Papers

Let $f$ be a dominant rational self-map of a normal projective variety over an algebraically closed field of arbitrary characteristic, or a dominant meromorphic self-map of a compact Kähler manifold. We prove that in every codimension, the degree sequence of $f$ differs from its dynamical-degree growth by at most a polynomial factor, with an explicit universal bound on the polynomial exponent. In particular, subexponential but superpolynomial degree growth cannot occur. At a strict dynamical-degree break, we prove the sharper asymptotic \[ \mathrm{deg}_{i,\omega}(f^n)=C_{f,\omega,i}\,\lambda_i(f)^n+O(\eta^n), \] for every $\sqrt{\lambda_{i-1}(f)\lambda_{i+1}(f)}<\eta<\lambda_i(f)$. This extends the degree-asymptotic theorem of Dang–Favre from codimension one in characteristic zero to arbitrary codimension and arbitrary characteristic, and recovers the surface theorem of Boucksom–Favre–Jonsson. Both results rely on a uniform estimate for the temporal discrete logarithmic Laplacian of the degree sequences, obtained by applying the Ptolemy inequality to ordered Hodge–Lorentz packages on varying finite-time orbit spaces.

Dynamics of biochemical systems are dictated by the stoichiometry and kinetics of reaction networks. These systems may admit positive steady states that could undergo qualitative change in response to varying rate parameters. These changes are called steady-state bifurcations, and they correspond to biologically meaningful processes. Classical bifurcation analysis relies on exact kinetics and rate parameters, which are often unknown or difficult to measure for complex systems. To address this, structural bifurcation analysis (SBA) was introduced to predict bifurcations using only the system's associated reaction network. In this work, the computational package NetBiF (Network Bifurcation Finder) is introduced to implement SBA. NetBiF is applied across diverse biochemical systems, including fundamental metabolic pathways, signaling and gene regulatory networks, and a large-scale carbon cycle model, to predict steady-state bifurcations. Crucially, for biochemical systems governed by power law kinetics, the implementation of NetBiF also classifies the steady-state bifurcations as saddle-node, transcritical, or pitchfork via a structural analog of Sotomayor's theorem. To this end, explicit formulations for the bifurcation criteria were derived for these biochemical systems. Determining and classifying bifurcation behaviors provide insight into biochemical systems' abilities to adapt to changing and fluctuating conditions. Collectively, the study presents NetBiF as a scalable, structure-based, and consistent computational tool to implement SBA. Our findings enable researchers to systematically study dynamical behaviors of biochemical systems through their reaction network structures without requiring explicit kinetic information.

Let $M$ be a finite-dimensional complete Riemannian manifold of bounded geometry. We consider a time-dependent vector field $X : \mathbb{R} \times M \to TM$ with $X_t:=X(t,\cdot) \in \mathfrak{X}^r(M)$ for all $t \in \mathbb{R}$ that satisfies some further uniform continuity assumptions. We suppose that $X$ admits a globally defined integral curve $\gamma : \mathbb{R} \to M$. To any isolated subset $\Omega$ of the Sacker-Sell spectrum of $t \mapsto \nabla X_t(\gamma(t))$, Sacker-Sell theory associates a subbundle $\mathscr{N}^\Omega$ of $T_\gamma M$ invariant under $\nabla X_t$. Under additional nonresonance conditions on $\Omega$, we prove, by an application of the parameterization method (Cabré et al., 2003), the existence of an invariant topological submanifold $\mathscr{W}^\Omega \subset \mathbb{R} \times M$ with $C^r$ fibers $W_t^\Omega:= \mathscr{W}^\Omega \cap \{t\} \times M$ that is unique in a sufficiently smooth subclass of invariant manifolds uniformly tangent to $\mathscr{N}^\Omega$ (made precise later). We moreover show regularity of $\mathscr{W}^\Omega$ with respect to parameters, and characterize the attraction towards $\mathscr{W}^\Omega$ via a foliation in the case $\Omega$ is the slowest part of the spectrum.

