Papers

Let $\Gamma<\mathsf{SL}(d,\mathbb R)$ be a Zariski-dense Borel-Anosov subgroup and let $\varphi$ be positive on its limit cone. For every $\chi \in \mathrm{Hom}(\Gamma, \mathbb R)$, we construct twisted $(\varphi,\chi)$-conformal densities and prove their uniqueness, atomlessness, and ergodicity. For every normal subgroup $\Gamma_0\lhd\Gamma$ with nilpotent quotient, we classify the ergodic $\varphi$-conformal densities of $\Gamma_0$ in terms of characters of $\Gamma/\Gamma_0$.

In this paper we consider area-preserving twist maps $f:\mathbf{T}^2\to\mathbf{T}^2$ with vertical rotation sets reduced to a single irrational number $\alpha$. Our first result states that under these hypotheses, there exists a unique $f$-invariant foliation $\mathcal{F}_G=\{\operatorname{Graph}(\phi_t)\}_{t\in\mathbf{T}^1}$, where each function $\phi_t:\mathbf{T}^1\to\mathbf{T}^1$ is $K$-Lipschitz for some constant $K=K(f)$. Moreover, we also prove that if $\mathcal{F}_G$ is a Lipschitz foliation, then $f$ is minimal.

We prove a sharp effective equidistribution theorem for closed geodesics on a compact hyperbolic surface. This is achieved by carefully analyzing the Selberg trace formula twisted by a Maass eigenfunction, which we refer to as Zelditch's trace formula, following Zelditch's observation that the geometric side can be expressed as a sum of period integrals of eigenfunctions on closed geodesics. The main technical input is a sharp upper bound for the matrix coefficients $\langle \phi, \varphi^2 \rangle$ uniformly in $t_\phi\to \infty$ and $t_\varphi \to \infty$, where $t_\phi$ and $t_\varphi$ are the eigenparameters corresponding to Maass forms $\phi$ and $\varphi$ on $X$.

Public health measures at mass gatherings work only if people follow them, and adherence depends both on how instructions are communicated and what others do. We link these behavioural factors to pathogen transmission in a local-scale, agent-based exposure model combining pedestrian dynamics with airborne transmission. Rather than fitting a parametric behavioural model, agents draw their adherence at run time from survey respondents (2,170 attendees of UK sports and music events) who reported the same context: we vary stewards' communication effectiveness and adherence of role models, as well as agents' shared social identity with each, assuming that strong shared social identity increases their influence. In a ticket-checkpoint queue of 101 agents, effective communication combined with strong shared social identity with stewards reduces the number of highly exposed agents by 82% for mask wearing. In contrast, strong shared social identity with non-adhering role models gives almost nine times as many highly exposed agents as with adhering role models. Physical distancing was not monotonically beneficial, because adherence changes how agents move: an infectious agent that kept distance moved along the edge of the queue, leading to similar results across conditions. Our pipeline transfers to any survey that includes behavioural measures and contextual variables.

We prove that every bounded-to-one action of a finitely generated commutative monoid has asymptotic dimension at most the torsion-free rank of its group completion, with control uniform for a fixed monoid, generating set, and generator fiber bound. The estimate is sharp. Together with the equality of classical and Borel asymptotic dimensions for these actions, it gives the sharp Borel rank bound and recovers hyperfiniteness for bounded-to-one Borel actions of countable commutative monoids.

We develop a contraction-based perspective on impact-time guidance that augments a baseline homing command with a timing bias. The proposed perspective treats the prescribed schedule as a moving time-to-go isochron and regulates motion normal to that set through velocity-normal lateral acceleration while the interceptor's speed remains constant. We derive a transport equation that characterizes homing-compatible time-to-go coordinates and define a predictor defect that quantifies the mismatch of approximate maps. We show that the scalar timing channel induces a coordinate-invariant rank-one metric on the normal quotient. To account for bounded lateral acceleration, we formulate a robust scalar filter and derive a necessary and sufficient condition for pointwise feasibility. We then show that terminal calibration and funnel invariance establish first interception at the prescribed time under the stated assumptions. We also develop a preterminal alignment and homing handover that avoids singular inversion as lateral timing authority vanishes near collision-course alignment. The proposed perspective accommodates analytic, numerical, and learned time-to-go maps that satisfy the required calibration and regularity conditions.

