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Wednesday, 4 November 2026: 11 new papers, 10 cross-lists and 18 replacements in Dynamical Systems (math.DS), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
A naturalization, in the sense of Deprit, of Vinti's dynamical system makes the construction of a first order analytical satellite perturbation solution without critical divisors possible. The process gives hints on how to recast Brouwer's classical solution in a form that is free from small divisors too. The functionality of classical perturbation solutions is in this way extended to the close vicinity of the critical inclination. The new analytical solution bears terms that may grow unbounded, but their negative effects do not disclose in time intervals in which this kind of solution may be useful in practice.
We investigate connections between operator spectra, complex dynamics, and determinants for wreath product groups. For a nontrivial finite abelian group $G$, the natural action of $G\wr\mathbb Z$ on a rooted tree produces a family of operators with a recursive structure. Applying Schur complements to this family yields a rational map on complex projective space. We determine the projective spectrum--the parameter values for which the operator pencil fails to be invertible--as the union of the Julia set of this map and an explicitly described subset of its Fatou set. We also prove that the associated Koopman representation and the left regular representation are weakly equivalent and therefore give the same projective spectrum. We then compute Fuglede--Kadison determinants for a family of operators on more general wreath products. These formulas yield injective operators with zero determinant for wreath products $K\wr\mathbb Z$, where $K$ is a discrete group admitting a nontrivial projection $e\in\mathbb C[K]$. This answers negatively a question of Li and Thom about whether injectivity implies positive determinant for operators defined by matrices over complex group rings of amenable groups. For $G_{tf}\rtimes_\sigma\mathbb Z$, where $G_{tf}$ is any countable torsion-free abelian group and $\sigma\in\operatorname{Aut}(G_{tf})$, we prove that injectivity does imply positive determinant in every matrix size.
Our main objective in this paper is to prove a large deviation result for the distribution of preimages of a point under the iteration of a holomorphic correspondence with respect to the unique equilibrium state associated to a H\"{o}lder continuous potential defined on the support of the Dinh-Sibony measure of the correspondence. As a consequence, we obtain a convergence result which generalises the result on equidistribution of preimages by Dinh and Sibony. We also study a situation where the Dinh-Sibony measure happens to be the unique measure of maximal entropy for an appropriate holomorphic correspondence and also record some comments that connect this work with some recent works in the literature.
We prove that if the entropy of generating forbidden words of a subshift over a finite alphabet is strictly smaller than its topological entropy, then the subshift has only finitely many ergodic measures of maximal entropy, each of which is Bernoulli up to a finite period, and admits a characteristic measure of maximal entropy. In particular, these conclusions hold when the entropy of minimal forbidden words is strictly smaller than the topological entropy, giving a negative answer to a question of Buzzi.
We study the dependence of canonical invariant probability measures of suspension semiflows on simultaneous perturbations of the invariant base density and the roof function. We compare two natural representations of the suspension measure: physical-height coordinates, in which the support depends on the roof function, and normalised-height coordinates, in which all measures are defined on a fixed cylinder. In physical-height coordinates, we obtain an exact fibrewise formula for the $L^1$ discrepancy between suspension densities and derive quantitative total-variation estimates. In normalised-height coordinates, we establish corresponding stability estimates through the product of the base density and the roof function. These results transfer quantitative statistical-stability estimates for the base densities to the associated suspension measures. For one-parameter families whose base densities are differentiable in $L^1$ and whose roof functions are differentiable in $L^\infty$, we derive an $L^1$ linear-response formula on the fixed cylinder and a bounded-Lipschitz first-variation formula in physical height. The latter contains a signed measure supported on the graph of the unperturbed roof. We further compute the exact first-order total-variation speed and prove that total-variation differentiability holds if and only if the first-order roof velocity vanishes almost everywhere with respect to the unperturbed fibre density. The results are measure-theoretic and do not require spectral or mixing assumptions on the suspension semiflow.
