a fixed point, the celebrated center-stable manifold argument used in deterministic analyses is not directly applicable. Instead, we use a path-dependent change of variables together with a pathwise Lyapunov–Perron-based proof strategy. As applications, we obtain strict saddle avoidance for stochastic mirror descent (including SGD) and for random reshuffling. For nonsmooth composite objectives, we prove avoidance results for a proximal-type stochastic gradient method. Combining these insights with suitable iterate convergence guarantees, this allows establishing convergence to local minimizers of the original objective function.

In this paper, we give a memorial tribute to Prof. Erik M. Bollt (1967–2025).

Extending recent work on refractive billiards, we introduce and study the corresponding refractive outer billiards system about a convex polygon. Gutkin and Simanyi showed in 1992 that for regular outer billiards, orbits about a certain class of polygons called quasi-rational polygons are bounded, and that orbits about rational polygons are periodic. Tabachnikov and Culter later proved in 2007 that every outer billiard system about a convex polygon admits a periodic trajectory. We generalize both results to the refractive setting.

Intensity of attraction quantifies the largest amplitude of a persistent bounded disturbance that an attractor can withstand without loss of controlled confinement in its basin. Although intensity has been formulated separately for flows and maps, its behavior under temporal sampling has remained unresolved. We establish an explicit correspondence between the intensity $\mu(A)$ of a continuous-time attractor and the intensity $\mu_h(A)$ of its exact time-$h$ map. For an $L$-Lipschitz vector field, \[ \frac{\mu(A)}{1+Lh} \leq \frac{\mu_h(A)}{h} \leq \mu(A)\frac{e^{Lh}-1}{Lh}, \] and hence $\mu_h(A)/h\to\mu(A)$. The resulting first-order rate is sharp in general, while smooth scalar escape geometries can exhibit second-order convergence. We extend the framework to one-step numerical methods through a stability theory for block intensity and to attracting invariant graphs over compact invertible nonautonomous bases, obtaining uniform sampling convergence over the forcing phase. For bounded-support random perturbations, normalized discrete intensity is identified with the pathwise safety threshold; above it, finite escape follows under an explicit finite-exit condition, while escape probabilities require additional assumptions on the noise law. We also show that the discrete state–normal boundary map converges to the normalized Pontryagin boundary system governing extremal reachable-set boundaries. Exact scalar benchmarks, a grazing resilience model, planar Duffing escape, anisotropic disturbances, periodic and quasiperiodic forcing, and transfer-operator computations illustrate the theory. These results give intensity estimated from discrete observations or simulations a sampling-independent continuous-time meaning.

We prove a non-lattice local central limit theorem and Edgeworth expansions for the logarithm of the norms of products of invertible independent random matrices. Our conditions include a contraction assumption, an assumption that supports of the matrices are “large enough" and their distributions are sufficiently regular. As a byproduct of the proof we are also able to provide a different proof to the optimal rates in the CLT proved in \cite{MatBE}. Like in \cite{MatBE} we provide several sufficient conditions for contraction.

We study sets of (measurable) returns in countable groups $G$, namely sets of the form $\{g\in G:\mu(A\cap T_gA)>0\}$ arising from measure-preserving actions. Extending a result of Bergelson, we show that sets of returns in $G\times G$ contain subsets of the form $B\times B$, where $B$ is large with respect to suitable notions of largeness that remain meaningful even for non-amenable groups. As a consequence, if $G$ is amenable, then every sufficiently large subset $A\subseteq G\times G$ satisfies $B\times B\subseteq AA^{-1}$ for some large set $B\subseteq G$. We also investigate when sets of returns in $G$ contain product sets $BB$ with $B$ large. In contrast with the Cartesian-product phenomenon above, this problem is considerably subtler in non-abelian groups and is closely connected to `symmetric correlation functions', namely functions of the form $g\mapsto \mu(T_g^{-1}A\cap T_gA)$. We use this connection to show that, for broad classes of amenable groups - including finitely generated nilpotent groups and certain solvable non-nilpotent groups, every sufficiently large set $A\subseteq G$ contains a large subset $B$ satisfying $BB\subseteq AA^{-1}$. Finally, we establish polynomial analogues of these results for finitely generated nilpotent groups, extending earlier work of Bergelson and Ruzsa.