We give a complete classification of algebraically primitive Teichmüller curves in $\Omega\mathcal M_g(2g-2)^{\mathrm{hyp}}$ for $g\ge5$. They are precisely the curves generated by the double regular \((2g+1)\)-gon when \(2g+1\) is prime, and by the regular \(4g\)-gon when \(g\) is a power of two.

This paper investigates the ambiguity of noisy short-arc angles-only orbit determination for a Molniya-orbit case study. Conventional Gauss Initial Orbit Determination followed by Gaussian Least-Squares Differential Correction is first shown to be unreliable under high-noise short-arc conditions, frequently failing to converge or converging to hyperbolic local solutions. To explore the candidate solution space, a Lambert-based initialization procedure is used over a grid of assumed ranges and angular-observation pairs, and the resulting states are refined using nonlinear least-squares optimization. The unconstrained solution set reveals multiple orbit families, including reentry elliptic, bounded elliptic, xGEO elliptic, and hyperbolic trajectories, all of which can reproduce the observed angular arc with comparable residuals. The solutions exhibit a structured range-velocity relationship, indicating that the angular measurements primarily constrain apparent line-of-sight motion rather than absolute range. Finally, physically motivated constraints are introduced to isolate Molniya-like bounded elliptic solutions. The results demonstrate that short-arc angles-only orbit determination is fundamentally non-unique without a priori information and that constraints should be interpreted as orbit-family selection tools rather than proof of uniqueness.

We classify compact Kähler surfaces with semipositive anticanonical bundle. In particular, we resolve a question of Demailly–Peternell–Schneider: any blowup of $\mathbb{P}^2$ at 9 points on a smooth cubic has semipositive anticanonical bundle.

We study a passive scalar on the two-dimensional torus driven by an arbitrary finite set of white-in-time divergence-free Fourier modes. We prove that the exponential $L^2$ dissipation rate of the advection-diffusion equation is uniformly bounded as the diffusivity tends to zero as long as there exist two nonparallel forced modes. If not, then the dissipation rate can be arbitrarily fast depending on the initial data, but the top Lyapunov exponent remains uniformly bounded below. The proof uses a projection onto a finite set of modes which depends on the forcing set. Part of our argument follows the structure of the four-mode case investigated by Chemnitz and Chemnitz, but uses a strictly weaker condition than their matrix inequality between Frobenius and operator norms. Combined with an accessibility argument, this allows us to consider an arbitrary set of forcing modes and initial data.

We study the topological and geometrical aspects of non-determinism in group actions, generalizing the notion of non-expansive direction for abelian actions. Under suitable conditions on the acting group, we establish restrictions on the collection of non-deterministic horoballs (or half-spaces) for any action. We illustrate our results in the case of free and nilpotent groups, expanding upon known cases for abelian groups. We then apply these findings to standing problems concerning minimal self-joinings in topological dynamics, which are of independent interest.

The theory of the basic reproduction ratio $R_0$ is established for the almost periodic non-densely defined Cauchy problems. Firstly, we present some dynamical properties of the linear almost periodic non-densely defined Cauchy problems. Then we define the evolution semigroups on the space of almost periodic functions, investigate their infinitesimal generators and define the basic reproduction ratio $R_0$. Finally, our developed theory is applied to age-structured models and functional differential systems.

The five-compartment tumour-immune model introduced by Kuznetsov et al. captures critical clinical phenomena, including tumour dormancy, immune evasion (`sneaking through'), and immunostimulation. Recent studies by Osojnik et al. highlighted the model's excitable and oscillatory dynamics, providing a mechanism for tumour relapse. In this paper, we recast these findings through the lens of geometric singular perturbation theory (GSPT) and the blow-up method, establishing a formal mathematical foundation for these phenomena. Because the model features a multiple-time-scale structure with dynamics separated by up to three orders of magnitude, standard Tikhonov-Fenichel theory is insufficient. To resolve this, we employ the parametrisation method to systematically compute higher-order approximations of the slow vector fields. We introduce a methodological novelty by demonstrating that the parametrisation method is indispensable within the geometric blow-up analysis. Specifically, we show that higher-order correction terms to the local centre manifolds are required to match the vector field on the corresponding slow manifold. Using these combined tools, we rigorously prove the existence of stable relaxation oscillations and transient large excursions. We uncover a rich collection of singular bifurcations, including a saddle-node on invariant circle and two distinct singular Andronov-Hopf bifurcations that drive complete and incomplete canard explosions. At the highly degenerate origin, our blow-up analysis reveals the spatial collision of a singular transcritical bifurcation and a nilpotent pseudo-singularity. We conjecture that this collision forms a novel singular transcritical Bogdanov-Takens organising centre. Ultimately, these geometric findings yield a precise bifurcation diagram in the parameter space of effector cell supply and death rates.

