Papers

We perform a linearized stability analysis of the Vlasov–Fokker–Planck equation obtained as the mean-field description of a stochastic spontaneous aggregation model with memory. Memory is represented by a chain of $K$ internal variables. We characterize the spatially homogeneous equilibria and derive a scalar dispersion relation for spatially inhomogeneous perturbations. When all relaxation rates are equal, we show that every Fourier mode that is unstable in the memoryless model possesses a unique critical relaxation parameter: sufficiently long memory stabilizes the mode, whereas it remains unstable for short memory. Although memory may destabilize individual modes associated with negative Fourier coefficients of the sensing kernel, we prove that, for radially symmetric distance-decreasing kernels, it cannot destabilize a homogeneous equilibrium that is stable in the memoryless model. Thus, memory cannot create an overall instability of an otherwise stable homogeneous state, although it may change the set of the unstable modes. We present a numerical example for the normalized top-hat kernel, demonstrating that increasing the effective memory length successively stabilizes the Fourier modes, with higher frequencies being stabilized before lower ones. This is consistent with the coarsening effect observed in recent particle simulations. Finally, under suitable nondegeneracy and regularity assumptions, we use the Crandall–Rabinowitz theorem to obtain branches of spatially inhomogeneous stationary solutions bifurcating from the homogeneous state.

The Dynamic Mode Decomposition (DMD) has been consolidated as a basic tool for data-driven analysis of dynamical systems, allowing simultaneous identification of coherent structures and their dynamics from time-resolved measurements. However, with a linear regression at its core, DMD is unable to produce accurate models from recordings of dynamics that are inherently nonlinear, such as the response to large perturbations and the evolution on chaotic attractors. Recent approaches attempt to simultaneously fit the linear and nonlinear contributions to the dynamics by performing a regression onto a physically motivated model structure. However, although the resulting nonlinear models can produce accurate short-term predictions, their linearization does not necessarily agree with that of the original system. In this work, we introduce a novel data-driven method — nonlinearity-subtracted DMD (NSDMD) — that focuses on producing an accurate linearization of a system when the nonlinear contribution to its dynamics are available while the linear part is not. This scenario is encountered, for example, when the nonlinear terms in the governing equations are known, while the linear operator contains uncertain material properties or it accounts for the closure of unresolved dynamics. This also arises when the data is generated by a black-box simulation code that is able to output the nonlinearity, but not the action of the linear operator on the snapshots. NSDMD leverages data snapshots of the nonlinearity to explicitly account for the purely nonlinear contributions to the dynamics and formulate a regression problem that finds a low-rank approximation of the underlying linear operator. We demonstrate the approach on several numerical examples, showcasing its improved capabilities for data-driven linear analysis of chaotic, partially observed, advection-dominated, and high-dimensional dynamics.

Input-to-state stability (ISS) provides a useful tool for analyzing the robustness of time-varying chemical reaction networks (CRNs). This paper investigates the ISS property for CRNs in two cases: one is for weakly reversible CRNs with nonzero deficiency and multiple linkage classes, which is beyond the limit of existing results requesting a weakly reversible CRN to be zero-deficiency and single-linkage-class; the other is for non-weakly reversible CRNs by leveraging the notion of linear conjugacy. These results, on the one hand enrich the studies of the ISS property of broader classes of CRNs, on the other hand serve for designing CRN-based molecular computing systems. The latter application suggests the ISS theory can justify the parallel mechanism of biomolecular computations. We use some CRNs of practical relevance to demonstrate our results, including the p53 signaling and dimerization network, the p21-activated kinase 1 network, etc.

