Papers

In 1965, Chowla conjectured that the signs of the Liouville function become asymptotically uncorrelated at any fixed collection of distinct shifts. In 2015, Matomäki, Radziwiłł and Tao proved an averaged form of Chowla's conjecture. In 2022, Lichtman proved a variant of this conjecture over primes on average. In the same year, Lichtman and Teräväinen proved the Hardy–Littlewood–Chowla conjecture on average. In this article, motivated by the recent work of Bergelson and Richter on the dynamical generalizations of the prime number theorem, we will establish the dynamical generalizations of these three results related to Chowla's conjecture. In the proofs, we will show a uniform pretentious-distance estimate on the prime Omega function. Then exponential sum estimates are used to establish the averaged shift invariance of the distributions related to the shifted sum of the prime Omega function.

We introduce high-order Winfree models for oscillatory associative memory. We characterize memory performance through the local-stability dichotomy between stored and unstored patterns. Using this criterion, we show that crosstalk noise drives a cascade of ferromagnetic, mixed, and spin-glass transitions as the memory load increases. For both polynomial and exponential Winfree models, we derive the phase boundaries and develop auxiliary-field modulation schemes adapted to the different phases. We further present corresponding circuit designs and numerical simulations that support our theoretical predictions.

A chemical transition state is a phase-space bottleneck: trajectories approach along stable directions and leave along unstable directions. Lagrangian betweenness (LB), introduced in finite-time transport theory for fluid flows, combines backward and forward deformation in a way suggestive of this gather-and-disperse geometry. We ask what LB measures when the transition-state geometry is known. For a linear rank-one saddle, LB is spatially constant even though the stable and unstable manifolds are present. Relative nonlinear corrections remain small on shrinking initial neighborhoods, including observation times on the logarithmic local escape scale. At fixed spatial resolution, however, increasing observation time can concentrate the spatial variation of LB near a stable or unstable manifold. A separable quartic Hamiltonian makes these two limits explicit. A nonseparable Hamiltonian shows local flattening near a hyperbolic periodic orbit. We then use HCN/CNH isomerization, where dividing surfaces and gap times are established by phase-space transition-state theory. Before any exit, larger LB generally accompanies longer eventual gap times. The incoming reactive action provides a dynamical interpretation: initial conditions closer to the normally hyperbolic invariant manifold have longer local passages and greater accumulated stretching during passage. Among trajectories still inside at 0.5 ps that exit before 5 ps, early LB does not usefully rank exit times. LB therefore characterizes transition-state-organized transport in this example; a large value is neither an intrinsic manifold marker nor a general measure of molecular residence time.

Nested invariant cones (NICs) encode hierarchical order and oscillation structures in many finite and infinite dimensional dynamical systems. We investigate how this hierarchy constrains the Birkhoff center and the supports of invariant probability measures. For eventually compact, dissipative semiflows that are uniformly eventually strongly monotone with respect to NICs, we prove that every connected component $B$ of the Birkhoff center lies in a single cone layer: there exists $j>0$ such that $B$ is unordered with respect to all lower-level cones and strongly ordered with respect to all cones at level $j$ or higher. The same conclusion holds for every connected component of the support of an invariant probability measure. Under an additional transversality condition involving a codimension-$d$ linear subspace, each such component admits a homeomorphic embedding into $\mathbb R^d$. If $d=1$ or $2$ and the restriction of the semiflow to its global attractor extends to a flow, then the system has zero topological entropy, independently of the dimension of the original phase space. We apply the theory to bidirectional cyclic feedback systems and scalar parabolic equations on the circle. In the parabolic case, the natural infinite family of zero-number NICs fails to be uniformly eventually strongly monotone, even for the heat equation; we overcome this obstruction by constructing a finite family of perturbed NICs. In both applications, every connected component of the Birkhoff center admits a planar embedding, and the topological entropy is zero.