We give an explicit geometric and algebraic description of the index map for AF-by-discrete groupoids. The index records transport between infinite tiles and abelianized returns in the groups of germs, subject to a defect in the dimension group of the associated Bratteli diagram. We determine the kernel of the index map and construct transposition factorizations of zero-index elements with trivial germs at singleton orbits. In particular, in every minimal case, this kernel equals the subgroup generated by dynamical transpositions. We also characterize approximation by the fixed AF core and prove almost finiteness for minimal systems that are compactly generated. We make higher homology explicit in terms of the groups of germs. We express the secondary parity differential by lifted surface relations and by an extension class constructed from balanced boundary data. Under minimality and comparison, lifting finite relations gives a split AH sequence. Explicit boundary and matrix calculations, together with a nonminimal example with nonzero parity differential, illustrate the theory.

We establish a criterion of arithmetic $k$-polynomial spectral localization and $k$-polynomial dynamical localization in expectation for quasi-periodic long-range operators on $\ell^2(\mathbb{Z}^{d})$ with power-law hopping based on the quantitative $C^{k}$-reducibility of the dual Schrödinger cocycle. As the application, we prove both localization properties for power-law long-range perturbations of the Almost Mathieu Operators with sufficiently large couplings and Diophantine frequencies.

We develop a nonlinear thermodynamic formalism for correspondences on compact metric spaces. A continuous energy is evaluated on empirical measures of successive pairs of states. The measure-theoretic pressure of a stationary transition pair is the sum of its kernel entropy and the energy of the joint distribution of two successive states. For forward expansive correspondences satisfying an ergodic-abundance condition, we prove a variational principle and the existence of nonlinear equilibrium pairs. We further prove that limits of the two-coordinate projections of asymptotically maximizing Gibbs ensembles are averages of the two-coordinate distributions of equilibrium pairs. When the energy depends on finitely many potentials on the space of admissible pairs, Legendre regularity assumptions allow us to identify nonlinear equilibrium pairs among linear equilibrium pairs through a self-consistency equation and a global maximization condition. Under these assumptions, real analyticity in one dimension and uniqueness of linear equilibrium pairs imply finiteness of the nonlinear equilibrium set. Finite-state examples exhibit continuous and first-order transitions, metastability, and phases that have identical one-coordinate marginals but distinct two-coordinate distributions. A criterion based on Taylor expansions determines the associated critical exponents

The propagation failure phenomenon in bistable lattice equations connects spatial discreteness and weak diffusion to the large number of stationary patterns and the inability of waves to move. We show that there naturally exist bistable lattice equations that do not admit propagation failure for intermediate values of diffusion parameters. The collapse of propagation failure is connected with the existence of a continuum of pinned waves and a continuum of bounded nonmonotone stationary solutions. It also enables us to find explicit forms of traveling waves of families of bistable lattice equations.

We study a geometric singular perturbation theory (GSPT) approach to optimal control problems involving entry-exit phenomena in planar fast-slow dynamical systems. In contrast to previous work, we assume that the control acts only on the slow dynamics and we consider minimum-time optimal control problems (i.e., we aim at minimizing the time to reach a certain target). We apply this method to a simplified fast-slow toy model, characterized by a repetition of entry-exit processes, enabling a transparent analytical exploration of the control dynamics and allowing us to derive the key results explicitly. First, we analyze each non-autonomous entry-exit phenomenon separately, and we derive the optimal way to control them. Second, by artificially generating a stable limit cycle, we determine under which conditions it is possible to reach the target. Finally, under such conditions, we combine all entry-exit processes together and we demonstrate the existence of an optimal control. Our approach mainly exploits techniques of GSPT and dynamic programming. Indeed, we show that our problem can be formulated in a way resembling the Bellman equation, even though not in its standard setting. Numerical simulations, built on such Bellman-like formulation, illustrate our analytical results. In particular, the optimal control is derived with an algorithm that converges in a finite number of steps. Lastly, we extend this novel approach to more complex planar fast-slow dynamical systems.

