Papers

We prove that if the first Lyapunov exponent of a compactly supported random matrix product is simple, then it is pointwise Hölder continuous for the Wasserstein–Hausdorff distance. By passing to exterior powers, we obtain pointwise Hölder continuity of the first $k$ exponents whenever they are simple.

We study products of independent, identically distributed random matrices with an exponential moment and a simple top Lyapunov exponent, without assuming irreducibility. Outside an exceptional invariant subspace called the equator, we establish contraction on average for a weighted projective distance and a spectral gap for the associated Markov operator. We obtain a central limit theorem, a Berry–Esseen bound, deviation estimates near the mean, and an almost sure invariance principle both for the growth of a fixed vector outside the equator and for the operator norm. For fixed vectors, a non-arithmeticity assumption also yields a local limit theorem and a renewal theorem. Our contraction criterion applies more generally to mostly contracting random maps on locally compact spaces with a Lyapunov function, including certain random interval maps.

We study the decision problem of global asymptotic stability (GAS) of the origin for homogeneous polynomial vector fields with rational coefficients. In the first part of the paper we give, for every fixed dimension $n$ and odd degree $2k+1$, a computable map which sends a homogeneous vector field of degree $2k+1$ on $\mathbb{R}^n$ to a homogeneous cubic vector field on $\mathbb{R}^m$, $m=\binom{n+k-1}{k}$, and preserves both GAS and Lyapunov stability. The map passes to new variables $y_\alpha=x^\alpha$ (one for each monomial $x^\alpha$ of degree $k$), in which the vector field becomes cubic, and adds a cubic penalty term transverse to the set of points of the form $(x^\alpha)_{|\alpha|=k}$; the main point is that this controls every initial state of the new system. As a consequence, the GAS problems for all degrees, for each fixed odd degree $d\ge3$, and for degree three are pairwise many-one equivalent. In the second part we prove that there are a fixed dimension $N_*$ and a fixed odd degree $d_*$ such that the set of homogeneous vector fields on $\mathbb{R}^{N_*}$ of degree $d_*$ whose origin is GAS is many-one complete. The reduction encodes an undecidable family of polynomial equations in integer unknowns into the constant term of a product of Jacobi theta series, realizes these series as outputs of a polynomial differential system with rational coefficients, and turns the existence of an integer solution into the sign of a maximal invariant average, which homogeneous GAS detects. Combining the two parts shows that GAS of rational homogeneous vector fields on $\mathbb{R}^N$ of degree $d$ is many-one complete for every odd $d\ge3$ and every sufficiently large $N$; in particular it is undecidable for cubic vector fields in a fixed dimension. Building on recent work, some analogous results are proven for Lyapunov stability of homogeneous vector fields.

Koopman operator theory provides a linear framework for analyzing nonlinear dynamical systems and has become a major tool for data-driven modeling. A central challenge, however, is that finite-dimensional approximations computed by methods such as extended dynamic mode decomposition (EDMD) require the dictionary to be specified a priori. Recent machine-learning approaches address this limitation by learning the dictionary from data, predominantly using artificial neural network (ANN) autoencoder architectures. Although kernel methods offer an alternative with greater interpretability and tractability for theoretical analysis, they have received little attention in this setting. We introduce extended dynamic mode decomposition with kernel-based dictionary learning (EDMD-kDL), a kernel-based method for learning finite-dimensional Koopman embeddings directly from data. The method combines ideas from collocation methods and bilevel optimization to simultaneously learn a kernel dictionary and the corresponding Koopman approximation. We evaluate EDMD-kDL against state-of-the-art ANN-based approaches on a range of numerical experiments, including global sea-surface-temperature forecasting and learning directly from video data. Across all tested settings, EDMD-kDL achieves performance comparable to or better than the ANN-based methods. Moreover, in contrast to standard kernel methods, the proposed approach is scalable to large datasets by design since the size of the required kernel matrices depends on the number of collocation points rather than the size of the training dataset.