We introduce Contractor Filters (CFs), a set-membership architecture for nonlinear continuous-time state estimation with bounded uncertainties and sporadic outliers. CFs represent feasible states symbolically by compositions of contractors rather than rematerializing a geometric enclosure after each update. This provides an abstraction layer between the set-theoretic filtering equations and their numerical realization, so prediction, correction, ambiguity, and outlier handling can be expressed within one algebra without prescribing a state geometry. We derive an exact nominal recursion and a guaranteed robust enclosure, with validated flow inclusions preserving soundness. Optional sub-paving compression then provides finite outer representations. CFs are evaluated against a robust particle filter on range-only and terrain-based localization under outliers, probabilistic model mismatch, association ambiguity, and multimodality. CFs retain the simulated truth in all tested configurations, while particle-filter reliability degrades under persistent bias and spatial aliasing. Contractor strength and compression budget control the trade-off between paving tightness and computational cost, while preserving the enclosure guarantee.

This paper develops theoretical and numerical extensions of Birkhoff pseudospectral transcription for aerospace optimal control problems. Building on the recently proposed body of knowledge, a Birkhoff integration operator is derived for flipped Legendre Gauss Radau (fLGR) grids, drawing connections to regular LGR grids and to the favorable conditioning properties that make Birkhoff interpolants particularly appealing. We also propose the enforcement of differential constraints via pseudospectral operators, and the primal and dual equivalence of this approach to state augmentation is established for control-rate limits. The numerical developments include the formulation and solution of heterogeneous multiphase optimal control problems, parameterized non-affine time mappings, composition of affine scaling and equilibration methods, and a hybrid derivative strategy to combine linear pseudospectral operators with numerical derivatives for nonlinear functions. These methods are implemented in the RAPTOR (Rapid Algorithm for Pseudospectral Trajectory Optimization and Robustification) transcription framework and assessed through feature-oriented benchmarks: a one-dimensional landing problem, a planar orbit-raising scenario, an atmospheric entry problem, and a multiphase reusable launch vehicle problem with simultaneous booster-recovery and upper-stage-insertion trajectories.

Let $k=\overline{\mathbb{F}}_p$ and let $f\in k[X]$ be a polynomial of degree at least $2$ such that $f-f(0)$ is non-additive. For $\alpha,\beta$ in an algebraically closed extension $L$ of $k$, we prove that there are infinitely many $\lambda\in L$ for which both points are preperiodic under $f(X)+\lambda$ if and only if $\alpha,\beta\in k$ or $f(\beta)-f(\alpha)\in E_f$. Here $E_f=\{c\in G_f:(f-f(0))^n(c)\in G_f\text{ for every }n\ge0\}$, where $G_f=\{c\in k : f(X+c)-f(X)\in k\}$. We also provide a necessary and sufficient criterion for the corresponding colliding orbits set to be infinite.

We prove that for every $(X,\mathcal{B},\mu,T)$, an aperiodic, probability preserving system and $\mathcal{P}$ a finite measurable partition of $X$, there exists an Alpern Multitower whose base is independent of $\mathcal{P}$. This removes the ergodicity assumption in the original proof of the independent Alpern Lemma of \cite{MR3373823}. As a consequence, we remove ergodicity assumption from recent results on limit theorems in deterministic systems \cite{Kosloff_Volny_2022,kosloffVolnyStableCLT,MR4955410}.

Developing resilient crops requires models that accurately capture how plant growth dynamics emerge from the continuous interplay between genetic and environmental factors. While high-throughput phenotyping platforms generate data for dynamic growth models, current modelling approaches show limitations: process-based crop growth models (CGMs) capture environmental dynamics but are difficult to scale across many genotypes because they require calibration of many parameters, while quantitative genetic models allow modelling many genotypes but often summarise time-varying environments using static covariates. To address this gap, we propose a data-driven framework to identify reduced systems of ordinary differential equations (ODEs) describing crop growth dynamics as functions of environmental variables and a controllable number of genotype-specific coefficients (GSCs). Using the Sparse Identification of Nonlinear Dynamics (SINDy) method, we infer model structure directly from data via sparse regression on a physiologically informed library of candidate functions. In this framework, GSCs parameterise cultivar-specific differences in response to environmental conditions. The resulting models are linear in these GSCs, making them compatible with mixed-model and genomic prediction approaches. We evaluate the approach on an in silico dataset generated with WOFOST for five potato cultivars across twelve growing seasons. Our identified ODEs generalise to held-out years and achieve $R^2 > 0.9$ for end-of-season yield prediction across held-out years when compared to full-scale WOFOST simulations. These results provide a proof of concept for SINDy-based identification of parsimonious dynamic crop models from longitudinal crop-growth data.