We study the problem of designing optimal damping for a system of $n$ coupled linear oscillators. Using the Weyl–Horn theorem, we construct an explicit damping matrix that realizes the optimal asymptotic decay rate of the system. We prove that this decay rate is sharp and is determined by the geometric mean of the natural frequencies. The resulting damping strategy is shown to outperform the classical Rayleigh (proportional) damping approach. We conclude by characterizing the parameter regimes in which the optimal damping matrix necessarily ceases to be positive definite and discuss the implications of this phenomenon.

An unusual object in the sky raises two related questions: what kind of object it is and what physical process drives its motion. Departures from a gravity-only trajectory can provide evidence, but the same apparent acceleration may be generated by distinct dynamical mechanisms. We formulate this as an identification problem and establish conditions for identifiable uniqueness among three reference classes: inert dynamics, impulsive forcing, and active stabilization such as a controlled spacecraft. The distinction rests on how each class responds to disturbances. Although different mechanisms can produce similar average trajectories, the specified reference models imply different stochastic responses around those trajectories. We show when these differences uniquely determine the dynamical class from the observed trajectory and when several classes remain observationally compatible.

For the unicritical family $f_{c,j}(z)=z^j+c$, the Multibrot sets $M_j$ converge geometrically to the closed unit disk as $j\to\infty$, but this limit does not quantify their finite-degree area behavior. We estimate Area($M_j$) using escape-time pixel counting while separating the effects of the iteration threshold $K$ and spatial sampling density $\rho$. The finite-iteration sets form a nested decreasing sequence, while grid refinement is treated empirically. Across $K\in\{25,100,500,1000\}$ and $\rho\in\{128,256,512,1024\}$, the final grid refinement changes the reported areas by at most $1.1\times10^{-3}$. At $K=1000$ and $\rho=1024$, the estimates are well described by $C(j)\approx\pi-2.49216j^{-0.57103}$. Independently, Laurent coefficients of the exterior conformal map and Gronwall's area formula yield finite-truncation upper bounds $U_{1000}(j)\approx\pi-2.07405j^{-0.643427}$. Across all tested degrees, the grid estimates remain below the corresponding conformal bounds, and both sequences exhibit similar power-law decay of the area deficit from $\pi$.

We compare the Feng–Huang and covering definitions of weighted topological entropy for equivariant continuous maps between dynamical systems. The two quantities agree on the whole space by their variational principles, but need not agree on non-invariant subsets. We prove that the Feng–Huang entropy is bounded above by the covering entropy on every subset, without assuming invariance. The proof does not rely on measure theory. Our main result provides a new perspective on the two relative weighted variational principles.

We show that any stable hypersurface of a symplectic $4$-manifold, on which the cohomology class of the symplectic form restricts to a multiple of a rational class, can be $C^\infty$-approximated by (possibly unstable) hypersurfaces whose closed characteristics equidistribute. The cohomological condition is necessary due to a famous example of Herman. The proof combines KAM theory with recent quantitative closing lemmas for Reeb flows and area-preserving maps. As a further application, we prove that every geodesible volume-preserving vector field on a closed three-manifold can be $C^\infty$-approximated by volume-preserving vector fields with equidistributed periodic orbits.

We study ergodic averages over Gaussian integers in shrinking angular sectors, which impose an increasingly restrictive directional relation between the real and imaginary coordinates. In uniquely ergodic systems, averages of continuous functions along $\Omega$ converge to the invariant integral uniformly in the base point and sector location for sectors whose widths shrink subpolynomially in the norm cutoff. For finitely generated multiplicative actions, we prove a quantitative sector–disk comparison in an explicit logarithmic shrinking range, with a power-saving error in $\log N$, uniformly in the base point and sector location. The exponents depend only on the number of distinct prime transformations, and the result requires neither ergodicity nor angular-distribution hypotheses on the sets of primes inducing the individual transformations. Strong unique ergodicity then yields convergence to the invariant integral at each fixed base point. Consequences include asymptotic independence of relative angular position and $T^{\Omega(n)} x$, Gaussian Liouville cancellation, and joint equidistribution modulo integers of prime-factor counts associated with two prime classes having divergent reciprocal-norm sums.

Previous studies of network dynamics have suggested that heterogeneity among nodes inhibits stability, at odds with the ubiquity of inherently heterogeneous natural and engineered systems. Here, we show that this conclusion arises from model reductions introduced for mathematical tractability and breaks down when nodal dynamics are higher-dimensional, yielding non-Hermitian Jacobians. In such systems, including neural, power-grid, and material networks, nodal heterogeneity can instead enhance stability, even when parameters are randomly disordered. Non-Hermiticity also underlies the stabilizing effects of network heterogeneity, which can arise even in one-dimensional nodal dynamics through nonreciprocal interactions, as shown for ecological networks. Our framework reveals disorder not as a liability but as a general resource for stabilizing complex systems.

