Papers

We use centered differences to construct Chernoff families and derive even asymptotic expansions for their iterated products. Richardson extrapolation is then used to cancel polynomial part and thereby raise the convergence order. As illustrations, we apply the construction to uniformly parabolic, Schrödinger-type second-order operators and to a third-order dispersive operator, obtaining, at extrapolation depth $m$, the rates $ O(n^{-2m}) $ on the Sobolev scales $H^{2m+\delta}$, $H^{4m}$, and $H^{5m}$, respectively, with arbitrary $\delta>0$. In the analytic category, quantitative all-order estimates allow the extrapolation depth to grow with the base resolution and yield the super-fast rates $e^{-cn^2}$ for the two second-order models and $e^{-cn} $ for the third-order model. The constructions are explicit in terms of the coefficients of the underlying differential operators and do not use the target semigroup in their definition. This provides a constructive positive answer to the challenge of super-fast Chernoff approximation formulated by Remizov in Problem 30 of arXiv:2410.00416.

We generalize Moser's stability theorem for minimal hypersurface foliations to arbitrary codimension. Let $\mathbb{T}^N=\mathbb{R}^N/\mathbb{Z}^N$ be the standard torus, and let $P\subset\mathbb{R}^N$ be an $m$-plane satisfying a Diophantine condition relative to $\mathbb{Z}^N$. We prove that, under every sufficiently small smooth $\mathbb{Z}^N$-periodic perturbation of the Euclidean metric, the affine foliation of $\mathbb{T}^N$ induced by $P$ persists as a smoothly conjugate foliation by minimal $m$-dimensional submanifolds. The normalized conjugating displacement depends smoothly and tamely on the metric and is locally unique. The proof uses Hamilton's formulation of the Nash-Moser implicit function theorem.

We consider the problem of computing Taylor approximations to the optimal feedback law and value function for infinite-horizon finite-dimensional nonlinear optimal control problem—also known as Al'brekht's method. The scalability of existing Al'brekht-based approaches to computing approximate solutions to the Hamilton-Jacobi-Bellman equation associated with the optimal control problem has been severely limited by computational resources. The computational time issue has been partially mitigated by recent works leveraging tensor-structure to develop efficient numerical methods. However, RAM requirements still have remained a major hindrance to scaling beyond state dimensions of a few hundred or low thousands. We propose an acceleration to compute the expensive higher-order terms in a reduced space, mitigating the curse of dimensionality. A fast and scalable Matlab implementation is provided open-source for general purpose use, with the ability to accommodate arbitrary polynomial nonlinearities in the control-affine state equation, as well as polynomial state-dependence in the cost function, and the ability to handle systems in generalized form with a non-identity mass matrix, such as finite element models. Combined with the exploitation of sparsity and modern low-rank Riccati solvers, this permits computing approximations to polynomial feedback laws based on Al'brekht's method in hundreds of thousands of states.

The periodic tiling conjecture asserts that if a bounded set $\Omega\subset \mathbb R^d$ of finite measure tiles $\mathbb R^d$ by translations, then it admits at least one fully periodic tiling. This conjecture was proved to hold in $\mathbb R$ by Lagarias and Wang, and was recently disproved by Greenfeld and Tao in sufficiently high dimensions. Its discrete analogue in $\mathbb Z^2$ was proved by Bhattacharya, whereas the conjecture in $\mathbb R^2$ remains open. In this paper, we analyze the structure of tilings by axis-parallel polygonal sets: bounded open sets in $\mathbb R^2$ whose boundary is a finite union of line segments parallel to the axes. We prove that every such tiling that is topologically minimal is weakly periodic. Moreover, we show that if an axis-parallel polygonal set $\Omega$ tiles $\mathbb{R}^2$ then it admits a quasi-periodic tiling (i.e., a finite union of translated lattices). The proofs combine geometric decompositions of tilings with rational approximation and discrete tiling theory. As a consequence, we obtain that the translational tiling problem for axis-parallel polygonal sets with real algebraic vertex coordinates is decidable.