Neural ordinary differential equations (NODE) and hybrid NODE models provide flexible continuous-time representations of complex dynamic systems, but their expressive capacity can exceed the information content of the available data. Consequently, these models may reproduce observed trajectories while retaining weakly identifiable parameters, poorly constrained neural components, and unreliable extrapolation. Here we introduce identifiability-aware neural ordinary differential equations (iNODE), a framework that incorporates practical identifiability into neural differential equation design. iNODE models embed neural components as explicit analytic functions within the governing equations, enabling direct sensitivity analysis, Fisher-information-based confidence intervals, and identifiability-aware architecture selection. Candidate architectures are generated under data-support constraints, jointly calibrated, and ranked according to predictive accuracy, parsimony, and parameter identifiability. We evaluate the complete iNODE workflow against conventional NODE and hybrid NODE workflows representative of current practice using four controlled ground-truth benchmarks spanning fully data-driven and hybrid formulations, latent time-varying parameters, partial observability, and sparse or noisy measurements. Using the same training data and evaluation scenarios, the iNODE workflow selected more compact architectures, reduced parameter uncertainty, and improved extrapolation and recovery from latent-dynamics. These results establish practical identifiability as a model-design principle for parsimonious and reliable neural differential equations.

We provide a rigorous mathematical proof of the Bosch-Simó conjecture ( M. Bosch, C. Simó (1993), Physica D, 217–229) regarding the abundance of strange attractors in the unfolding of a Shilnikov-Hopf bifurcation. By constructing a general class of dissipative return maps that capture the essential global geometry of this codimension-two scenario, we demonstrate that "large" strange attractors – in the sense of Broer-Simó-Tatjer – are a persistent and statistically robust feature of the dynamics. We prove that, for a parameter set of positive Lebesgue measure, the system admits attractors supporting a unique Sinai-Ruelle-Bowen (SRB) measure. The proof relies on a dimensional reduction to a rank-one model, enabled by strong transverse contraction and singular angular expansion. Furthermore, we show that this chaotic regime is interspersed with sequences of parameters yielding superstable periodic orbits, providing a characterisation of the transition between ordered and chaotic motion in the singular limit of the bifurcation.

We prove dynamical comparison for Deaconu–Renault groupoids associated to minimal surjective non-injective local homeomorphisms of compact metrizable spaces with finite Lebesgue covering dimension. As a corollary, the associated $C^*$-algebras are UCT Kirchberg algebras, recovering results by Carlsen–Thomsen via dynamical methods. In the zero-dimensional case, our result also verifies Matui's AH-conjecture for these groupoids using a recent breakthrough of Xin Li. Our proof combines techniques from both the purely infinite and stably finite regimes: We construct partial actions of non-abelian free groups as suitable “large subgroupoids” and establish comparison properties for these using the paradoxical towers technique developed by Gardella–Geffen–Kranz–Naryshkin. The boundary of the subgroupoid is controlled by a groupoid version of the topological small boundary property which we deduce from finite covering dimension of the unit space. As a byproduct, we prove the classical small boundary property for minimal actions of countable discrete groups on finite-dimensional compact metrizable spaces without any freeness assumption.

Highly pathogenic avian influenza (HPAI) outbreaks require rapid control during active transmission and evidence-based decisions on the safe restocking of depopulated farms. We developed a stochastic spatial SEIR-based metapopulation model for a synthetic HPAI outbreak on the fictional Jolly Island. Farms were classified as `Broiler-2', `organic duck', or `Other' production systems. The model incorporated local, environmental, movement-mediated, and distance-dependent transmission, together with reactive and preventive culling, production-specific confinement, and capacity-based restocking. The simulated epidemic was geographically concentrated and differed substantially among production classes. Preventive culling reduced mean cumulative burden from 16,362.7 to 13,631.9 infectious-farm-days, with an overall reduction of 16.7%. Earlier confinement substantially reduced epidemic magnitude, while stronger environmental transmission increased the epidemic peak. Restocking risk declined as the epidemic approached resolution. Under the model assumptions, 24 May 2026 was the first candidate date satisfying the predefined rebound-probability threshold of 0.20. For restocking on 15 March 2026, none of the tested restocking fractions met this criterion. Capacity-based restocking reduced cumulative burden by 8.45% and rebound probability from 0.780 to 0.533, compared with restocking relative to the baseline population. These findings demonstrate the value of integrating epidemic control and post-outbreak recovery within a single modelling framework. Timely confinement, targeted preventive culling, and phased capacity-based restocking may reduce both epidemic burden and resurgence risk, although operational decisions should also incorporate surveillance, biosecurity, economic considerations, and regulatory requirements.