We study rational valued counts of geodesic strings (reparametrization equivalence classes of closed geodesics) for complete Riemann-Finsler manifolds, based on the Fuller index of the geodesic flow. The main conceptual result is a product type formula for these counts. Combined with aspects of KAM theory, it yields the following sample phenomenon. Let $g$ be a generic Finsler metric on $T ^{2}$, sufficiently $C ^{\infty }$-close to a flat metric, and fix a prime $p$ and a nontrivial free homotopy class $\beta $. If there is a class $\beta$ $g$-geodesic string with multiplicity divisible by $p$, then there is another one. We also obtain arithmetic constraints on counts of closed geodesics in mapping tori and flat bundles, and constraints on the existence of negative sectional curvature metrics. These counts can be understood as a shadow of a conjectural orbifold Morse homology of the infinite-dimensional quotient stack $[LX/S^1]$.

We classify rank 2 rel 0 arithmetic affine invariant subvarieties in the minimal stratum H(6). The proof follows and extends Apisa's approach in genus three, reducing the analysis to the classification of rank 1 rel 1 cylinder rigid affine invariant subvarieties in lower-genus boundary strata. In particular, together with the classification of algebraically primitive rank 2 rel 0 orbit closures, this gives a complete description of rank 2 affine invariant subvarieties in H(6).

Forsythe conjectured in 1968 that the normalized residual directions of restarted conjugate gradients approach a two-cycle. We prove the conjecture for restart length two: unless the method terminates, its even and odd normalized residuals converge separately.

In this work we study a fractional wave equation with Balakrishnan–Taylor type damping posed on a bounded domain $\Omega\subset\mathbb{R}^n$. The model couples the fractional Laplacian with a nonlinear damping coefficient depending on the fractional energy, leading to a doubly nonlocal evolution equation that extends the classical wave equation with energy-dependent dissipation. By means of the theory of linear operators, we establish the global well-posedness of both mild and regular solutions. We then investigate the long-time dynamics of the associated semigroup and prove that it is gradient and asymptotically smooth. As a consequence, we establish the existence of a compact global attractor and show that it coincides with the unstable manifold of the set of stationary solutions. To the best of our knowledge, this provides the first characterization of the asymptotic dynamics for fractional wave equations with Balakrishnan–Taylor type damping.

In this paper, we establish quantitative Denjoy–Koksma type estimates for higher-dimensional quasi-periodic torus rotations. For Diophantine frequency vectors, we establish quantitative estimates on the discrepancy between Birkhoff averages and spatial averages for observables with various Besov-type regularities. The Besov framework is naturally adapted to the small-divisor structure of the problem, allowing a sharp analysis of resonant frequencies and yielding optimal rates throughout the entire parameter range. These estimates yield, to the best of our knowledge, the sharpest currently available convergence rates for Hölder continuous observables. As applications, we obtain substantially improved quantitative homogenization results for Hamilton–Jacobi equations in spatially quasi-periodic settings, as well as nearly optimal statistical regularity estimates for invariant measures under perturbations.

Let $T : X \to X$ be a continuous map of a compact metrizable space, and let $\phi : X \to \mathbb{R}$ be a continuous function. The ergodic optimization problem is to maximize the integral $\int \phi \, d\mu$ as $\mu$ ranges over all $T$-invariant Borel probability measures on $X$. In this paper we consider a constrained version of the ergodic optimization problem. Given a `constraint set' $\mathcal{C}\subset C(X)$, let $M_\mathcal{C}(X,T)$ be the set of $T$-invariant Borel probability measures $\mu$ on $X$ such that $\int g \, d\mu = 0$ for all $g \in \mathcal{C}$. We investigate the problem of maximizing the integral $\int \phi \, d\mu$ over the constrained set $M_\mathcal{C}(X,T)$. We address basic properties of this optimization problem, beginning with nonemptiness of $M_\mathcal{C}(X,T)$ and existence of optimal solutions. Additionally, we establish the generic and prevalent uniqueness of optimal measures, we provide a realization result, and we give a characterization of the dual problem. This framework provides a common generalization of several previously considered optimization problems in dynamical systems and optimal transport.