Peter Lax's 1968 discovery that certain nonlinear wave equations possess operator pairs (Lax pairs) opened the theory of completely integrable systems, yielding exact solutions, infinite conservation hierarchies, and perfectly elastic soliton collisions. When a nonlinear dispersive equation is provably non-integrable (admitting no Lax pair and only finitely many conservation laws), what mathematical structure governs its dynamics? The K(m,m) compacton equations, u_t + (u^m)_x + (u^m)_{xxx} = 0 for m >= 2, provide a testing ground: they support traveling waves with compact support, yet fail every known integrability test. We prove universal algebraic formulas determining the shape, width, and speed of every K(m,m) compacton. We develop a constraint-deficit mechanism explaining how the five independent conservation laws constrain emergent amplitudes and suppress the fundamental Fourier mode of the collision debris, while higher radiation modes remain unconstrained. The mechanism's conclusions rest on explicit hypotheses supported by numerical evidence. A phase-plane analysis reveals the compacton as a singular limit of smooth solitary waves through simultaneous regularity loss and support compactification. These results delineate intermediate territory between integrable order and generic nonlinear dynamics, organized by scaling universality and by the constraint deficit of a finite conservation structure.

Let $\mathcal{S}_c$ be the subgroup of $Aut(\mathbb{C}^2)$ generated by affine maps and shear automorphisms with constant Jacobian. We determine the possible conformal types of the rank-one limit set of an invariant recurrent Fatou component when the induced dynamics has infinite order. In particular we show that, under this infinite-order hypothesis, the limit set of a finite composition of Hénon maps (polynomial or transcendental), can only be conformally equivalent to a disk or an annulus. Thus we exclude the remaining three conformal types, i.e. $\mathbb{D}^*$, $\mathbb{C}^*$ and $\mathbb{C}$, from the classification result by Arosio, Benini, Fornæss and Peters. We also prove that the set of fixed points of a finite composition of Hénon maps (polynomial or transcendental) contains no entire curves.

We study the behavior of saddle connection directions on translation surfaces from the point process viewpoint. We define a direction point process for saddle connections of length at most $R>0$ on every translation surface. Given a general ergodic $\mathrm{SL}(2,\mathbb R)$-invariant measure $\mu$ on a stratum of translation surfaces, we also define a slope point process associated with $\mu$. By utilizing the dynamics and geometry of strata, we show that for almost every translation surface, the direction point processes, rescaled by $R^2$, converge in distribution to the slope point process as $R\to\infty$. Using Palm distribution theory, we prove the existence of limiting joint gap distributions and obtain new regularity results for gap distributions associated with the general $\mu$. Finally, we study how these point processes and joint gap distributions behave as the genus $g$ tends to infinity. In the principal stratum, we show that the slope point processes, rescaled by $g^2$, converge in distribution to a Poisson point process. By passing to the corresponding Palm distributions, we show that the rescaled slope gaps converge jointly to independent exponential random variables, which is in stark contrast to the known non-exponential gap distributions in a fixed genus.

We show that finite-order exponential mixing does not imply the central limit theorem for smooth diffeomorphisms. For every integer $R\geq2$, we construct a $\mathcal{C}^{\infty}$ diffeomorphism $T:M\to M$ of a compact smooth manifold $M$ and a $T$-invariant probability measure $\mu$ such that $(T,\mu)$ is exponentially mixing of order $R$ for $\mathcal{C}^1(M)$ observables, but not of order $2R+2$. For some centred observable $f_0\in\mathcal{C}^{\infty}(M)$, no subsequence of the normalised sums $\frac{1}{\sqrt n}\sum_{i=0}^{n-1}f_0\circ T^i$ converges in distribution to a Gaussian law. In particular, exponential decay of two-point correlations implies neither exponential mixing of all orders nor the central limit theorem. We realise $(T,\mu)$ in two ways. In the first, $T$ is a fixed linear Anosov diffeomorphism of $\mathbb T^2$ and $\mu$ is singular with respect to Lebesgue measure. In the second, $M$ can be any compact smooth surface without boundary and $\mu$ is a physical measure whose statistical basin contains a nonempty open set. The main theorems have been formalised in Lean 4.