Inspecting or servicing a non-cooperative spacecraft is a two-player game: the target can thrust to defeat the inspector's plan. Hamilton-Jacobi-Isaacs reachability answers such games exactly but its cost grows exponentially with state dimension, while learning-based controllers scale yet certify nothing. Convex formulations are the usual escape, typically cast as convex-concave saddle-point problems. That framing fails in the terminal-distance orbital game: with effort regularization, the payoff is convex in both controls, so no pure open-loop saddle point need exist. In its place we derive two exact one-sided guarantees read from the players' terminal reachable sets via support functions: an escape certificate proving the target can hold a guaranteed standoff, and a security strategy bounding the miss distance the inspector can force. Together they bracket the engagement without assuming player rationality, and over two hundred perturbed trials the escape certificate separated capture from escape without error. A projected extragradient method supplies a strategy pair in about 25 ms, certified in place by best-response gaps. The construction carries unchanged to elliptical reference orbits via Yamanaka-Ankersen dynamics, where the orbital phase moves miss distance by nearly a factor of two, reproduced by nonlinear Keplerian propagation. Under receding-horizon play the inspector captures in half of ten representative engagements. Angles-only navigation error, keep-out zones, multiple pursuers, and closed-loop play are treated as extensions.

We study the problem of identifying nontrivial, i.e., nonzero, periodic fixed-points in discrete-time Lur'e feedback systems. Using the circulant matrix constructed from the transfer function of the linear subsystem, whether stable or unstable, we introduce an algebraic framework that allows us to determine when such fixed-points exist. This framework yields a sector bound defined by two vectors, whose slopes correspond to the maximum and minimum positive singular values of the circulant matrix. Assuming that the nonlinear feedback function is memoryless, we show that a necessary condition for the existence of nontrivial $P$-periodic fixed-points is that the intersection of the continuous completion of the nonlinear feedback function with that sector bound contains at least one point other than the origin. Our characterization provides a unified condition valid for all periods $P$, and further enables us to derive upper bounds on the amplitudes of admissible periodic fixed-points with bounded feedback functions. In particular, for relay feedback systems with passive feedback functions, we derive both upper and lower bounds for the amplitudes of such periodic fixed-points.

We show that every Anosov diffeomorphism is topologically expansive in the sense of [9]. We also prove that Lipschitz shadowing forces the stable and unstable bundles to be uniformly transverse, with a quantitative bound in terms of the shadowing constant. Finally, we give two examples on complete Riemannian manifolds: one is expansive but does not have the shadowing property, while the other has finite volume, bounded sectional curvature, Lipschitz shadowing, and orthogonal invariant bundles, but is not expansive with respect to its Riemannian distance.

The paper focuses on the following problem: given a generalized Bratteli diagram $B$, determine conditions under which the tail equivalence relation $\mathcal R$ does not admit a tail invariant probability measure on the path space $X_B$. We give several sufficient conditions, formulated in terms of the incidence matrices, that guarantee the nonexistence of such a measure. We also discuss a new spectral method for finding tail invariant measures for a class of stationary generalized Bratteli diagrams. This method allowed us to answer affirmatively an open problem formulated in arXiv:2212.13803.

Let $M$ be a complex surface endowed with a holomorphic projective connection and let $C \subset M$ be a compact smooth holomorphic curve. Given two singular holomorphic foliations $\mathcal{F}$ and $\mathcal{G}$ near $C$, with $C$ not $\mathcal{G}$-invariant, we attach to every point $p \in C$ an index $\mathrm{Ind}(\mathcal{F},\mathcal{G},C,p) \in \mathbb{C}$ of $\mathcal{F}$ relative to the reference $\mathcal{G}$, and we prove that these indices add up to $T\mathcal{G} \cdot C$. When $C$ is $\mathcal{F}$-invariant the index is the Camacho–Sad index corrected by the order of tangency of $\mathcal{G}$ with $C$, and the index formula reduces to the Camacho–Sad formula. The main applications are two contact formulas. At a tangency point $p$ of $\mathcal{G}$ with $C$, let $k(\mathcal{G},C,p)$ be the ratio of the curvatures at $p$ of the leaf of $\mathcal{G}$ and of the curve; it is a projective invariant, although each curvature separately depends on a choice of metric. If $\mathcal{G}$ has no singular points on $C$ and only simple tangencies with $C$, then $$ \sum_{p} \frac{1}{1 - k(\mathcal{G},C,p)} = \frac{2}{3}\left( C \cdot C + g - 1 \right), $$ where $g$ is the genus of $C$: the number of tangencies depends on $\mathcal{G}$, but this weighted count does not. For a pencil of lines in the projective plane it is the Plücker formula for the class of a plane curve. The second formula asserts that, for a curve in general position with respect to $\mathcal{F}$ and $\mathcal{G}$ which is not a geodesic, the total index of the tangencies of $\mathcal{F}$ with $\mathcal{G}$ along $C$ equals the number of tangencies of $\mathcal{G}$ with $C$ minus one third of the number of inflection points of $C$; in particular, it does not depend on $\mathcal{F}$.