We introduce causal graph rewriting, a model of computation in which local rules are applied on directed acyclic graphs in an asynchronous manner. The non-determinism arising from asynchrony is disciplined by the oriented edges, which must be understood as both computational dependencies and locality constraints—and are themselves subject to the rewriting. We illustrate the model through two examples: a particle system, and a time-dilation example—reminiscent of general relativity. We study the well-definedness and properties of induced subgraphs and graph composition, which isolate and recombine the region affected by a rewrite. We then study locality with respect to these constructions, showing how a local rewrite preserves positions, borders, and context. Our main result concerns sequential composition: locality extends from single rule applications to arbitrary valid sequences, as any local rule is automatically $*$-local and $*$-extensive. We also formalise and prove the simulation of any one-dimensional cellular automaton.

We study the squared Rayleigh-residual functional on complex projective space and its negative gradient flow. For an arbitrary complex matrix, every positive-residual critical point has a strictly descending tangent direction; consequently, the local minima are precisely the eigenlines. For normal matrices, we describe the critical set in terms of circles through spectral points and identify the amplitude dynamics with a replicator equation whose payoff matrix has rank at most four. This gives explicit logarithmic first integrals and a reduced representation of interior trajectories. We compute transverse escape rates near critical components and analyze how polynomial reweighting changes unstable coordinates. Finally, we distinguish local saddle trapping from concentration of Haar-distributed initial states and derive the density of the filtered ensemble. Numerical examples with random spectra and a Jordan block illustrate convergence, nonmonotone Rayleigh motion, and the improvement of finite-time spectral exploration by low-degree random polynomial filters.

We consider balanced doubly intermittent full-branch interval maps with two neutral fixed points. Although the ordinary empirical measures fail to converge for Lebesgue-almost every initial point, we show that this non-statistical behavior persists under a broad class of deterministic observation schemes: if the sampling sequence is block-dominating, then both endpoint Dirac masses are accumulation points of the sparse empirical measures for Lebesgue-almost every orbit. This class includes sampling sequences of positive lower density, arithmetic and polynomial sequences, and finite-index regularly varying sequences. We also prove that every map in the balanced class has the strong natural-measure property, including at the boundary exponent $\beta=1$: the pushforwards of every absolutely continuous initial probability converge to a distinguished endpoint mixture $\nu_*$. Abstract consequences of this one-time limit include annealed convergence along every deterministic sampling sequence, the existence of arbitrarily sparse sequences with almost-sure convergence, and an almost-sure convergence criterion based on a power-saving variance bound. For the symmetric quadratic map $g_1\in\mathfrak F_*$, we verify the dyadic covariance condition on a countable uniformly dense endpoint-flat family: the double covariance sum is $O_\phi(n\log(n+2))$. Consequently its dyadic empirical measures converge for Lebesgue-almost every initial point to $\tfrac12(\delta_{-1}+\delta_1)$.

We prove that any two rank-one systems are sub-$L^1$ orbit equivalent, fully clarifying quantitative orbit equivalence for a generic class of transformations. As a corollary, we establish unconditional optimality of Belinskaya's theorem.

We develop and analyze angular maps as a numerical tool for visualizing and detecting rotational dynamics in nonlinear, nonautonomous dynamical systems in discrete and continuous time. An angular map assigns to each initial point the angular spectrum of the variational equation along the corresponding trajectory. This spectrum describes the long-time average rotation of subspaces transported by the linearized dynamics, measured by maximal principal angles. Building on the theory of angular spectra, we develop efficient numerical algorithms based on forward and backward subspace iteration. We prove that these algorithms asymptotically provide angular spectral values for subspaces that are dominant in either forward or backward time. Applications to Hénon maps, a planar flow, and the Lorenz system illustrate how angular maps reveal rotational features across phase space. For continuous-time systems, we prove that suitably rescaled angular spectra of exact time-step maps converge to the continuous angular spectrum in the Hausdorff metric as the step size tends to zero. For autonomous systems, we also justify a simplified algorithm that uses successive trajectory points to approximate the angular range associated with the flow direction.