Although non-dimensionalisation is a fundamental step in modelling, it is rarely obvious, especially for large dynamical systems, how to systematically construct optimal non-dimensionalisations. In this work, we develop a general mathematical framework for non-dimensionalising polynomial ODEs by interpreting them as chemical reaction networks (CRNs) and characterise the space of admissible scalings. Within this viewpoint, admissible scalings of time and species are encoded by Laurent monomials, and we derive exact formulae for the minimal number of multiplicative dimensionless parameters required by the reaction-channel coefficients and by the full initial value problem, together with characterisations of the scale assignments that attain structural lower bounds. These counts are expressed in terms of structural properties of the underlying CRN, and we show that the full initial value problem requires no more parameters than the corresponding Buckingham count. These results lead to constructive algorithms which we analyse rigorously and implement in the open-source Python package CRNNondim, available at https://github.com/georgio-hawi/crnnondim . Given a polynomial system or CRN, CRNNondim determines species and time scalings that bring variables as close to unity as possible while minimising the number of free parameters, and among the admissible scalings it chooses free parameters that result in the corresponding dimensionless system being sparse and simple. We apply CRNNondim to numerous well-known biological and biochemical models, yielding simpler dimensionless systems with fewer free parameters, and interpreting the parameters that govern their dynamics.
We describe an algorithm that, given a postcritically finite quadratic polynomial, outputs a finite state machine whose regular language is the set of external arguments for all points on the Hubbard tree, generalizing an example of Milnor. We do this by extracting orbit portraits from angled internal addresses, and constructing a suitable Markov partition of angle intervals. A main application is computing direct ray connections between Hubbard trees in matings of quadratic polynomials. We use this to understand ray connections in quadratic matings along veins of the Mandelbrot set, and prove that first ray connections between Hubbard trees are picked up at parabolic parameters or tips of veins.
In this paper we consider cycles of singular algebraic foliations on the real projective plane. Our first result shows that there exists a limit cycle of singular algebraic foliations such that the limit cycle cannot be realized in any affine chart, that is, it fails to appear in any affine plane via the usual dehomogenization. On the other hand, under an appropriate convexity assumption, we verify that a cycle becomes fully visible, lying entirely in some affine chart. We then analyze the accumulation behavior of limit cycles of singular algebraic foliations on the real projective plane. Last we prove that algebraic foliations of degree at most two on the real projective plane have only finitely many limit cycles, and the maximum number of limit cycle is zero, one and at least six for algebraic foliations of degree zero, one and two on the real projective plane, respectively. This result extends Bamon's finiteness theorem for quadratic affine vector fields in [Publ. Math. IH\'ES, (64)1986, pp. 111--142].
We investigate conditions and obstructions for the existence of ergodic full measures in the context of countable observables on compact spaces. Extending the theory in a unified way, we are able to show a complete picture via thermodynamic formalism for symbolic, uniformly hyperbolic and related systems. We prove conditional variational principles for the topological entropy, showing good approximations by ergodic full measures, and we give conditions where we precisely get ergodic full measures. In contrast with the case with a finite number of observables, new phenomena appear: we exhibit explicit examples of level sets with a null measure with respect to any invariant measure but having large entropy. On the other hand, based on induced ergodic systems in the sense of Kakutani-Rokhlin and heavily utilizing the formalism for countable Markov shifts, we go beyond the classical thermodynamic approach and show a different way of building ergodic full measures. Additionally, we study the regularity and shape of the entropy spectrum and, in the end, we also give some applications of the main conditional variational principles.
In this paper, we consider a planar system of nonlinear delay differential equations and prove the existence of the star-shaped periodic solutions conjectured by Hara (Nonlinear Analysis: Theory, Methods \& Applications, 30:4555--4563 (1997)) for seven parameter values. These solutions cannot be obtained by a direct application of the standard Fourier spectral method, since the corresponding linearized operator is degenerate. From numerical observations of the Fourier coefficients, we derive an ansatz that writes a star-shaped periodic solution as a periodic profile in a rotating frame. Restricting the profile to real Fourier coefficients, we show that the resonance between the delay and the period of the profile fixes the rotation frequency exactly. We then validate a zero of the resulting real system with the radii polynomial approach and interval arithmetic, with bounds that cover all Fourier modes. For each parameter value, we also determine the fundamental period and the winding number of the solution.