We investigate the regularity of Lyapunov exponents for products of random invertible matrices of arbitrary dimension $d\ge 2$. We prove that the top Lyapunov exponent is pointwise Hölder continuous with respect to the Wasserstein-Hausdorff distance in the space of probability measures with compact support, in the case where this exponent is simple and the equator has dimension equal to or less than one.

We prove sharp arithmetic localization for one-dimensional quasiperiodic potentials with jump discontinuities, beyond uniform transversality. Our results include every non-constant $C^1$ monotone sampling function, even with arbitrarily large flat regions, and also certain non-monotone potentials. A key new mechanism is a symbolic unique-continuation principle: first-return words to the region where $f'>0$ have no common Dirichlet resonance, forcing eigenfunction mass to transfer away from flat or negatively sloped regions and recovering quantitative monotonicity of finite-volume eigenvalue branches.

We provide several results showing that sums of “small” Cantor sets can turn into an interval or a finite union of intervals. Specifically, we show that 1) There exists a Cantor set $C\subset [0,1]$ of zero logarithmic capacity such that $C+C=[0,2]$. 2) For any $n\ge 2$ there exists a Cantor set $C\subset [0,1]$ of zero logarithmic capacity such that a Minkowski sum of $n$ copies of $C$ is an interval $[0,n]$, while a sum of any smaller number of copies of $C$ is still a Cantor set. 3) For any $n\ge 2$ there exists a collection of Cantor sets $C_1, C_2, \ldots, C_n\subset [0,1]$ of zero lower box-counting dimension such that a Minkowski sum of any $n-1$ sets from this collection is a Cantor set of zero lower box-counting dimension, while the sum of all $n$ sets is an interval $[0,n]$. 4) There is an uncountable family of Cantor sets of zero lower box-counting dimension such that the sum of any two different sets from this family is a finite union of intervals. 5) For each of the properties above, there are examples of Cantor sets with the same property having zero Hausdorff $h$-measure for an arbitrary given gauge function $h$.

We introduce a new criterion which guarantees that a conservative local diffeomorphism is both non-uniformly expanding (in particular has all its Lyapunov exponents positive) and stably ergodic. The criterion involves averaging expansion rates over preimages and allows us to construct explicit examples in a variety of homotopy classes, including those represented by linear endomorphisms with contracting directions.

We study the asymptotic behavior of extreme events in multivariate stationary sequences, with particular emphasis on stochastic processes generated by dynamical systems. We develop a general framework for describing the dependence structure of extremes through two complementary objects: the tail process and the spectral measure. We first extend the enriched point-process convergence of the univariate theory to the multivariate setting, proving that the decorated point process of exceedances converges to a marked Poisson cluster process whose marks retain the componentwise and temporal structure of extreme clusters. We then construct a spectral measure for stationary sequences, which assigns weight to distinct extremal profiles and characterizes the asymptotic dependence between components, and establish a de Haan-Resnick type representation. We also relate the spectral measure to the tail process, showing that its normalized profile distribution is recovered from the angular component of the tail process. Finally, we recover the classical univariate and multivariate exceedance processes as special cases. We illustrate the theory through applications to dynamical systems, using examples in which different dynamical mechanisms give rise to distinct extremal clustering patterns and cross-dependence structures. These applications demonstrate the ability of the proposed framework to capture genuinely multivariate features of extreme events and to extract information about the underlying dynamics that is not accessible through classical exceedance processes.