We prove mixing local limit theorems for dynamical systems in the superdiffusive regime where there is convergence to a multivariate stable law (including the nonstandard domain of the central limit theorem). We apply our results to d-dimensional observables for intermittent maps and solenoids, as well as intermittent semiflows and flows.

We construct a smooth autonomous divergence-free vector field on $\mathbb{T}^3$ whose flow is exponentially mixing with respect to Lebesgue measure. This answers an open question about smooth autonomous exponential mixing in three dimensions. Starting from an exponentially mixing Anosov suspension flow, we perform surgery near four periodic orbits and show that the resulting manifold is diffeomorphic to $\mathbb{T}^3$. The resulting vector field has exactly sixteen hyperbolic equilibria. We also prove a general transfer theorem that preserves exponential mixing under certain singular changes of coordinates.

We prove that two completely different oscillators synchronise through arbitrarily weak reciprocal impulses. Any two planar oscillators with hyperbolic attracting limit cycles and equal natural periods lock stably one-to-one under translation impulses with generic directions and a suitable range of positive strength ratios, whatever their vector fields, limit cycles and transverse contraction rates. The mechanism is topological: the winding of the asymptotic phase gradient forces phase advances and delays, and the relative winding number of an effective impulse field gives lower bounds on attracting and unstable locked states. Equal periods are not essential: locking persists under small nonzero mismatches, with persistence and loss governed by a shifted phase balance. The analysis covers the full state-dependent impulsive dynamics, including transverse stability. Nearly simultaneous emissions may make the return discontinuous; under an explicit regularity condition the global return has a normally attracting Lipschitz invariant circle with topologically Morse–Smale dynamics.

We study multidimensional entry-exit phenomena in slow-fast systems whose fast linearization contains a nilpotent component. We show that the classical spectral balance still determines the exit point as $\varepsilon \to 0$, however the nilpotent component induces an algebraic amplification invisible to the spectrum, producing a $\varepsilon|\log\varepsilon|$-shift of the exit point for $0 <\varepsilon \ll 1$. We further extend the analysis to systems with several spectral branches and show that mode dominance follows a hierarchical, lexicographic selection principle: accumulated spectral growth determines the leading order, while the nilpotent component resolves degeneracies between spectrally equivalent modes.

For any $n\geq 1$, we construct a continuum of non-conjugate actions by homeomorphisms of $\operatorname{SL}_{n+1}(\mathbb{Q})$ on the $n$-sphere $S^{n}$ that do not extend to a continuous action of $\operatorname{SL}_{n+1}(\mathbb{R})$. We do this by adapting a blow-up construction of Bowden and Mann. We motivate studying these actions and note down some related observations.

Let $\{f_i(x)=M_ix+a_i\}_{i\in I}$ be an affine iterated function system on $\mathbb{R}^d$ with attractor $\Lambda$, and let $\mu$ be an associated self-affine measure. Assume that the $M_i$ generate a Zariski dense subgroup of $\mathrm{GL}_d(\mathbb{R})$. Under the strong open set condition, we show that $\dim_{\mathrm{H}}\Lambda$ equals the affinity dimension and that $\dim_{\mathrm{H}}\mu$ equals the Lyapunov dimension. The above results are consequences of the following projection theorem, which requires no separation condition. For every $1\leq n\leq d$ and for Furstenberg measure a.e. $(d-n)$-dimensional subspace $V$, \[ \dim_{\mathrm{H}} (P^V)_*\mu=\min\{n,\,\dim_{\mathrm{H}}\mu\}. \] This projection theorem in fact holds under hypotheses weaker than Zariski density: simplicity of the Lyapunov spectrum and a mild transversality condition (the Oseledets non-concentration property). The source of dimension increment in our proof is the positive dimension of the fiberwise conditional measures, exploited through Wu's one-dimensional entropy increase theorem. A key ingredient in reducing the higher-dimensional problem to this setting is a new structural result: a slab-filling property of the measure.