We study the relative periodic orbits in the Gutzwiller-type anisotropic Kepler problem, which is a generalized model derived from the classical Gutzwiller anisotropic Kepler problem. By reducing the rotational symmetry of the $z$-axis, we obtain a reduced system with two degrees of freedom and its energy surface forms a compact and regular three-sphere in a certain parameter range. Combining the estimation of the Seifert rotation number of the planar Kepler orbit and the CHHL formula introduced in \cite{CHHL23}, we prove that this system admits infinitely many periodic orbits on every compact regular energy surface. This model can be applied to the $(1+2n)$-body problem to find infinitely many relative periodic orbits with hip-hop symmetry as well as relative periodic orbits in the $n$-pyramidal problem.

Fractional partial differential equations describe nonlocal dynamics, but discovering them from noisy data is difficult because fractional differentiation amplifies high-frequency measurement noise and the derivative orders are unknown. We propose Weak-Pareto, which combines an adjoint-consistent weak formulation of fractional terms with Pareto-based subset selection over discrete term types and continuous fractional orders. For linear right-hand-side terms, the adjoint transfers fractional operators from measured fields to smooth test functions, replacing noise-sensitive pointwise differentiation with smoothing integration; for nonlinear terms, the noise-suppression effect is partial yet useful. Coefficients are fitted by ridge regression within a branch-aware differential-evolution search over the orders. The support size is then selected at the validation-error-complexity elbow. We show that the variance of fixed linear right-hand-side weak features vanishes under grid refinement, whereas noise amplification in pointwise fractional features increases with derivative order. Across fractional advection-diffusion, reaction-diffusion, and Burgers benchmarks, Weak-Pareto recovers parsimonious structures from clean and noisy measurements. In controlled advection-diffusion and Burgers comparisons, it retains the correct support at every tested multiplicative-noise level, whereas the unregularised strong-form counterpart largely fails once noise is introduced; this advantage persists under additive Gaussian noise. Ablations show that the weak library drives noise robustness and that continuous-order Pareto search avoids the support-selection failure of a dense fixed dictionary. On the advection-diffusion benchmark, Weak-Pareto yields more consistent operator recovery and substantially lower measured runtime than a contemporary neural baseline.

We prove a linear operator of a finite-dimensional Banach space has the topological shadowing property, as introduced in \cite{lny}, if and only if its spectrum lies either within the open unit complex disk or outside the closure of that disk. Moreover, the spectrum of every uniformly expansive linear operator with the topological shadowing property lies either within the open unit complex disk or outside the closure of that disk. Lastly, we prove that a normal operator with the topological shadowing property on a Hilbert space does not have any nonzero nonwandering points.

Different long-time patterns can prevail in different regions of a reaction–diffusion system's parameter space. We study the transition curves between such regimes for a one-dimensional Brusselator in the two-parameter plane $(\sigma,b)$, focusing on wave/stripe-like, spiral/source-defect-like, and target-like states. We develop a feature-based continuation framework built on time-dependent PDE simulations. Scalar observables extracted from late-time solution data distinguish the regimes and define threshold level sets where the feature crossings are regular. A secant predictor and a local one-dimensional sweep corrector are used to trace these level sets. For the spiral transition, we introduce a branch-adapted spacetime symmetry-defect feature that separates asymmetric source-like patterns from more symmetric wave patterns. For the target transition, we use the minimum of a core spatial-variance score and a tail temporal-variance score to detect the characteristic structure of half-target states. The method recovers robust side portions of both transition curves. In lower parameter regions, where mixed and irregular patterns make a single scalar feature less specific, we instead report transition estimates obtained from vertical parameter sweeps and direct inspection of spacetime plots. These results show how simulation-based continuation and direct pattern classification can be combined to map regime boundaries while preserving the different levels of numerical evidence.