Let $M$ be a closed manifold of arbitrary dimension. The length spectrum of a negatively curved metric $g$ on $M$ is the set of lengths of all possible closed geodesics of $g$. We prove that metrics whose length spectrum is exponentially sparse, i.e., the gaps between distinct lengths are bounded below by a quantity decaying exponentially in the length, are prevalent, and hence dense, among $C^k$-smooth negatively curved metrics on $M$ for sufficiently large $k$. Moreover, unlike in all known constructions, the exponent in our bound grows only sublinearly in the regularity $k$. This work is motivated by our recent result on prevalent unmarked length spectral rigidity for expanding circle maps and a very recent result by DeWitt, Durham, Reber, and O'Hare that $C^k$-smooth negatively curved metrics with exponentially sparse spectra are locally rigid.

Rate-induced tipping ("R-tipping") in ecological modelling is characterised by too-rapid change of an environmental parameter causing collapse of a population without any bifurcation being crossed ("B-tipping"). A well-known example of a slow-fast predator-prey model in which R-tipping is observed [Vanselow, Wieczorek, Feudel, Journal of Theoretical Biology, 479, 64-72 (2019)] suffers from unecological "resurgence", whereby populations driven to functional extinction recover towards stable coexistence. We propose an analytically tractable model, incorporating a strong Allee effect into the prey dynamics in a slow-fast Lotka-Volterra-type predator-prey system, which induces bistability and thus renders the extinction state a genuine attractor. Applying geometric singular perturbation theory (GSPT), we describe the dynamics of the extended model that is obtained by "ramping" of the inverse prey carrying capacity. We show the presence of a parabolic-shaped, folded critical manifold which admits a folded saddle singularity, the strong canard of which separates solutions that track the moving coexistence state from those that tip to extinction. The analytical simplicity of our model allows us to derive explicit expressions for the critical rate that separates "tracking" from "tipping" dynamics. Finally, we argue that our system represents a minimal model for rate-induced tipping to extinction, in that it incorporates three essential ingredients: bistability, a folded critical manifold, and a folded-saddle-type canard.

We develop an algorithm that takes as input a veering triangulation V and a flow cycle c of V, and outputs a veering parent V_c of V. Constructing veering parents combinatorializes drilling out closed orbits of transitive pseudo-Anosov flows. The algorithm relies on locally approximating the structure of the loom space associated to V in sufficient detail to carry out a certain modification of the Agol-Guéritaud construction. We use an implementation of the drilling algorithm to construct layered parents of non-layered veering triangulations (and hence Birkhoff sections for transitive pseudo-Anosov flows) and geometric parents of non-geometric veering triangulations. We also find closed orbits of (drilled) pseudo-Anosov flows that are not ambiently isotopic to the geodesics in their free homotopy classes, and show that certain veering triangulations encode almost orbit equivalent flows. In particular, we provide new computational evidence for a conjecture of Ghys asserting that all transitive Anosov flows with orientable invariant foliations are almost orbit equivalent.

In the first part of this series the $\sigma$-ring spanned by the periodic solutions of a generalized linear cellular automaton in a discrete group was shown to be Hopf–Galois, with pro-algebraic Galois group, while a Galois theory for the $\sigma$-ring spanned by the finite support solutions was left open, with parameterized difference Galois theory suggested as the tool. We apply that tool to automata over $\mathbb Z$. The Fourier transform turns the automaton into a rank one equation $\sigma(\hat x)=\hat\alpha\hat x$ over a field on which $\delta=z\,d/dz$ acts, and we compute, and algorithmically decide, its parameterized Galois group: for genuine automata with symbols rational in the time variable it is either $\mathbb{G}_m(\mathbb C)$ or $\mathbb{G}_m$, the dividing line being separability of the symbol, $\hat\alpha(z,t)=\gamma(z)\psi(t)$. Finally, the parameterized group in the time direction decides holonomy in the space direction; as a corollary, the Stirling numbers of the first kind are not spatially holonomic.