Let $\omega$ be a concave modulus of continuity that is weaker than Lipschitz, meaning $\omega(t)/t$ diverges as $t$ approaches $0$. We construct a diffeomorphism of the circle with irrational rotation number, in the regularity class $C^{1+\omega}$, with a wandering interval. This construction implies that Denjoy's 1932 theorem is sharp in regularity, unless additional restrictions are imposed on the rotation number. The construction in the special case $\omega(t) = t\log(1/t)$ settles an open problem dating back to Herman's 1979 work on circle diffeomorphisms, which gave constructions for $\omega(t) = t\log(1/t)^{1+\varepsilon}$ for every $\varepsilon > 0$. Our examples arise as limits of periodic circle diffeomorphisms with rapidly converging rotation numbers.

We study the spiral product $G_\gamma(t,x,y)=(z_1(t)f_1(x),z_2(t)f_2(y))$, which couples two spherical $\mathscr C$-totally real immersed factors through a profile curve $\gamma=(z_1,z_2)\subset\mathbb S^3$. Its minimality is governed by a weighted-geodesic system: $G_\gamma$ is minimal precisely when both factors are minimal and $\gamma$ is an unparametrized geodesic of $|z_1|^{2k_1}|z_2|^{2k_2}g_{\mathbb S^3}$. This flow is Liouville integrable and, when $k_1+k_2>0$, its phase map has an open dense full-rank locus. Consequently, closed profiles of arbitrarily large primitive order occur densely. For compact minimal factors, Routh reduction and Sturm oscillation give $\text{Ind}(G_\gamma)\geq \text{Ind}(f_1)+\text{Ind}(f_2)+2m_\gamma-3$ for a closed oscillatory profile of primitive closing order $m_\gamma$. On the contact level, a factor-adapted choice of profiles produces, from any prescribed pair of compact connected embedded special Legendrians, embedded special Legendrian products of every sufficiently large prime closing order. Finally, canonical finite horizontal lifts and Hopf projection give Delaunay-type minimal Lagrangians in complex projective spaces. For compact embedded inputs and ordinarily closed profiles, the primitive spherical quotient is embedded, while its projective quotient is embedded exactly when the reduced relative winding is one.

We construct two explicit countable discrete groups $\Gamma_1$ and $\Gamma_2$ that are both ICC and have Kazhdan's property (T). Although $\Gamma_1$ and $\Gamma_2$ are not isomorphic as groups, their group von Neumann algebras are isomorphic: $L(\Gamma_1)\cong L(\Gamma_2)$. This provides a counterexample to Connes' rigidity conjecture for ICC property (T) groups. This result was obtained with assistance from GPT-5.6 Sol, independently of and concurrently with work by OpenAI.

We study uniform rigidity and topological mild mixing through continuous observables. For a fixed sequence of times, the observables rigid along that sequence form a closed unital $T^{\pm1}$-invariant algebra and determine the maximal factor uniformly rigid along the prescribed sequence. We then give functional forms of the classical ${\rm SIP}^{*}$- and ${\rm IP}^{*}$-return-time criteria: a topological dynamical system is mildly mixing exactly when it has no nonconstant locally SIP-rigid observable, and in the minimal category the same property is equivalent to the absence of nonconstant locally IP-rigid observables. Finally, a locally IP-rigid observable yields a canonical orbit-name factor carrying marked local data. For fixed local data, the $T^{\pm 1}$-invariant core of the local rigidity algebra determines a uniformly rigid factor whenever the core is nontrivial.

In this paper we prove that every multi-singular hyperbolic set of a $C^1$ flow is Komuro expansive. This is the strongest natural form of expansivity for flows with singularities accumulated by regular orbits, allowing arbitrary orientation-preserving time reparametrizations. We then use this to establish a two-sided asymptotic counting bound of the form $e^{ht}/t$ for periodic orbits, and prove that the normalized orbit measures converge to the unique measure of maximal entropy. We also establish the same results for a $C^1$ open and dense subset of star vector fields.