We study the number of limit cycles of the piecewise smooth differential system \[ \dot{x}=y, \quad \dot{y}=-x-\varepsilon\bigl(f(x)y+\operatorname{sgn}(x)g(x,y)\bigr),\] where $f(x)$ is a polynomial of degree $n\geq 1$, $g(x,y)$ is a bivariate polynomial of degree $m\geq 2$, and $\varepsilon$ is a sufficiently small parameter. Using the first-order averaging theorem for discontinuous systems, we obtain that the number $H_{m,n}=\lfloor n/2\rfloor+\lfloor m/2\rfloor$ is a lower bound for the number of limit cycles bifurcating from the linear center $\dot{x}=y, \ \dot{y}=-x$. The bound is sharp, and we provide an explicit example that attains $H_{m,n}$ limit cycles.

In this paper we introduce relative mean sensitivity both in topological and measure-theoretical settings under group actions. It turns out that relative positive entropy is relative mean $n$-sensitive between minimal systems; relative positive $\mu$-entropy for an ergodic measure $\mu$ is relative mean $\mu$-$n$-sensitive.

We investigate solitary states in coupled oscillator networks, where a single oscillator detaches from a synchronized cluster and evolves at a distinct mean frequency. We focus on resonant solitary states, in which the mean frequency of the detached oscillator is shaped by resonance with a mode of the cluster. Here, we introduce a normal-form and symmetry-based approach to identify such states. We reduce the high-dimensional network dynamics to a two-dimensional second-order system that couples a rotator representing the solitary oscillator to a harmonic oscillator representing the dominant Laplacian mode of the cluster. We use normal-form transformations to introduce symmetry into this system. Using this symmetry, we reformulate the frequency selection problem as the search for relative equilibria of the normal form symmetry. We derive explicit algebraic conditions for the existence of these relative equilibria, and hence for resonant solitary states and their effective frequencies.

We examine a 4-dimensional generalization of predator-prey models describing interactions among four species. We show that these systems exhibit a rich variety of asymptotic behaviors, with solutions that are either integrable, leading to non-sustainable dynamics characterized by unbounded population growth or collapse, or chaotic, where all species coexist over time. Moreover, we review the integrable cases, which are classified into two-parametric families, and examine three-parametric families that behave similarly to the known two-parametric integrable cases. We also analyze chaotic Lotka-Volterra models in terms of their Lyapunov exponents and Poincare' surfaces of section, which turn out to reveal underlying structures that are compatible with what is expected from sustainable dynamics.

Integrable partial differential equations, such as the Korteweg–De Vries (KdV) equation, admit infinite hierarchies of commuting higher symmetries. Their symmetry reductions give rise to integrable finite-dimensional dynamical systems solvable in terms of Abelian functions. This underlies the finite-gap integration method for the KdV equation, introduced by S.P. Novikov and subsequently extended to many other integrable systems. In this paper, I introduce the concept of reduction ideals and a new class of reductions for integrable differential–difference equations, leading to integrable finite-dimensional systems in both commutative and noncommutative settings. This class includes periodic reductions, symmetry reductions, and a broad family of other reductions defined in terms of the associated Lax operator. The reduction constraints define integrable maps that enable solutions of the reduced systems to be extended to solutions of the corresponding differential-difference equations. The construction is illustrated using the Volterra hierarchy.