This paper is the second in a series about (relatively) free profinite semigroups and S-adic representations of minimal shifts. The first installment introduced the key notions of profinite images and saturating properties. The present paper examines the relationship between relative invertibility conditions (on a directive sequence) and charging properties (of a shift). The main result shows that the two properties are equivalent under mild conditions. Moreover, charging properties are shown to be equivalent to density of subgroups generated by return words in relatively free profinite groups. The main result can then be used to deduce purely discrete statements about subgroups generated by return words in relatively free groups. We also highlight applications of the main result to the study of maximal bifix decoding, following earlier work by the second author. Finally, we establish some results on the effect of direct images on charging properties and explain how to apply these results to the case of episturmian shifts.
Let $\lambda_L \approx 1.17628$ be Lehmer's number. We prove that no pseudo-Anosov homeomorphism of a closed orientable surface of genus six with orientable invariant foliations has stretch factor $\lambda_L$. This gives a negative answer to a question of Lanneau and Thiffeault on the minimal stretch factor in even genus. As a consequence, the minimal stretch factor $\delta_6^{+}$ of such maps is the largest root of $x^{12}-x^{9}-x^{7}+x^{6}-x^{5}-x^{3}+1$, approximately $1.20189$. The proof combines the Lefschetz method of Lanneau and Thiffeault, the standardly embedded train tracks of Hironaka and Tsang, an elementary analysis of the transition digraph, and an obstruction from periodic conjugacy classes in the fundamental group.
We study the concentration of logarithmically interacting defects, such as Ginzburg--Landau vortices and screw dislocations, on closed curves in two dimensions, subject to a confining field. We prove that a measure supported on a smooth closed curve solves the mean-field transport equation exactly when, in Lagrangian labels, the curve moves with the negative induced velocity and each label keeps its mass. This contour law is a gradient flow of the mean-field energy. We prove that the gradient of the energy on a curve has a leading term proportional to the operator $|D|$ with Fourier symbol $|k|$, which prevents every smooth equilibrium curve from being a local minimizer, in a gradient-flow analogue of the Kelvin--Helmholtz instability of vortex sheets. In a radially symmetric case with quadratic confinement, we find an explicit circular equilibrium whose linearization is diagonal in Fourier modes, and we show that, as $N\rightarrow \infty $, the linear spectrum of the regular $N$-gon converges, mode by mode, to that of the circle. Finally, we compute a critical point on two concentric circles with lower energy than the single circle, indicating a rich family of multi-circle equilibria.
An invariant random subgroup of a discrete group is a conjugation-invariant probability measure on the space of its subgroups. We confirm a conjecture of Simon Thomas and classify the invariant random subgroups of the finitary special and general linear groups over a finite field. These are shown to be stabilizers of random tuples of vectors and functionals. The classification uses the character theory of the finitary special linear group established by Skudlarek and the theory of the invariant random subgroups of the finitary permutation group established by Vershik. In addition, we classify the invariant probability measures for several natural actions of these finitary groups.
In this paper, we study Egorov's theorem for Laplacian eigenfunctions on compact hyperbolic surfaces in Zelditch's quantization. In this quantization, the remainder in Egorov's theorem for eigenfunctions is given explicitly by the second derivatives of the symbols along the weak stable foliation of the geodesic flow. Forward in time, we show that Egorov's theorem for eigenfunctions is valid to all logarithmic times, far beyond the Ehrenfest time T_E=|log h|/2, where h going to 0 is the semiclassical parameter. Backward in time, we introduce the Schrodinger group along the stable horocycles, which absorbs the exponentially growing part of the remainder. We show that Egorov's theorem for eigenfunctions is valid up to -T_E+T_h for any Th going to infinity with 0<=T_h<=T_E. Moreover, the limit of the remainder in Egorov's theorem at the Ehrenfest time -T_E is exactly the difference between the semiclassical measure and the Liouville measure, tested against the symbols propagated by the Schrodinger group. Consequently, Egorov's theorem for a sequence of eigenfunctions holds at -T_E (and in fact, at -T_E-T for any fixed T in R if and only if the sequence tends to be equidistributed in the phase space. Thus, the quantum unique ergodicity (QUE) conjecture is equivalent to this Egorov property holding for every eigenbasis. Furthermore, QUE holds with a polynomial rate if and only if Egorov's theorem holds at -T_E with a polynomial remainder. In this case, Egorov's theorem holds beyond -T_E with a polynomial remainder, and the stronger the rate is in QUE, the longer the time is in Egorov's theorem.