In this paper, we establish the exponential convergence of Laskar's pioneering frequency map analysis, strictly improving upon classical polynomial bounds. By utilizing appropriately chosen weighting functions, we achieve such exponential rates in the analytic quasi-periodic regime for frequency vectors satisfying Diophantine or Brjuno nonresonance conditions. Furthermore, we extend the theoretical framework beyond the analytic quasi-periodic regime into the analytic almost periodic setting. By employing and generalizing Bourgain's framework, we accommodate a much broader class of anisotropic spatial structures. These results yield the first unified theory of exponential convergence for frequency map analysis, revealing the interplay among analyticity, spatial structures, nonresonance conditions, the choice of weighting functions, and convergence rates. This framework has important implications for applications in fields including celestial mechanics. Moreover, the novel techniques developed herein yield new insights into the study of weighted Birkhoff averages.

Let $f:\mathbf{P}^n\to\mathbf{P}^n$ be a $q$-polarized endomorphism, where $q>1$, and let $R_f$ be its ramification divisor. We study the singularities of the ramification pair $(\mathbf{P}^n,R_f)$. We show that, for a general $f$, the pair $(\mathbf{P}^n,R_f)$ is log canonical. When $n=2$, we prove that there exists an integer $s\geq1$ such that the log canonical threshold $\mathrm{lct}(\mathbf{P}^2;R_{f^s})\geq1/(q^s-1)$. The passage to an iterate is necessary in general, and the lower bound is optimal. In particular, $(\mathbf{P}^2,R_{f^s}/(q^s-1))$ is a log Calabi–Yau pair, completing the proof of Gongyo's conjecture for smooth projective surfaces.

In this work, we analyze a completely degenerate germ of a real vector field at the origin of $\mathbb{R}^3$. We construct a generic three-parameter unfolding of this germ and identify the associated local bifurcations. We prove that the phase portrait of the unfolded system contains a one-parameter family of heteroclinic orbits connecting two hyperbolic equilibrium points.

Attention-only dynamical theories model Transformer residual directions as particles aggregating on a sphere. We extend this framework by incorporating the feed-forward network (FFN) term as a local steering field acting on each token state. The resulting theory predicts that the tangential component of the FFN field is necessary for motion in residual-direction space, that critical residual directions correspond to nonlinear projective equilibria, and that a commutator defect determines when a finite attention–FFN block can be accurately approximated by a parallel, additive flow. Across GPT-2, Pythia, Mistral, and Llama models, the extended theory improves one-step angular prediction relative to an attention-only baseline, with the contribution of the FFN increasing from GPT-2 to Llama-3-8B. Intervention experiments show that retaining only the tangential FFN component preserves most model quality, whereas retaining only the radial component causes performance to collapse. The tangential component also preserves output diversity under aggregation pressure. As a practical application, layers with small commutator defects can be approximately parallelized with only a modest increase in loss, whereas layers with large defects degrade rapidly. These findings support the interpretation of FFN layers as directional steering fields that shape Transformer residual geometry and govern the feasibility of block-level interventions.

We determine the minimum hyperbolic area of Teichmuller curves arising from holomorphic quadratic differentials on closed Riemann surfaces of genus two. The minimum is 3\pi/5, and it is attained precisely by quadratic differentials q=\omega^2 for which the translation surface (X,\omega) lies in the GL_2^+(R)-orbit of the double-pentagon translation surface. Equivalently, the extremal projective Veech group is the triangle group \Delta(2,5,\infty). The proof combines a small-area classification of noncompact hyperbolic orbifolds with a derivative-preserving affine descent construction for nonsquare quadratic differentials. The three possible nonsquare zero patterns are then excluded by arithmetic, marked-point, and covering obstructions.

We establish exponential mixing for the randomly forced and weakly damped KdV equation in $L^2(\mathbb{T})$. The noise is bounded, localized, and degenerate in high frequencies. Our proof relies on a general probabilistic framework in [11,33], nonlinear smoothing for KdV and its linearization via normal form transformation, and stabilization of the system by localized force. This paper continues a series of works connecting asymptotic compactness, control theory, and ergodicity and mixing for randomly forced dispersive PDEs.