The basic reproduction number $R_0$ is the central dimensionless quantity in mathematical epidemiology, characterizing the threshold for disease outbreak and the early growth rate of an epidemic. The algebraic form of $R_0$ varies widely across models of distinct transmission mechanisms, and its interpretation can yield further biological insight into transmission dynamics and disease intervention. We present a structural taxonomy of $R_0$ for compartmental ordinary differential equation models spanning a range of disease transmission features, including staged progression, differential infectivity, competing strains, vector-borne transmission, sexually transmitted infection, recurrent infection, maternal transmission, and gene drive inheritance. For each structural class, we provide a worked example, deriving $R_0$ via the next-generation matrix approach and interpreting the resulting algebraic expression in terms of the underlying transmission pathway. By unifying disparate derivations under a single structural framework, we clarify when and why particular algebraic forms of $R_0$ arise. This review serves both as a research synthesis and as a self-contained pedagogical reference for students and researchers entering the field.

We establish the first pointwise convergence result for ergodic averages with iterates along explicit and deterministic sequences of intermediate growth, that is, growing faster than any polynomial but slower than any exponential. In particular, we show that the sequence $(\lfloor \exp((\log n)^c)\rfloor)_{n\in\mathbb{Z}+}$, with $c\in(1,8/7)$, is universally $L^p$-good for every $p\in(1,\infty]$. This gives an affirmative answer to an open problem dating back to the mid 1980s and contributes to Bellow's program, initiated in the earlier part of the same decade, on the characterization of $L^p$-good sequences in pointwise ergodic theorems. The proof combines the so-called one-frequency circle method with a delicate application of Vinogradov's method for estimating exponential sums whose phases involve $\big(\lfloor \exp((\log n)^c)\rfloor\big)_{n\in\mathbb{Z}_+}$. An interesting feature of our analysis, reminiscent of estimates arising in the study of the zero-free region of the Riemann zeta function, is that the argument relies on the classical Vinogradov method, in the sense that it necessitates estimates on the number of solutions for the Vinogradov system of Diophantine equations with explicit dependence on the system's parameters.

We study skew product measure-preserving systems driven by an irrational rotation and the step function with values $1$ and $-1$ on each half of the circle. These systems can be seen as a deterministic version of Kalikow's $[T,T^{-1}]$ system, and are particular instances of Rokhlin cocycle extensions. Under the assumption of ergodicity, we show that these systems obey an interesting rigidity property for isomorphisms. That is, we show that two skew products are isomorphic if and only if the rotations in the base are isomorphic, and the fiber transformations are flip isomorphic. Under mild extra assumptions, we are able to characterize all isomorphisms. We prove that by choosing the rotation angle suitably, these systems can realize arbitrarily low measure-theoretic complexity in the sense of lower slow entropy. We apply this result to obtain new examples of systems that fail the variational principle for slow entropy. These examples can have a set of invariant measures as rich as desired (any metrizable Choquet simplex, up to affine homeomorphism).

We prove the dynamical uniform boundedness conjecture for unicritical polynomials $z^d + c$ over every number field $K$ when $d \geq 4$, and over every number field $K$ not containing $\mathbb{Q}(\sqrt{-3})$ when $d = 3$. Furthermore, our unconditional bounds on perodic points are effectively computable. In the remaining cases, we show that the dynamical uniform boundedness conjecture is implied by the boundedness of Mordell-Weil ranks over $K$ for a specific family of elliptic curves when $d = 3$ and for Jacobians of dynatomic curves when $d = 2$.

In this paper, we study a stochastic Susceptible-Infected-Removed (SIR) model where the infection and the recovery rates depend on individual covariates for susceptibility and infectiousness of the infector and the infectee. Such models allow explicit nonlinearity in the incidence term. They are also important from a practical perspective, as they allow for the incorporation of individual heterogeneity into the epidemic process. Statistical estimates for crucial epidemiological parameters, such as the basic reproduction number, herd immunity threshold, could be vastly different, and even biased, when the population heterogeneity is ignored in the mathematical model. We describe our epidemic model as an Interacting Particle System (IPS) of Stochastic Differential Equations (SDEs) driven by Poisson Random Measures. Our main mathematical contributions are a Functional Law of Large Numbers (FLLN), which approximates the empirical random measure of the IPS by means of a deterministic measure-valued function, and the propagation of chaos phenomenon, which establishes asymptotic independence of the particles as the population size goes to infinity with an explicit construction of McKean–Vlasov type Kac's “nonlinear process”. We also briefly mention how the propagation of chaos phenomenon leads to a product-form likelihood function, which forms the basis of the so-called Dynamic Survival Analysis (DSA) method for parameter inference based on sparse data.