Let $W$ be a Weinstein manifold and let $f: W \to W$ be a compactly supported symplectic automorphism. Using Floer theories, one can define two different entropies of $f$, called "Floer-theoretic" entropy and "categorical" entropy. The current article compares these two entropies and show that the former bounds the latter from below. As an application of the comparison, using that these entropies are invariants up to the conjugate relation, we count conjugacy classes in symplectic mapping classes. Especially, we show an exponential abundance of conjugacy classes that are smoothly isotopic to the identity, when $W$ is a Weinstein manifold of dimension $4n \geq 6$, satisfying a local condition.
We establish an analogue of the Hilbert irreducibility theorem for all K3 surfaces over number fields that admit an automorphism of positive entropy. Our method draws ingredients from both algebraic geometry and complex dynamics.
For every \(c>1\) sufficiently close to \(1\), we prove that there is \(\kappa>0\) such that every sufficiently large \(A\subset[N]\) of density at least \((\log N)^{-\kappa}\) contains \[ x,\ x+(p-1),\ x+(p-1)^2,\ldots,\ x+(p-1)^k, \qquad p=\lfloor n^c\rfloor\in\mathbb P. \] We also prove pointwise almost-everywhere convergence of the corresponding multiple ergodic averages.
A two-species aggregation-diffusion system on the one-dimensional torus with antisymmetric cross-interactions modeling pursuit-evasion dynamics is studied. The self-interaction strength is used as a bifurcation parameter, and the linear stability of the homogeneous state is characterized by Fourier analysis. It is shown that non-reciprocal coupling enlarges the stability region. In a symmetric diffusion setting, exponential convergence to the homogeneous state is established for sufficiently weak self-interactions by means of an entropy argument, with a decay condition independent of the cross-interaction strength. Under suitable nondegeneracy and spectral assumptions, the first loss of stability is shown to occur through a Hopf bifurcation associated with a single spatial Fourier mode. A branch of time-periodic patterns is thereby obtained, along which the two species oscillate with a fixed phase difference. An explicit formula for the cubic Hopf coefficient is derived, and an essential dependence of the direction and stability of the bifurcating branch on the second spatial harmonic is identified. In the case of matching diffusion coefficients and interaction kernels, sufficiently strong non-reciprocity is shown to transform a subcritical stationary instability of the corresponding reciprocal system into a supercritical Hopf bifurcation with stable periodic solutions. An exchange-of-stability result is proved, and the theory is illustrated for Kuramoto, two-mode cosine, and top-hat interaction kernels.
We study structural defects in fixed points of block substitutions, building on arXiv:2604.12637 and providing its technical supplement. We develop an automata-based framework to characterize their occurrence and derive verifiable combinatorial criteria. We also introduce simplifications for the construction and analysis of these automata, including tensor-product decompositions, that clarify the mechanisms underlying structural defects and their emergence.
The basic reproductive number, $\mathcal{R}_0$, is widely used to determine whether an infectious disease can persist in a population. However, its interpretation may become unclear when the underlying disease-free demographic dynamics exhibit multiple types of stable attractors. In this study, we investigate how demographic bistability between fixed points and periodic cycles affects the calculation and interpretation of $\mathcal{R}_0$. We consider two discrete-time host-parasite models with infection: a tick-host model motivated by Borrelia burgdorferi transmission and a Nicholson-Bailey host-parasitoid model with intraspecific competition and infection. For each model, we calculate reproductive numbers associated with different disease-free attractors using the next-generation operator method and compare their invasion thresholds. In the tick-host model, the reproduction number calculated on the stable demographic cycle differs from that calculated at the positive coexistence fixed point, resulting in parameter regions where infection can invade at one demographic attractor but not the other. In contrast, in the host-parasitoid model, the host-driven invasion threshold is necessarily the same at the positive coexistence fixed point and on the host-only periodic cycles. These contrasting results demonstrate that demographic bistability alone does not necessarily produce attractor-dependent invasion thresholds. Rather, the effect of demographic dynamics on invasion depends on the structure of pathogen transmission. In particular, a complete host--vector--host transmission pathway can allow differences between demographic attractors to alter the invasion threshold. Our results emphasize the importance of considering both demographic dynamics and transmission structure when interpreting $\mathcal{R}_0$ in populations with multiple stable disease-free states.