Does evolutionary entropy depend on the width of the age classes used to describe a population? We show that the apparent dependence on temporal resolution is a consequence of the loss of information about reproductive timing within discrete classes. When this information is retained, corresponding continuous and discrete representations yield the same cycle entropy, generation time, evolutionary entropy, and reproductive potential in a common unit of time. When within-class timing is unknown, a uniform local reconstruction yields a logarithmic correction. For constant post-maturity hazard and fertility, continuous exponential and discrete geometric descriptions correspond exactly at every time scale. Thus, for a fixed reference duration, evolutionary entropy is a property of the reproductive process, not of its temporal partition. These results provide a mathematical basis for comparing demographic models across different temporal scales by identifying the information required to separate differences in reproductive organisation from effects of the chosen age-class representation.

We study closed-set stabilizers in topological full groups of minimal effective ample groupoids with Cantor unit space. For almost finite or purely infinite groupoids, we extend the maximal stabilizer theorem of Grigorchuk and Vorobets to every subgroup containing the commutator subgroup. For a nonempty closed nowhere dense set, maximality of its stabilizer is equivalent to minimality of its stabilizer action, minimality of the regular reduction, and primeness of a canonical wide open subgroupoid. Without assuming almost finiteness or pure infiniteness, we prove spatial rigidity for stabilizers acting minimally on infinite closed nowhere dense sets. Their isomorphism types determine the ambient Kakutani equivalence class and, when Hausdorff, the effective reduction. We also realize every AF groupoid with compact unit space as the full restriction, with its relative topology, of any minimal effective ample groupoid with Cantor unit space to a closed nowhere dense subset. Combining rigidity and realization yields continuum many pairwise nonisomorphic stabilizers in every subgroup containing the commutator subgroup; these stabilizers are maximal when the ambient groupoid is almost finite or purely infinite. The realization also gives explicit AF subquotients of the ambient reduced groupoid $C^*$-algebra, including the CAR algebra, with compatibility on the canonical diagonal subalgebras.

For endomorphisms of the complex projective plane of a given degree, we prove that the existence of a special curve of bounded degree, in the sense of the dynamical Manin–Mumford conjecture, is confined to a proper algebraic subvariety of the parameter space. This confirms a conjecture of DeMarco and Mavraki in dimension two. For general regular polynomial endomorphisms of $\mathbb A^k$, we obtain a uniform dynamical Manin–Mumford bound outside an exceptional hypersurface. These results arise from a common strategy, valid in every dimension and codimension, which reduces such questions to the geometry of exceptional loci in the space of pairs formed by subvarieties and maps. We control these loci by combining arithmetic equidistribution, blender constructions, dynamical directions, monodromy arguments, and degenerations at the boundary of parameter space.

We study the attractor lattice of a class of hybrid dynamical systems on compact metric spaces. Under the trapping guard condition, we prove that hybrid attractors form a bounded distributive lattice isomorphic to the attractor lattice of an associated semiflow. The regularized preimages define a lattice epimorphism from a sublattice of the regular closed, attracting neighborhoods of the semiflow onto those of the hybrid system, and from the whole lattice when the guard set and its reset image have empty interior. Then, using grids on the state space, we construct a convergent cofiltration of outer approximations of a fixed-time map of this semiflow. This construction realizes every finite sublattice of hybrid attractors by a lattice of attracting neighborhoods that are unions of grid elements at sufficiently fine resolutions, and provides index pairs for computing Conley indices. We illustrate these constructions computationally by extracting the Conley–Morse graphs of the bouncing ball, the rimless wheel, the spiking neuron, and exhibiting the attractor lattice for an impacting oscillator.

In this paper, sufficient conditions are established for diagonal perturbations of weighted shifts to be supercyclic, hypercyclic, and frequently hypercyclic, extending several classical results for weighted shifts to this more general setting. In addition, a complete analytic characterization of the shadowing property for these operators is obtained. Consequently, examples of frequently hypercyclic perturbed weighted shifts with the shadowing property are constructed.