In this paper, we develop the thermodynamic formalism for intermittent full-branch interval maps with finitely many neutral fixed points and for a broad class of possibly discontinuous potentials. Under a hyperbolicity assumption on the potential, we prove the existence of a unique equilibrium state with positive entropy and exponential decay of correlations. Furthermore, we characterize the phase transitions for the one-parameter family $\beta\varphi$, with $\beta>0$, and prove that, at the critical parameter, multiple ergodic equilibrium states may coexist, at most one of which has positive entropy. Our approach relies on a detailed analysis of conformal measures.

A similarity pair is the dynamical system in $\mathbb{C}$ generated by two maps $f:z \to sz-1$ and $g:z \to sz+1$ for $|s|<1$. Associated to the dynamical system is an attractor $\Lambda$. The Barnsley–Harrington Mandelbrot set $\mathcal{M}$ is the set of $s\in \mathbb{D}$ for which $\Lambda$ is connected. Let $K$ denote the filled set of a connected $\Lambda$. For $s\in \partial \mathcal{M}$ we show that the (partially defined) action of the semigroup on $\partial K$ is topologically conjugate to a (discontinuous) piecewise linear action of constant slope. Conditional on a conjecture (satisfied for `most' $s\in \partial \mathcal{M}$) we give a necessary and sufficient condition in terms of the dynamics on $\partial K$ for $K$ to contain cut points, and we describe the set of all such cut points in terms of infinite walks in a directed graph $\textrm{IG}$ obtained by an explicit recursive algorithm. The structure of the `dynamical cut point set' for a 2-dimensional family of piecewise linear actions (containing those coming from $s\in \partial \mathcal{M}$) recovers and generalizes the Douady–Hubbard–Thurston quadratic minor lamination for the abstract Mandelbrot set.

Parameter estimation is a fundamental challenge in the calibration of ordinary differential equation (ODE) models, where repeated numerical integration can lead to high computational cost. In this work, we investigate whether quantum algorithms can be leveraged to assist parameter estimation in nonlinear dynamical systems. We develop a hybrid classical-quantum framework that reformulates a data-assimilation-augmented parameter estimation problem as a combinatorial optimization task. Model dynamics and data assimilation are enforced entirely on the classical side, while the resulting parameter estimation cost functional is discretized and approximated by a quadratic unconstrained binary optimization (QUBO) surrogate. This surrogate is mapped to an Ising Hamiltonian, and quantum optimizers are used to search for low-energy configurations corresponding to candidate parameter estimates. We apply the framework to SIS and SIR epidemic models, the chaotic Lorenz-63 system, and a high-dimensional two-layer Lorenz-96 system. In this setting, the method is used to recover classical system parameters from partial state observations across steady-state, chaotic, and high-dimensional multiscale dynamical systems. Numerical experiments with synthetic data show that the proposed approach accurately recovers parameters while requiring data-assimilation solves only on a prescribed coarse grid. The framework avoids quantum state tomography, illustrating a viable pathway for integrating quantum optimization into data-driven parameter estimation for nonlinear dynamical systems.

We prove that in a semi-dispersing Minkowski billiard, that is a billiard in the complement of a collection of convex sets in a normed space with a smooth strictly convex norm, any billiard trajectory of finite length has only finitely many reflections off the walls.

The dissipative dynamics of a pair of wedge disclinations with opposite Frank angles confined to a circular domain is investigated under the assumption of radially symmetric motion. Two distinct dynamical regimes are identified: a diverging-disclination regime and an annihilating-dipole regime. The stationary points associated with both regimes are determined and fully characterized in terms of their stability, and quantitative estimates for the characteristic evolution times in their vicinity are derived. In the annihilating-dipole regime, it is shown that, after a suitable time rescaling, the resulting law of motion for the dipole coincides with that of an edge dislocation.