The maintenance of continuous variable information in working memory is thought to rely on persistent patterns of cortical activity. In delayed-estimation tasks, neural activity can form localized activity peaks, or “bumps,” whose positions track the remembered variable. Such activity is well described by continuous-attractor neural field models, but most existing models collapse cortical inhibition into a single homogeneous population. Here, we introduce a stochastic neural field model with distinct excitatory, parvalbumin-expressing (PV), and somatostatin-expressing (SST) populations to examine how inhibitory subtype structure shapes persistent activity. Using a Heaviside firing-rate approximation, we derive stationary bump solutions and reduce their linear stability to separate shifting and scaling modes. We show that population thresholds and inhibitory timescales determine both bump stability and the mechanism by which stability is lost, while inhibitory connection strengths and spatial scales substantially reshape the stable parameter region. In particular, broader SST connectivity promotes stable bump states. Finally, we derive an effective diffusion coefficient for noise-driven bump wandering and show that increasing the SST spatial footprint reduces the rate of memory diffusion. Together, these results demonstrate how inhibitory subtype structure can shape both the deterministic stability and stochastic precision of continuous-attractor memories.

This paper proves the integrability of a nonholonomic billiard defined by a knife-edge in a disk. The paper begins by parametrizing the impact space using coordinates that reveal the system's rotational symmetry. It then constructs a map on the space of post-impact states that encodes the billiard dynamics through a recurrence relating each post-impact state to the next. In addition, the paper shows that the knife-edge billiard admits a family of generalized caustics.

Let $n\ge 2$, and let $M$ be either the $n$-dimensional sphere $\mathbb{S}^n$ or the real projective space $\mathbb{RP}^n$. Given $\alpha\ge 1$, we construct a finitely generated group $G_{n,\alpha}$ that embeds into the group of $C^\alpha$-diffeomorphisms of $M$, but does not embed into the group of $C^\beta$-diffeomorphisms of $M$ for any $\beta>\alpha$.

We introduce a notion of typical horseshoe, consisting of a pair of rectangles admitting Markov returns at times of positive lower density, with controlled hyperbolic geometry and orbits that shadow typical trajectories. We prove the abundance of typical horseshoes for the random standard map and establish several statistical properties of the system, including exponential mixing and large deviation estimates for the associated projective and two-point processes.

We use the Labourie–Wentworth's formula and the thermodynamic formalism to show that, along the slice constructed by Bronstein–Davalo, the logarithmic top-eigenvalue length spectrum has a uniformly positive second variation near the Barbot representation. As a major application, we show that every closed surface group admits a non-Hitchin Borel Anosov representation into $\mathrm{SL}_{3k}\mathbb{R}$ for every $k\geqslant 1$. In particular, we obtain the first such examples in the even dimensions $6k$. We also study the local behavior of related objects of this slice around the Barbot representation, including the Lyapunov exponent of the flat bundle, the Hausdorff dimension of the limit set, and the Hilbert entropy of the representation.

In 2015, Liu, Qu (Commun. Math. Phys., Vol. 338) show that the Hausdorff dimension of the spectrum of the Thue-Morse Hamiltonian admits a positive lower bound independent of the coupling constant lambda: dim_H sigma(H_{tm,lambda}) ge (log 2) / (140log 2.1). In this paper, we prove that the spectrum of the Thue–Morse Hamiltonian has full Hausdorff dimension for any lambda in mathbb{R}: dim_H sigma(H_{tm,lambda})=1.

A weighted dynamical version $\mathbf{Z}_f$ of the classical Ihara zeta function is presented on connected finite regular graphs, expressed in terms of weighted periodic orbit data. We establish its meromorphic continuation, with poles given by the resonances of the associated non-backtracking transfer operator acting on a suitable Banach space, and compute its residues. At simple spectral parameters, the residue of $\mathbf{Z}_f$ is identified with the invariant Ruelle distribution on the graph phase space. Combining this result with a relation obtained by Arends-Palmirotta, we further derive Patterson-Sullivan and Wigner residue formulae. This provides a finite-graph analogue of residue formulae for weighted dynamical zeta functions for geodesic flow on rank one locally symmetric spaces, in the spirit of Schütte-Barkhofen-Weich.