Let $f$ be a homeomorphism of a closed orientable surface of genus at least one that is isotopic to a pseudo-Anosov homeomorphism, possibly relative to a finite invariant set. We construct an intrinsic monotone factor by collapsing the maximal continua whose iterates have uniformly bounded diameter in the universal cover. The quotient is homeomorphic to the original surface and the factor is isotopic to $f$ and has several strongly chaotic dynamical features. It is infinitely continuum-wise expansive, tight, and topologically mixing, and it has a dense family of rotational horseshoes, dense periodic points, and a single topological homoclinic class. The factor can be taken as an area-preserving homeomorphism. The quotient preserves stable translation length on the fine curve graph and the applicable homological rotation sets. When $f$ is $C^\infty$, a companion paper gives a unique measure of maximal entropy for the factor; it has full support and is Bernoulli.

In this note we show that all interval exchange transformations of three intervals (3-IETs) are spectrally disjoint from all mixing and disjoint from lightly mixing systems. Until now, it was known that almost every 3-IET is rigid, which implies this property. However, not all 3-IETs are rigid, hence our result is a strict improvement of current knowledge. We then use this result, among many others, to prove that although some 3-IETs are mildly mixing, they are always disjoint in the sense of Furstenberg from lightly mixing systems.

This paper studies the topological entropy of real-analytic Liouville integrable Hamiltonian systems. We prove that the Hamiltonian flow on a compact connected regular energy hypersurface has zero topological entropy whenever a complete family of commuting real-analytic first integrals is generically independent in a connected neighborhood of that hypersurface. The proof uses finite normal expansions along minimal analytic carriers and an ergodic version of Ziglin's leading-term argument to obtain independent invariant normal polynomials, whose Poisson geometry makes all tangent Lyapunov exponents of the carriers vanish. In particular, we prove that a real-analytic Liouville integrable geodesic flow on a closed real-analytic Riemannian manifold has zero topological entropy, giving a positive answer to the conjecture proposed by Paternain in 1992.

We prove that the singular locus of the geodesic flow on the unit tangent bundle of a nonpositively curved surface enjoys an additional regularity property: its characteristic function belongs to the geometric anisotropic Sobolev space $H^{5/8-}_xH^{-13/16}_\theta$. In particular, this implies that if the geodesic flow is not ergodic with respect to the Liouville measure, then the zero set of the curvature must be thick in the sense that it supports a nonzero, nonnegative function in $H^{5/8-}$. The proof relies on semiclassical analysis and a new radial estimate for parabolic flows.

While the robustness of collective dynamics in static networks is well established, how agent mobility and network topology jointly shape dynamical robustness remains largely unexplored. In this letter, we investigate the robustness of mobile Stuart-Landau oscillators performing random walks on networks and interacting diffusively when co-located. In the rapid-mobility regime, we reduce the dynamics to a mean-field framework linking network degree statistics to an effective global coupling strength. We show analytically that increased degree heterogeneity in the network enhances the effective coupling strength, and strictly lowers the critical threshold of functional degradation for amplitude death. Numerical simulations confirm the predicted heterogeneity-dependent ordering of the transition thresholds, showing an earlier collective collapse in scale-free networks than in random or regular networks. Our results establish mobility and network heterogeneity as key determinants of robustness in mobile oscillator populations, with implications for resilient biological and engineered collectives.

A set $S$ of positive integers is $2$-large if every two-coloring contains arbitrarily long monochromatic arithmetic progressions with common difference in $S$. We prove that every such set meets every nil-Bohr neighborhood of $0$. This proves a conjecture of the first author and gives a sharp answer to a question of Host, Kra, and Maass. We also show that, for every fixed nil-Bohr neighborhood \(B\) of \(0\) and every \(\eps>0\), restricting the common difference to \(B\) increases the extremal number for \(k\)-term arithmetic progressions by at most a factor of \(1+\eps\), provided \(k\) is sufficiently large.

We study the Fatou components of holomorphic skew products $F \colon \mathbb{C}^2 \to \mathbb{C}^2, F(z,w) = (f(z) + r(z,w), g(w))$ near an attracting invariant fiber, introducing a refined classification of components on the fiber: bulging, partial bulging, and nowhere bulging. For polynomial skew products, we give a complete characterization of when the basin at infinity is non-bulging, and show that in this case non-bulging implies nowhere bulging, ruling out partial bulging entirely. In the transcendental setting, the behavior is richer: we construct partially bulging wandering domains and Baker domains, show that any orbitally unbounded Fatou component can be made nowhere bulging by choosing an appropriate function $r$, and prove that for any nonconstant $r$ there exists an entire function $f$ with an escaping wandering domain that fails to be bulging. Bulging thus depends on the interplay between fiber dynamics and perturbation.