We prove the following “Torus Trichotomy Theorem”: every finitely generated subgroup of $\pls$ is either virtually free, a central extension of a cocompact Fuchsian group by a finite cyclic group, or a one-ended, non-acylindrically-hyperbolic group containing $\mathbb{Z}^2$. In particular, we obtain the “free or surface” alternative for word-hyperbolic groups of PL homeomorphisms of the circle.

We study two shadowing properties for flows that differ in the allowed reparametrizations of time: oriented shadowing permits arbitrary increasing reparametrizations, whereas standard shadowing requires their distortion to be uniformly close to one. We prove that these notions are distinct already for $C^\infty$ flows on every closed oriented surface. Moreover, such examples are $C^0$-dense among $C^1$ flows with a singularity and consequently, on closed oriented surfaces with non-zero Euler characteristic, they are dense among all $C^1$ flows. We then relate local standard shadowing to recurrence and entropy. A non-trivial chain-transitive set with local standard shadowing forces positive topological entropy unless it is an irreducible almost heteroclinic set. Consequently, for a zero-entropy flow with standard shadowing, every non-trivial chain-recurrent class has this form, and every non-singular one is minimal. For surface flows, we further prove that oriented shadowing together with finitely many singularities forces every chain-recurrent class to be minimal.

We prove that a Hausdorff topological group is tame if and only if it is precompact.

We consider Delone point patterns with self-similarity. Under mild conditions, the similarity factor is a Pisot number if and only if the pattern is uniformly discrete. The classical case is a Meyer set $\Lambda$ with $\Lambda\supset \theta\Lambda$ for some $\theta>1,$ for which $\theta$ must be a Pisot number or a Salem number. When $\Lambda$ contains several similar copies of itself, the case of a Salem number drops out for $\theta<2.$ On the other hand, strictly self-similar patterns with a Pisot factor must be Meyer sets. Various examples are given.

The pentagram map sends a polygon to the intersections of consecutive short diagonals. We study its shape dynamics after translating and positively rescaling each iterate to restore unsigned area and barycenter. For polygons whose dynamics is generated, after passing to a finite iterate and cyclic relabelling, by one projectivity, we prove a spectral dichotomy. A dominant real projective line yields flattening and unbounded diameter, whereas a dominant real eigenvalue with a subdominant non-real pair produces asymptotic motion on concentric homothetic ellipses. We apply this framework to pentagons and hexagons via Glick's operator and to Poncelet polygons via the Darboux–Schwartz projectivity. We obtain spectral diagrams and a Jacobi-function formula for the Poncelet return spectrum. Consequently, we recover Schwartz's long-and-thin result for strictly convex non-projectively-regular pentagons, prove a line-or-ellipse dichotomy for convex hexagons under explicit nondegeneracy assumptions, and establish flattening for strictly convex non-projectively-regular Poncelet polygons.

We study quantitative edit-distance asymptotics for symbolic codings of irrational rotations $x \mapsto x+\alpha$ on $\mathbb{T}$ in terms of the irrationality exponent $\mu(\alpha)$. For the binary coding determined by an interval $[0,\beta)$, let $d_N(x,y)$ be the edit distance between the codings of the orbits of the points $x$ and $y$ up to time $N-1$. For every $\alpha \notin \mathbb{Q}$, almost every $\beta \in (0,1)$, and almost every $x,y \in \mathbb{T}$, we determine the edit-distance exponents $\displaystyle (*) \; \; \limsup_{N\to\infty}\frac{\log d_N(x,y)}{\log N} = \frac{\rho(\alpha)}{\rho(\alpha)+1} \; \;$ and $\; \; \displaystyle\liminf_{N\to\infty}\frac{\log d_N(x,y)}{\log N} = \min\Bigl\{\frac{1}{2},\frac{1}{\rho(\alpha)}\Bigr\}$, where $ \rho(\alpha):=\mu(\alpha) - 1$. We find a surprising transition at the golden mean $\varphi= \frac{1+\sqrt{5}}{2}$: If $ \rho(\alpha) \le \varphi$, then both asymptotics in $(*)$ hold for all $\beta$. However, for every finite $\rho>\varphi$, there is an uncountable set of $\alpha$ with $\rho(\alpha)=\rho$ for which both edit-distance exponents are strictly smaller than the corresponding values in $(*)$ for uncountably many $\beta$. We also derive consequences for aperiodic circle homeomorphisms and Sturmian sequences. For rotations of $\mathbb{T}^d$ coded by boxes, we prove that for almost every rotation vector, the common edit-distance exponent is $d/(d+1)$.