We solve a well-known question of Polterovich from 2002 regarding the rigidity of Hamiltonian diffeomorphisms in Hofer's metric in dimension two: every non-trivial Hamiltonian diffeomorphism of a surface of genus at least one has all positive iterations separated from the identity in Hofer's metric. In higher dimensions, we make progress on the case of autonomous Hamiltonian flows: we prove Hofer non-recurrence for symplectically hyperbolic manifolds, and completely characterize Hofer recurrence for functions of the moment polytope on monotone toric manifolds. These results apply equally well to Viterbo's spectral metric, settling cases of the $\gamma$-rigidity conjecture, and therefore in many cases also to the $C^0$-metric. Our approach relies on the philosophy that symplectically visible invariant sets govern symplectic non-recurrence phenomena. In dimension two, we use invariant annuli provided by low-dimensional dynamics, and apply methods of $C^0$ symplectic topology to approximately invariant Lagrangian submanifolds. For symplectically hyperbolic manifolds, the invariant sets are provided by sublevel and superlevel sets of autonomous Hamiltonians, while in the toric case, they comprise Lagrangian torus fibers. Across the arguments, we use quantitative Lagrangian Floer theory and its relation to notions of classical dynamics.

Let $G$ be a locally compact group, let $\Gamma$ be a lattice in $G$, and let $\eta$ be a spread-out probability measure on $G$. A Furstenberg discretization of $\eta$ to $\Gamma$ is a probability measure $\mu$ on $\Gamma$ such that every $(G,\eta)$-stationary space is also $(\Gamma,\mu)$-stationary. For cocompact lattices $\Gamma$ and under mild regularity conditions on $\eta$, we construct Furstenberg discretizations satisfying finite moment conditions of arbitrary order with respect to arbitrary prescribed subadditive functions. Our approach relies on a rather general study of differential entropy and Lyapunov exponents in the setting of locally compact groups. We relate Lyapunov exponents to spectral radii in weighted $L^{1}$-algebras and prove vanishing results for weights coming from length functions. Additionally, we prove that if $G$ has the property of rapid decay or the Kunze–Stein property, this yields entropy formulas for stationary actions whose Koopman representation is weakly contained in the left-regular representation. As consequences, we obtain preservation results for Poisson boundaries under discretization and characterize Zimmer amenability of stationary actions in terms of weak containment and maximal Furstenberg entropy.

We study a class of expanding Markov maps on the unit interval, which includes sofic \(\beta\)-transformations, Gauss maps, tent maps and cookie-cutter maps. We investigate the quantitative recurrence property for points that are dynamically badly approximable, namely, points whose orbits eventually remain a positive distance from a given point. We show that the Hausdorff dimension of the intersection of a recurrence set and a badly approximable set is determined by the same pressure function as that of the corresponding recurrence set. We further obtain the Hausdorff dimensions of two related intersections: recurrence sets with non-recurrent sets, and shrinking target sets with badly approximable sets.

Let $\Gamma<\operatorname{PSL}_2(\mathbb R)$ be a torsion-free, non-elementary, finitely generated Fuchsian group with parabolic elements, and let $\rho:\Gamma\to G$ be a type-preserving, Zariski dense, positive representation into a connected split semisimple real algebraic group. We prove quantitative local mixing for Bowen–Margulis–Sullivan measures on $\rho(\Gamma)\backslash G/M$, where $M$ is the centralizer of a maximal split torus in a compatible maximal compact subgroup. For every $u$ in the interior of the limit cone of $\rho(\Gamma)$, correlations of smooth compactly supported functions along $\exp(tu)$, normalized by $t^{(\operatorname{rank}_{\mathbb R}G-1)/2}$, admit a complete asymptotic expansion in integer powers of $t^{-1}$. We also obtain a uniform Gaussian asymptotic for transverse displacements, with an integrable remainder. We deduce asymptotics for matrix coefficients of the quasi-regular representation on $L^2(\rho(\Gamma)\backslash G)$ and for orbital counting in Riemannian balls. More generally, these conclusions hold for Zariski dense, relatively Borel Anosov representations of $\Gamma$ satisfying a peripheral root-balance condition. This includes relatively Borel Anosov representations into split simple groups of rank at most three and self-joinings of type-preserving geometrically finite real hyperbolic representations. To prove these results, we develop an intrinsic cusp geometry on the limit set of $\rho(\Gamma)$ in the Furstenberg boundary and use it to construct a countable Markov coding. We establish exponential moment bounds and uniform distortion estimates for the unbounded vector-valued return cocycle. Using peripheral Cartan growth and Zariski density, we establish the quantitative non-integrability needed for the spectral analysis.