Let $G$ be a locally compact, second countable group admitting a Reiter sequence, and suppose that $G$ acts on a von Neumann algebra $\mathcal{M}$ equipped with a semifinite trace $\tau$ via trace-preserving automorphisms. In this paper, we prove a maximal inequality for ergodic averages along such Reiter sequences. Our approach relies on approximating the Reiter sequence by a linear combination of characteristic functions of Følner sequences using the layer-cake representation. Furthermore, we establish the bilateral almost uniform convergence of these ergodic averages. As an application, we deduce a non-commutative Wiener-Wintner type theorem along Reiter sequences.

We introduce a general methodology to construct globally unstable trajectories near the normally hyperbolic invariant manifold (NHIM) of a saddle-center-center Hamiltonian equilibrium. The proposed approach builds upon the classical geometric mechanism of searching for trajectories near heteroclinic connections between invariant tori of the NHIM. The main ingredient is the Birkhoff normal form near the NHIM, which allows us to control the dynamics of the hyperbolic and center components. We construct transition maps on the NHIM that encode the dynamics along heteroclinic connections. By combining the transition maps with the inner dynamics, given by the restriction of the Hamiltonian flow to the NHIM, we show that these connections can be efficiently (in terms of the travel time) shadowed by true trajectories of the full Hamiltonian system. Finally, the efficiency of this methodology is illustrated in the case of the NHIM associated to the libration point $L_1$ of the Spatial Restricted 3-Body Problem (SR3BP) for the Sun-Earth system.

In 1965, Chowla conjectured that the values of the Liouville function become asymptotically uncorrelated at consecutive natural numbers. Recently, the two-point case is resolved by OpenAI. In 2026, Charamaras and Richter conjectured a generalization of the Chowla conjecture to the asymptotic independence of the consecutive prime Omega functions. In this article, we prove the two-point case of their conjecture. As a consequence, we obtain a dynamical generalization of the two-point Chowla conjecture.

This paper deals with the pullback measure attractor for a non-autonomous distribution-dependent stochastic lattice wave equation with nonlinear noise on $\mathbb{Z}$. We establish well-posedness in the energy space $\ell^{2}\times\ell^{2}$ and show that the solution generates a continuous non-autonomous dynamical system on $\mathcal{P}_{4}(\ell^{2}\times\ell^{2})$, which possesses a unique pullback measure attractor $\mathcal{A}$. Furthermore, we prove the convergence of the attractor $\mathcal{A}^{\varepsilon}$ as $\varepsilon\to 0^{+}$ to the attractor of the corresponding distribution-independent limit system.

Motivated by a question of Hochman, we investigate whether two deterministic time measures or time sets can be distinguished almost surely from a single planar Brownian image. Using rescaled correlation integrals, we give a criterion for mutual singularity of the laws of the image measures, even when the time measures have the same dimension. For self-similar time sets satisfying the open set condition, we also recover the natural occupation measure from the compact image. Together, these results allow us to distinguish lattice and non-lattice self-similar sets and their natural measures within this class. For a naturally parametrised family with two contractions, with fixed dimension, each parameter can be distinguished from all but finitely many others. If the dimension is sufficiently small, every distinct parameter can be distinguished.

We prove that for every choice of five positive masses the planar five-body problem has only finitely many central configurations up to rotation, translation and scaling. This answers Smale's sixth problem for five bodies. Albouy and Kaloshin proved the statement for all masses outside a closed algebraic subset of codimension two of the mass space. We keep their classification of singular sequences and replace the construction of the exceptional set by a direct exclusion: we use finite boundary charts for seven of the eight diagrams that can carry an unbounded four-product, and an exhaustive order census for Diagram 5. We derive the necessary mass conditions where needed and show that no germ with an unbounded four-product lies on a curve of normalized central configurations through a positive configuration. Five diagrams are excluded for all complex germs, two on curves through real configurations, and one on curves through positive configurations. Parts of the argument are computer-assisted: exact algebra, interval arithmetic and exact linear programming.