Given a Riemann surface {\Sigma}, let {\Gamma} {\subseteq} Mod({\Sigma}) denote the monodromy subgroup of a family of complex curves homeomorphic to {\Sigma}, arising from a sufficently ample Lefschetz pencil. We establish that the group {\Gamma} acts with Zariski dense orbits or ergodically on certain character varieties for {\pi_1}({\Sigma}). This answers a version of a Conjecture of Katzarkov, Pantev, and Simpson appearing in [KPS03].

For mirror descent generated by a Legendre kernel, perhaps one of the most basic question in optimization is this: must every accumulation point of a bounded mirror descent sequence be Karush–Kuhn–Tucker (KKT) stationary under proper stepsizes? We show that the answer is no. A longstanding obstacle to resolving this question is the boundary blow-up of the Legendre gradient: it keeps every mirror step in the interior, while at a boundary limit, the inverse entropy metric vanishes on active coordinates and can erase the dual-feasibility in the KKT system. We construct $C^\infty$ objectives and bounded sequences generated by the Shannon-entropic mirror descent on the nonnegative orthant $\R_+^n$, for every $n\geq 3$, and on the probability simplex $\Delta_n$, for every $n\geq 4$, such that, in each case, the set of accumulation points is a smooth boundary circle containing a nonempty relatively open arc of non-KKT points. The steps satisfy $\alpha_k\asymp k^{-\beta}$ with $\beta\in(1/2,1)$, the objective values are nonincreasing, and the objectives are entropy-relatively smooth. Hence the pathology stems from the degeneracy of the Bregman geometry at the boundary, rather than from failure of descent, or improper stepsizes. To the best of our knowledge, these provide the first counterexamples to KKT accumulation for bounded mirror descent sequences with nonincreasing objective values.

We consider the Riemann problem for a 3x3 system of conservation laws with generalized Chaplygin pressure $p(\rho,v)=-\frac{A(v)}{\rho^\alpha}$, $0<\alpha\leq 1$, where the pressure depends on an additional transported variable. We analyze the system's wave structure and classify the Riemann solutions. Whenever classical solutions consisting of shocks, rarefaction waves, and contact discontinuities fail to exist, singular solutions arise. We verify that these satisfy the conservation laws in the distributional sense within the classical Dirac delta framework, and compare them with Nedeljkov's shadow-wave construction, giving two complementary descriptions of the same singular solution. We further study admissibility via the Dafermos maximum entropy dissipation principle, with several examples showing how it selects the physically relevant solution. Lax-Friedrichs simulations illustrate the Riemann wave patterns and provide a comparison with the analytical results. To construct viscous profiles for the isolated overcompressive $\delta$-shock, we assume $\alpha\in\mathbb{Q}\cap(0,1]$ and apply the Dafermos regularization together with a spherical blow-up. Working in three directional charts, we construct the reduced singular concatenation consisting of the left outer orbit, the middle orbit on the blown-up boundary, and the right outer orbit. We then prove that, for sufficiently small positive viscosity, this singular concatenation perturbs to a heteroclinic orbit. Consequently, the isolated overcompressive $\delta$-shock is realized as the zero-viscosity limit of a family of self-similar Dafermos viscous profiles.

We study the dependence of the top Lyapunov exponent of a mixed Markov quasi-periodic cocycle on the transition probabilities. The transition matrix is assumed primitive, the Markov extension on the torus is assumed non-resonant along directed cycles, and the top Lyapunov exponent is assumed simple. We first prove the result in the irreducible case by a forward Markov transfer operator and contraction estimates uniform over the initial Markov state. The reducible case is then obtained, as in the Bernoulli argument of Bezerra-Sánchez-Tall, by decomposing along measurable invariant sections into restricted and quotient bundle cocycles and inducting on the fiber dimension. A measurable invariant section need not admit a continuous trivialization, even when the original matrices cocycle are continuous. We therefore formulate the irreducible analytic case for essentially bounded measurable bundle cocycles with essentially bounded inverses.