The Shnirelman-Zelditch-Colin de Verdière theorem and its weak mixing extension relate quantum ergodicity and quantum mixing with ergodicity and weak mixing of the geodesic flow. Integrable systems, such as the flat torus, do not satisfy either in general. Restricting to position-dependent observables, Marklof and Rudnick established equidistribution for almost all eigenfunctions of rational polygons. In this note, we extend this result to off-diagonal elements for a subset of rational polygons not including the torus using weak mixing of the directional billiard flow for almost all directions. Additionally, we provide a different proof that establishes the same result for $2$-tori.

Given a non-atomic standard probability space $(\Omega,\mu)$, we exhibit examples of non-amenable closed topological subgroups of $\mathrm{Aut}(\Omega,\mu)$ whose natural near-action on $(\Omega,\mu)$ is whirly. This answers a 2010 question by Pestov in the negative. The argument proceeds via constructing Gaussian near-actions from topologically faithful unitary representations of non-amenable whirly Polish groups.

In this work, we establish the quantitative unique continuation on both $n$-dimensional simplex lattice $\Delta^{(n)}_{N}$ and $\mathbb{Z}^n$ for arbitrary $n\geq 3$.

The Koopman-von Neumann formulation of classical statistical dynamics maps the isometric evolution of probability densities in $L^1$ under the Liouville equation to a unitary evolution of quantum mechanical wavefunctions in $L^2$ generated by the antisymmetric part of the Liouville operator. This approach enables the use of Hilbert space techniques to model the statistical evolution of observables. However, being an $L^2$ method, it is not suitable for representing pointwise evolution along dynamical trajectories. We propose a Koopman-von Neumann framework that replaces the $L^p$ spaces associated with a volume measure on the state space manifold, $X$, by a reproducing kernel Hilbert space (RKHS), $\mathcal H$, of continuous functions on $X$ that satisfy joint analyticity conditions with respect to the generator (vector field), $V$, of the dynamics and its RKHS adjoint. Our scheme employs a dilation of the antisymmetric part of $V$ to an essentially skew-adjoint operator, $L$, on the tensor product Hilbert space $\mathfrak H = \ell^2(\mathbb N) \otimes \mathcal H$, followed by a dilation of $L$ to an essentially skew-adjoint, free derivation, $\mathcal D$, acting on a weighted symmetric Fock space $\mathfrak F$ generated by $\mathfrak H$. We show that the unitary evolution generated by $\mathcal D$ consistently recovers the (generally, non-unitary) Koopman evolution of observables in a dense subalgebra of $\mathcal H$ generated by jointly analytic vectors, over a time interval that is uniformly bounded away from zero. We then build a family of weak spectral triples $(\mathcal A_n, \mathfrak F, -i \mathcal D)$, $n \in \mathbb N$, wherein $\mathcal A_n$ are non-abelian $*$-subalgebras of $B(\mathfrak F)$ generated by creation and annihilation operators and $-i \mathcal D$ plays the role of a Dirac operator inducing extended pseudometrics on the state spaces of $\mathcal A_n$.