We study uniqueness for parabolic motions in the Newtonian $N$-body problem with a prescribed minimal normalized central configuration at infinity. A McGehee compactification identifies their tails with the strong stable manifold of the corresponding outgoing zero-energy equilibrium. The configuration projection of this manifold is locally a smooth graph, yielding a smooth initial-velocity field and uniqueness among all parabolic motions confined to a sufficiently narrow truncated cone, without any minimizing assumption. For parabolic geodesic rays, a renormalized action excess and a quantitative no-exit argument remove the confinement hypothesis when the initial configuration lies sufficiently far inside a smaller cone. We then prove that the set of collision-free initial configurations admitting more than one such ray has Lebesgue measure zero. The global argument propagates the local stable manifold backward, discards critical values of the configuration projections by Sard's theorem, and separates distinct regular branches through an explicit differential formula for their renormalized excess. No nondegeneracy assumption on the prescribed minimal central configuration is required. The proofs combine invariant-manifold theory with direct action comparisons and do not use weak KAM theory, Busemann functions, or the differentiability theory of Hamilton–Jacobi solutions.

Within the framework of multivalued dynamical systems, we compute the Conley index for a class of limit cycles in planar Filippov systems. We classify these cycles according to their interaction with the switching manifold, including crossing, sliding, and mixed regimes. Using the Conley index for multivalued admissible maps developed by Mrozek, we construct the appropriate index pairs and determine the corresponding indices for each class. These computations enlarge the class of Filippov invariant sets for which the Conley index is known and extend previous results on low-codimension singularities to periodic dynamics.

This paper studies non-autonomous iterations of $\lambda e^z$, where at each step the parameter $\lambda$ is chosen from a bounded interval. Urbański and Zdunik introduced and developed a thermodynamic formalism for random dynamics of exponential maps, extending the autonomous result established independently by Rees and Lyubich to typical parameter sequences in $[A, B]^{\mathbb{N}}$ with $1/e<A<B<\infty$. In this paper, we show that the same conclusion indeed holds for all parameters in $[A, B]^{\mathbb{N}}$, Our proof is based on carefully crafted uniform versions of pullback arguments, inspired by the original methods of Rees and Lyubich. In a related direction, Urbański and Zdunik established a non-autonomous analogue of Misiurewicz's theorem for the exponential map, later refined by Lech. We further extend their work by providing a refined condition for the Julia set to be the entire complex plane.

Let $T$ be the Gauss map. For each $\alpha\in [0,1]$, let $E(\alpha)$ be the survivor set consisting of all $x\in [0,1]$ whose orbit $\{T^k(x):k\ge 0\}$ never enters the interval $(\alpha,1]$. We prove that the set-valued function $\alpha\mapsto E(\alpha)$ is increasing and locally constant almost everywhere. Moreover, we explicitly characterize all intervals where the function $\alpha\mapsto E(\alpha)$ remains constant. We also describe the points at which the map $\alpha\mapsto E(\alpha)$ is not locally constant. Further, we obtain the Hausdorff dimension of $E(\alpha)$ for all $\alpha\in [0,1]$. We prove that the function $\alpha\mapsto \dim_H(E(\alpha))$ forms a devil's staircase on $(0,1]$, but is not continuous at $0$. This discontinuity implies that the result of Raith [Acta Math. Univ. Comenian. 63 (1994), 39-53] fails for piecewise monotone interval maps with infinitely many branches. Finally, we give an application quantifying the set of points sharing identical $\alpha$-continued fraction expansion and regular continued fraction expansion.

We define a class of flows on compact metric spaces that we call \emph{$C^\alpha$ metric Anosov flows}, $0 < \alpha < 1$ or $\alpha = \mathrm{Lip}$. We give verifiable criteria for a metric Anosov flow to be $C^\alpha$. We show that $C^\alpha$ metric Anosov flows have Hölder continuous local product structure, and in particular Hölder continuous holonomies. We deduce that topological conjugacies between $C^\alpha$ metric Anosov flows are automatically Hölder continuous. We show that any $C^\alpha$ metric Anosov flow admits a strong (i.e. Hölder continuous) Markov coding. This yields strong ergodic properties for equilibrium states for Hölder potentials for $C^\alpha$ metric Anosov flows. The regularity of the local product structure is proved in a non-compact setting which applies for the geodesic flow on the universal cover of a locally CAT(-1) space. This allows us to obtain Hölder regularity results for the holonomy maps for pinched negative curvature manifolds with respect to the Sasaki metric. This contrasts with the result of Ballmann, Brin and Burns that there are examples in this class for which the dynamical distributions are not Hölder.