Let $f$ be a postcritically finite endomorphism of the complex projective space $\mathbb{P}^N$ of degree at least two. We prove that every Fatou component of $f$ is preperiodic to the basin of a superattracting periodic orbit.

The second part of Hilbert's 16th problem is one of the most fundamental open problems in the qualitative theory of ordinary differential equations and dynamical systems. A central unresolved question is whether the Hilbert number $H(d)$ is finite for arbitrary polynomial degree $d$. (A very recent manuscript from OpenAI asserts the finiteness of $H(d)$; see the remark at the end of this paper.) In the absence of a general upper bound theory, most existing work constructs explicit perturbed polynomial systems to obtain improved lower bounds for $H(d)$, while rigorous upper bound estimates remain largely unavailable. This paper focuses on improving the quantitative lower bound for polynomial systems through a systematic Hierarchical Melnikov Realization framework. We construct perturbed dynamics based on an anisotropic Chebyshev Hamiltonian, where hierarchically selected perturbation coefficients generate simple zeros of the first-order Melnikov function simultaneously across multiple disjoint period annuli. Two distinct 3-adic filtrations supply the necessary logarithmic correction factors, and a Borel–Gauss analysis establishes the required local rank condition under sufficiently strong anisotropy. As the main result of this work, we prove that $H(d)=\Omega\bigl(d^2\ln^2 d\bigr)$, which improves the previously known lower bound by a factor of $\ln d$. The full construction of limit cycles and the resulting dynamical bounds are formally certified using Lean 4 formal verification.

This work is concerned with a stabilizing adaptive dynamic programming (ADP) approach to approximate solution of a given infinite-horizon optimal control problem. Since the latter problem cannot, in general, be solved exactly, a parametrized function approximator for the infinite-horizon cost function is introduced in ADP (so called “critic”). This critic is used to adapt the parameters of the function approximator. The so called “actor” in turn derives the optimal input of the system. It is a notoriously hard problem to guarantee closed-loop stability of ADP due to the use of approximation structures in the control scheme. Since at least stabilizability is always assumed in the analyses of ADP, it is justified to invoke a respective Lyapunov function. The proposed ADP scheme explicitly uses the said Lyapunov function to simultaneously optimize the critic and guarantee closed-loop stability. A Hessian-free optimization routine is utilized for the actor and critic optimization problems. Convergence to prescribed vicinities of the optima is shown. A computational study showed significant performance improvement for the critic-based approach compared a nominal stabilizing controller for a range of initial conditions.

For every quartic Salem number $\beta$, we prove that the equal-weight Bernoulli convolution $\nu_{\beta^{-1}}$ has Hausdorff dimension strictly less than one. The main step converts large Fourier coefficients at geometrically separated frequencies into a deficit in the Shannon entropy of a finite convolution. A positive trigonometric product gives the entropy estimate, while the algebraic norm controls the number of distinct atoms in each spatial cell. For the two quartic Salem numbers in $(1,2)$, reciprocal differences at suitable return times produce the required Fourier coefficients. A finite block cover then gives the dimension bound. This proof is independent of numerical computation. Separate exact calculations give $\dimH\nu_{\beta_1^{-1}}<1-1.4\cdot10^{-23}$ and $\dimH\nu_{\beta_2^{-1}}<1-7.8\cdot10^{-22}$, where $\beta_1<\beta_2$ are the two parameters in $(1,2)$. The accompanying verification programs use rational interval arithmetic and the Python standard library.

We prove the hyperplane winning property of weighted badly approximable vectors inside nondegenerate analytic manifolds and affine subspaces satisfying almost optimal Diophantine conditions. This, in particular, answers a conjecture of Badziahin–Velani for higher-dimensional manifolds, including affine cases that were understood only in $\mathbb{R}^2$.