Machine Learning and more specifically Deep Learning involves solving large scale nonconvex optimization problems. Several algorithms have been proposed in the literature, that seem to achieve satisfactory practical efficiency for difficult instances, the Stochastic Gradient Method being the most rudimentary, while still outperforming more recent algorithms at a number of learning tasks. A major open question about the current methods used in deep learning is to understand their convergence properties. Following a line of previous works about the long time behavior of gradient-type algorithms, %and in particular the recent contributions from Azizian et al., we present a new approach for studying the asymptotic properties of a wide family of methods from an ergodic theoretical viewpoint. Our main results include a study of the expected time for a stochastic optimisation algorithm to reach a certain small neighborhood of a minimizer and show that this reaching time distributes exponentially around its average, which is given by the inverse of the stationary measure of the target. The assumptions on the Stochastic Gradient noise include the Gaussian and the Sub-Exponential assumptions.

We study the smooth topological equivalence between a nonautonomous linear system on the positive half-line and a quasilinear perturbation whose nonlinear part is not assumed to be globally bounded with respect to the state variable. The linear equation is assumed to admit a $(\mu,\nu)$–dichotomy, and the construction is carried out through the Green operator associated with this dichotomy. The usual global boundedness of the perturbation is replaced by locally uniform Green-integrability conditions along the relevant linear and nonlinear solutions. Under a smallness condition involving the global Lipschitz constant of the perturbation and the Green operator, we first construct Palmer-type maps which give a continuous topological equivalence on $\mathbb R^+$. We then impose Green-integrability conditions on the successive variational equations and a further first-order smallness condition which guarantees the invertibility of the derivative of the inverse map. Under these assumptions, the equivalence is of class $C^r$. We also provide a class of examples for which the perturbation is unbounded with respect to the space variable and all the hypotheses can be verified directly.

We study the nonlinear stability of the elliptic equilibrium points $L_3$, $L_5$, and $L_6$ in the planar equilateral restricted four-body problem with unequal primary masses. We provide a systematic classification of nonlinear stability over the full two-parameter mass plane by combining high-order Birkhoff normal forms, numerical continuation, and singular reduction of resonant normal forms. In nonresonant regions, Arnold's theorem is applied when the quartic nondegeneracy condition holds, while sixth-order normalization resolves the degenerate cases. The $2$:$-1$ and $3$:$-1$ resonances are analyzed using the theorems of Alfriend, Markeev, and Meyer, including a classification of the $3$:$-1$ resonant dynamics. Singular reduction provides the corresponding reduced orbit spaces and reveals saddle-center bifurcations associated with the appearance and disappearance of periodic-orbit families. These results extend previous stability analyses beyond symmetric mass configurations and provide a geometric description of the resonant dynamics in the unequal-mass equilateral restricted four-body problem.

Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicities. We study the set $Acc^\Omega(x)$ of weak-$^*$ limits of the sequence $\frac{1}{N}\sum_{n\leq N}\delta_{T^{\Omega(n)}x}$ in $\sigma$-compact dynamical systems $ (X,T)$, demonstrating that if $x \in X$ is quasi-generic for an ergodic measure $\mu$, then $\mu \in Acc^\Omega(x)$. This extends a result of Bergelson and Richter, who studied the problem in the setting of uniquely ergodic systems. We give a more precise description of the set $Acc^\Omega(x)$ in the case of the horocycle flow on non-compact quotients of $SL(2,\mathbb{R})$. We show that for every non-periodic $x\in X$, in addition to Haar measure, there exists sequences $(s_n), (c_n) \subseteq \mathbb{R}$ such that $$ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{r^2}{2}}\nu^{i}_{s_n-2\log|1+c_nr|} dr\in Acc^{\Omega}(x), $$ where $\{ \nu^{i}_{s} \}_{i \leq k}$ denotes the one parameter family of periodic measures in each of the $k$ inequivalent cusps. Depending on Diophantine properties of the non-periodic point $x$, we show that $Acc^\Omega(x)$ contains a full two parameter family of such periodic measures, as well as the Dirac measure at each cusp. In particular, these results yield almost-everywhere divergence of pointwise averages along $\Omega(n)$ for the non-compact horocycle flow.