Let $ \operatorname{MaxCol}(n,d) $ denote the maximal number of nonsimultaneous elastic collisions among $n$ identical hard balls moving freely in $\mathbb{R}^d$. We give a planar construction with exponential lower bound with respect to $n$: $ \operatorname{MaxCol}(n,2)\geq 2^{\lfloor n/2\rfloor}, \quad n\geq3 $.

As engineering systems are optimized to reduce weight and improve performance, vibration behavior, including nonlinear effects, is becoming increasingly important to analyze. While methods such as harmonic balance are popular for analyzing periodic nonlinear vibration behavior, these analyses often only consider deterministic loading. However, real applications experience additional noise in the excitation, and alternative approaches are frequently limited to smaller models. This work expands on a recent approach to calculate a time-varying covariance using a linearized system around a periodic nonlinear response. Specifically, numerical conditioning is addressed and improved for cases increasing from a few degrees of freedom to hundreds of degrees of freedom. The algorithm is then extended to capture correlations across both space and time via a period-spanning covariance matrix. Model reduction is proposed for this calculation to maintain computational efficiency. The period-spanning covariance matrix is used to sample full stochastic cycles and estimate fatigue-relevant statistics up to 1393 times faster than stochastic time integration. Importantly, the present methodology uses the same gradient information as stability analyses and continuation solutions, so it can easily be applied for many nonlinear analyses. The methodology is demonstrated for two beam systems: one with unsteady aerodynamic forces and another with a Bouc-Wen frictional tip force. For the aeroelastic model, statistics are accurately estimated via the present approach. For the frictional system, some errors are introduced compared to stochastic time integration and are attributed to the linear time-periodic nature of the system used in the present stochastic calculations. However, estimated stochastic statistics remain useful given the significant reduction in computation time.

In machine learning, a self-attention dynamics describes the attention mechanism of transformers, and is obtained treating the layer index of a transformer as time index. Even though naturally discrete-time, this dynamical system is most often studied in continuous-time. Scope of this paper is to provide a thorough investigation of the discrete-time version of the self-attention dynamics: a complete classification of its equilibria is given, and their stability properties are studied, in both the single-head and the multihead case. We show that in both cases the main attractor of the dynamics is a consensus equilibrium aligned with the principal eigenvector of the value matrix (in the single-head case, a corresponding matrix of weights in the multihead case), even though other locally asymptotically stable bipartite equilibria can be show to exist at least in the single-head situation. The behavior is confirmed by looking at the learned weight matrices of a real transformer.

We establish a structure theorem for smooth random walks on the circle and study the dimensional properties of their stationary measures, including exact dimensionality and a Ledrappier-Young-type dimension formula. These results provide the technical tools needed for the main theorems in Part II "Dimension theory of group actions by circle diffeomorphisms II: Minimal sets". They also yield an approximation theorem in the spirit of Katok, Avila-Crovisier-Wilkinson, and Morris-Shmerkin, together with a time-reversal symmetry for circle actions: every semigroup $T$ contained in $\mathrm{Diff}_+^{1+\alpha}(\mathbb{S}^1)$ without finite orbits has the same number of minimal sets as its inverse semigroup $T^{-1}$.