This paper concerns the planar Liénard system \(\dot x=y-F(x),\ \dot y=-x\), where \(F(x)\) is a piecewise linear function with exactly \(m\) jump points and no fold points. Tonnelier [SIAM J. Appl. Math. 63 (2002)] conjectured that the maximum number of limit cycles of the system is \(2m\). The conjecture was confirmed for \(m=1\) in [J. Lond. Math. Soc. 113 (2026)], and a lower bound of \(2m\) for arbitrary \(m\) was established by Chen et al. [arXiv:2608.19542]. In this paper, we construct systems with at least \(4m-2\) hyperbolic crossing limit cycles for every positive integer \(m\), thereby disproving Tonnelier's conjecture for \(m\geq2\). We also establish the uniform upper bound \(2^{224(m+1)^2}\) for the total number of crossing, grazing, sliding, and composite limit cycles.

In deep learning, approximation theory motivates increasing representation size. We ask whether this benefit extends to dynamics learning through autoregressive prediction. We analyze the learned time evolution through the eigenstructure of Koopman operators, using relative residuals to detect spurious eigenpairs arising even as one-step error falls. For bounded Koopman operators, we show that minimal residuals over learned dictionary spaces converge pointwise to their full-space counterparts as these spaces approximate the observable space in $L^2$. Our hypothesis is that Koopman spectral reliability helps explain how consistently rollout error decreases with increasing dimension. We compare two models of a shared Koopman autoencoder trained alternately for reconstruction and latent evolution, using latent-prediction loss (one-step prediction errors in latent coordinates) or spectral-residual loss (relative residuals of candidate eigenpairs). Across six chaotic systems, both models reduced median windowed rollout error from smallest to largest dimension. The spectral-residual model achieved lower medians than the latent-prediction model for all systems and dimensions, and its median fell by a larger factor in every system. Its median decreased monotonically with dimension in four systems, against one for latent prediction. Against four baseline families, its mean valid prediction times were nearly always longer. At the largest dimension under two-stage training, we compared eigenvalue positions with each learned dictionary's residual contours. Spectral-residual eigenvalues concentrated in low-residual regions, whereas latent-prediction eigenvalues also appeared in high-residual regions, consistent with the hypothesis.

Let $f:\mathbb{P}^N\to\mathbb{P}^N$ be a postcritically finite endomorphism of degree at least two. Combining Yamanoi's hyperbolicity results with the dynamical property of $f$, we prove that the complement of its postcritical divisor is taut unless $f$ is a monomial power map or admits a nontrivial equivariant rational fibration with a polarized divisorially postcritically finite base map. We establish analogous alternatives for polarized PCF endomorphisms of rationally connected projective manifolds and, for Kobayashi hyperbolicity, on normal rationally connected projective varieties. In dimension two, the non-taut case can be described, up to iteration and rational semiconjugacy, by monomial power maps and skew products with monomial fiber maps.

For the quadratic family $f_c(z)=z^2+c$, we study the non-escaping locus $\KM=\{(c,z):c\in\M,\ z\in K_c\}$, the model space for cubic capture straightening in the framework of Inou and Kiwi. Here $\M$ is the Mandelbrot set, $K_c$ is the filled Julia set of $f_c$, and $J_c=\partial K_c$. We prove that $\partial\KM$ has full Hausdorff dimension $4$. The set $\KM$ is locally connected at every $(c,z)$ with hyperbolic $c\in\M$. At parameters on $\partial\M$, we construct mixed puzzle pieces from Yoccoz puzzles and parapuzzles. For non-degenerate pairs, shrinking of both planar puzzles implies local connectedness of $\KM$ at $(c,z)$. However, local connectedness fails at some $(c,z)$ with $c$ real parabolic and $z\in K_c\setminus J_c$, even though both the parameter space and the filled Julia fiber are locally connected there.