We prove several necessary and sufficient conditions under which a point satisfies the Poincaré Recurrence Theorem for all computable (ergodic) measure-preserving transformations and all sets of a particular complexity. The necessary conditions are obtained by constructing specific measure-preserving transformations which violate recurrence. While some of these conditions pertain to standard notions of algorithmic randomness, others involve new notions of genericity developed in the context of a computable probability space, which we call $\Pi^0_n$-genericity and quasi-$\Pi^0_n$-genericity. We also provide conditions under which points recur at a positive frequency in sets containing them, regarding both simple and multiple recurrence.

In this paper, we study the ergodicity of geodesic flows on closed rank-one manifolds of nonpositive sectional curvature. We introduce the infinite-order vanishing set of the Gaussian curvature in dimension two and of the fiberwise second moment of the reduced Jacobi determinant in higher dimensions. We bound the Liouville measure of the singular set in terms of the volume of this subset and, for surfaces, obtain a corresponding Hausdorff dimension bound. In particular, if this subset has zero volume, then the singular set has zero Liouville measure, and the geodesic flow is ergodic. We also give a geometric criterion for finitely many exceptional regions whose fundamental groups have virtually Abelian images of rank smaller than the dimension of the manifold. Under strict negativity of sectional curvature outside these regions, we obtain a Hausdorff dimension bound for the singular set and prove that the geodesic flow is ergodic with respect to Liouville measure. By extending this method to real-analytic metrics, we characterize the singular set as the zero set of a nontrivial real-analytic determinant constructed from curvature operators, thereby establishing ergodicity in this setting without further assumptions.

Reservoir computing (RC) trains only a linear readout over a fixed recurrent layer, making it fast and data-efficient for online prediction. However, a static reservoir degrades under system drift, readout-only adaptation is then insufficient, and unconstrained reservoir adaptation can destroy the echo-state and incremental stability properties that make RC reliable. This paper proposes LoRA-RC, which adapts the recurrent matrix through a low-rank correction driven by streaming prediction errors. The base reservoir and adaptation bases are fixed offline; a small core matrix is adapted online, projected onto a spectral-norm ball, and low-pass filtered at each step. The projection guarantees that every applied recurrent matrix remains within a certified contraction set, and an incremental input-to-state stability bound is established for the reservoir along each online adaptation path, with path-independent rate and gain. On a Lorenz system with an abrupt parameter drift, LoRA-RC cuts post-drift prediction error by 56% versus a fixed RC and 51% versus readout-only adaptation; ablations over 20 seeds show that removing the projection inflates this error by more than a factor of 40.

Quasispecies theory describes how mutation and selection shape highly variable RNA virus populations, but most models explicitly consider neither RNA-dependent RNA polymerases nor the delays associated with their synthesis and functional activation. A recent minimal model showed that delayed polymerase availability can induce extinction by reorganizing basins of attraction, a mechanism termed {\tau}-tipping. Here, we extend this framework to a delay differential equation model in which master and mutant genomes encode distinct functional polymerases. We characterize its equilibrium and bifurcation structure, identifying master-mutant coexistence, mutant-only persistence (error catastrophe), and complete extinction governed by transcritical and saddle-node bifurcations. Although delays leave the equilibria unchanged, they reorganize their basins of attraction and can redirect populations that would otherwise persist at fixed mutation and replication parameters toward extinction, thereby extending {\tau}-tipping to systems with autonomous mutant replication. We also recover the previously studied complementation model as a limiting case in which defective genomes depend on master-derived polymerase, providing a comprehensive analysis of the bifurcation structure. Comparing the two models shows that the ability of mutant genomes to encode functional polymerases determines whether loss of the master sequence results in mutant replacement or complete population extinction. The occurrence of delay-induced extinction in both settings demonstrates the robustness of {\tau}-tipping and connects intracellular replication kinetics, complementation, and quasispecies extinction. Finally, we translate this mechanism into concrete virological predictions and propose experimental strategies to test the effects of replicase timing, RNA degradation, and functional complementation on viral persistence.