We study the Hausdorff dimension of attracting sets of three-dimensional flows whose one-dimensional quotient dynamics is non-uniformly expanding. Our main assumption is that the quotient map admits a full-branch uniformly expanding Gibbs–Markov induced map. We first prove that every such attractor contains a hyperbolic horseshoe of Hausdorff dimension strictly larger than two. As a consequence, after an arbitrarily small $C^2$ perturbation, the associated dynamical Lagrange and Markov spectra have robustly non-empty interior for an open and dense set of $C^1$-observables. We then establish intermediate dimension properties. Every admissible intermediate Hausdorff dimension is realized by an ergodic invariant measure supported on the attractor. Moreover, every intermediate stable and unstable dimension is realized by a strictly ergodic hyperbolic subsystem. The approach extends the dimension theory previously developed for geometric Lorenz attractors to more general non-uniformly hyperbolic attractors including singular-hyperbolic attractors, Rovella attractors, and broad classes of skew-product suspension flows modeled by intermittent and Benedicks–Carleson maps.

We prove that for every choice of five positive masses the planar five-body problem has finitely many central configurations up to similarity. This settles Smale's sixth problem for five bodies in the plane. Albouy and Kaloshin proved it for masses outside a closed algebraic subset of codimension two. We follow their method. Infinitely many central configurations would give an algebraic curve of complex solutions. At the places at infinity of the curve the valuations of the mutual distances must balance. We determine the possible valuation vectors at all positive masses, together with the mass conditions under which each can occur. We treat limits in which four bodies form a cluster of zero potential energy by expanding the potential, which is constant along the curve, at a place. This excludes, at every positive mass, one of the two diagrams in which such clusters occur, and part of the other. The same expansion excludes one further diagram at every positive mass. What remains is an exact analysis, by computer algebra, of the sets of mass conditions that can hold together at positive masses. Exact rational certificates show that the balance fails under each of them, except on a few one-parameter families of masses, which we treat separately. On the rays through $(1,4,4,4,4)$ and $(1,1,4,4,4)$ the certificates do not apply. On the first ray they cannot apply: there the complex equations have a continuum of solutions, as Albouy and Kaloshin observed. On these rays we count the real central configurations: there are 306 and 294 up to orientation-preserving similarity. The certificates, the programs that check them and a proof of the theorem in the Lean proof assistant are available.

Following the celebrated breakthrough in Bénard-He-Zhang, we prove Khintchine dichotomy for push-forwards of self-similar measures on $\mathbb{R}^d$ under uniform $C^{1+\alpha}$ maps. For this, we also study almost random walks on $\mathrm{SL}(d+1,\mathbb{R})/\mathrm{SL}(d+1,\mathbb{Z})$ and prove their effective equidistribution.

Let $G$ be a connected Lie group equipped with a compatible right-invariant metric and let $G$ act continuously on a compact metrizable space $X$. We prove that $X$ admits a $G$-equivariant embedding into the right argument shift action on the space of one-Lipschitz maps from $G$ to $[0,1]$ if and only if the set of fixed points topologically embeds into $[0,1]$. This generalizes a theorem proved by Gutman, Huo and Tsukamoto for $\mathbb{R}^k$-flows.

In this paper, we develop an abstract framework for the theory of uniform exponential attractors in non-autonomous dynamical systems. Our construction employs time-dependent absorbing sets while ensuring exponential attraction among the constructed sets. Moreover, we show that the resulting exponential attractors are uniform-that is, they remain stable under perturbations. As an application of this abstract theory, we study the weakly damped hyperbolic equation with sub-quintic nonlinearity. Using the proposed approach, we construct exponential attractors for the process generated by the Shatah-Struwe solution, which satisfies both energy and Strichartz estimates. Furthermore, we establish the Holder continuity of these exponential attractors with respect to the external forcing term.

Let $\lambda$ be the Liouville function. In this paper we prove an average form of Chowla Conjecture for even $k $. that is, for every fixed even integer $k\geq 2$, we prove that \[ \lim_{\min\{M,N\}\to \infty}\frac{1}{MN}\sum_{m=1}^M\sum_{n=1}^N\lambda(m)\lambda(m+a_1n)\cdots\lambda(m+a_kn)=0. \] We construct a stationary model for both averaging variables, derive the finite conditional-entropy estimates. A comparison of overlapping intervals on the logarithmic axis then forces this function to vanish.