Computing local Stackelberg equilibria in nonlinear sequential games is challenging because gradient-based best-response methods may diverge, cycle, or converge to stationary points that do not satisfy the required equilibrium conditions. These failures can result from curvature asymmetry, unstable response mappings, and lack of invariance with respect to the follower's best-response manifold. This paper introduces the Stackelberg Sliding-Mode Algorithm (SSMA), a control-theoretic framework that recasts Stackelberg equilibrium computation as a manifold-stabilization problem. The follower and leader first-order optimality conditions define Stackelberg sliding manifolds, and sliding-mode dynamics drive the iterates to these manifolds in finite time. The follower curvature and leader reduced-curvature conditions determine manifold attractivity, making valid local Stackelberg equilibria attractive while rejecting stationary points that fail the second-order equilibrium conditions. The resulting approach integrates finite-time manifold stabilization, Stackelberg-consistent dynamics, and curvature-based equilibrium selection within a unified framework. Across 500 nonconvex Monte Carlo trials, SSMA achieved 100% success, reached the prescribed tolerance in a median of 26 iterations, compared with 456 for the nearest fully successful baselines, and attained terminal residuals near \(10^{-16}\), while converging to a valid local Stackelberg equilibrium.

We introduce positively transverse reaction networks and show that, with mass-action kinetics, no bounded trajectory with positive initial condition has a boundary omega-limit point. These networks are characterized by a geometric property of a collection of polytopes defined by a subset of complexes of the network.

In this paper, we study $r$-neutralized Pfister–Sullivan entropy, defined by measuring the complexity of orbit segments whose empirical measures lie near a prescribed invariant measure. We mainly obtain the following three results: (i) We disprove a variational conjecture posed by C.~Dong and Q.~Qiao [On $r$-neutralized entropy: entropy formula and existence of measures attaining the supremum, Comm. Math. Phys. 406 (2025), Article No. 74.] (ii)] We obtain variational formulas over probability measures for both upper and lower $r$-neutralized topological entropies. We also introduce new measure-theoretic quantities through PS localization and establish some variational principles for upper $r$-neutralized topological entropy (iii) We obtain upper bounds for $r$-neutralized PS entropy under different smoothness assumptions and lower bounds for $C^{1+\alpha}$ diffeomorphisms, in terms of entropy, dimension and Lyapunov exponents. For $C^\infty$ diffeomorphisms and $C^{1}$ Anosov diffeomorphisms, the $r$-neutralized PS entropies equal $h_\mu(f)+r\dim M$ for every ergodic measure $\mu$. We also establish exact formulas on mixing surface hyperbolic basic sets and show that the whole-manifold measure formula can fail for $C^2$ surface diffeomorphisms.

Let $k$ be a complete, non-archimedean. We study the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over $k$ and over $\mathbb{C}$ equipped with the hybrid norm. In both settings, we show an analogue of Mané–Sad–Sullivan's theorem about the continuity of the Julia set (in the Hausdorff topology). The non-archimedean case relies on the maximum modulus principle, which we show for analytic functions over analytic spaces without boundary.

We study the calibration of interaction-driven ordinary differential equation (ODE) models for traffic and pedestrian dynamics by means of optimal control and stochastic gradient descent. The interaction forces are either prescribed by classical physics-inspired laws, whose parameters are identified from trajectory data, or represented by feed-forward neural networks whose weights act as control variables. For the neural-network-driven particle system we prove well-posedness of the state equation, Lipschitz-continuous dependence of the trajectories on the initial data and on the network parameters, existence of an optimal parameter set, and a first-order optimality system with an explicit backward adjoint equation. The resulting adjoint representation of the gradient is combined with a projected mini-batch ADADELTA scheme, so that each gradient evaluation requires one forward and one backward ODE solve, independently of the number of network parameters. Building on this framework we propose two new second-order models, one for vehicular traffic and one for crowds, that combine goal-seeking, learned interaction, repulsion-alignment or dissipation, and external control terms. Numerical experiments compare six models (a follow-the-leader model of Lighthill-Whitham-Richards type, a neural-network traffic model, the social force model, a neural-network pedestrian model, and the two new models) in terms of cost reduction and root-mean-square trajectory deviation. The neural-network pedestrian model achieves the lowest final cost, while the richer second-order models trade a higher calibration cost for individualised, behaviourally interpretable dynamics. We conclude by discussing limitations and directions for incorporating richer behavioural and environmental effects.

We develop a theory of bi-contact plugs on oriented $3$-manifolds to construct Anosov flows or, more generally, structurally stable nonsingular flows. For a hyperbolic plug with filling Morse-Smale boundary laminations, we prove that every strongly transverse gluing map can be isotoped through strongly transverse maps to identify a strongly adapted unoriented contact form. The resulting flow is hyperbolic and is Anosov when the quotient is closed. This answers a question of Béguin-Bonatti-Yu. The result also extends to gluing nonsingular partially hyperbolic flows. Anosov completion by reflection embeds each such hyperbolic plug in an Anosov flow on its oriented double. Further applications include a classification of structurally generic projectively Anosov flows without saddle periodic orbits and Morse-Smale examples with prescribed saddle count. Generalizing the construction of Bonatti-Bowden-Potrie, finite capping embeds any bi-contact plug in a closed projectively Anosov flow with the same topological entropy, preserving partial hyperbolicity for attracting plugs. We also construct flows with controlled invariant torus dynamics on torus bundles and on doubles of plugs with coorientable boundary bi-foliations. Finally, every cooriented partially hyperbolic bi-contact plug admits a Liouville structure on its thickening, with Liouville trajectories projecting to positively reparametrized flow lines. Repelling plugs yield four-dimensional Liouville domains, often with chaotic skeletons.

We prove that topological conjugacy of elements of the Brin–Thompson group $nV$ is undecidable for every $n\geq2$. From a Turing machine $T$, the reduction produces two elements of $nV$, which are conjugate by an involution in $nV$ if $T$ halts on the empty tape. If $T$ does not halt on the empty tape, the topological dynamics of the two group elements are distinguished by their visits to minimal subsystems of the set of (germ-)aperiodic points. In these groups, and any finitely-generated supergroups inside the homeomorphism group of Cantor space, we also obtain undecidability of the Whitehead problem (the problem of being in the same automorphism orbit), and undecidability of group-theoretic conjugacy.

We study discrete networks of $N+1$ identical phase oscillators with positive reciprocal weights and smooth odd periodic interactions, motivated by discrete models of Huygens synchronisation. The interaction laws define an intrinsic potential, and removing the common phase gives a gradient map on the $N$-torus. A weighted graph Laplacian yields explicit step sizes for strict Lyapunov decrease and global invertibility. Our main result is a Morse–Smale criterion for maps $z\mapsto z+hK\sin z$: if $K$ is symmetric, nonsingular and weakly diagonally dominant, the map is Morse–Smale for $0<h<1/\lambda_{\max}(K)$, and this interval is optimal. The proof combines invariant coordinate slabs with a maximum principle for the adjoint variational equation. The criterion applies to unit-weight paths of any length and to the three-leaf star, and a weighted version covers paths with discretely concave weights. By structural stability, it extends to every graph containing a spanning path with sufficiently weak additional links. Conversely, we construct infinitely many trees with Morse potentials and an almost-global sink, whose maps nevertheless have non-transverse heteroclinic intersections.

Let $f:(\C^n,0)\to(\C,0)$ be a holomorphic germ with an isolated critical point at the origin, and let $\varphi_f$ be the negative gradient flow of $\re f$ with respect to a Riemannian metric. We show that the origin is an isolated invariant set of $\varphi_f$ and that its Conley index is the suspension of the Milnor fiber, \[ h(\varphi_f,\{0\})\;\simeq\;\Sigma F_f\;\simeq\;\bigvee_{\mu(f)}S^n , \] so that $\mu(f)=\rank \CH_n(\varphi_f,\{0\};\Z)$: the Milnor number is the rank of the homological Conley index of the flow in degree $n$. The proof uses continuation to a holomorphic morsification of $f$. Since $\im f$ is a first integral of the flow, after a rotation of the target through an angle avoiding finitely many values there are no connecting orbits between the $\mu(f)$ nondegenerate rest points that result, and the sum property of the Conley index applies. McCord's theorem then recovers the Poincaré–Hopf index of $-\nabla\re f$.