Papers

Using differential Galois theory—specifically, Magid's theory of the complete Picard-Vessiot closure—we show that a system of first-order differential equations in $n$ variables admits a Liouvillian first integral if and only if it admits a first integral in a Liouvillian Picard-Vessiot extension, if and only if it admits a first integral in its Picard-Vessiot ring. Consequently, a system with a Liouvillian first integral admits one in a differential field obtained from the field of rational functions by first taking a finite algebraic extension, then adjoining an exponential of an integral, and then an integral. This yields a new proof of Singer's theorem on Liouvillian first integrals of autonomous planar systems (Trans.\ Amer.\ Math.\ Soc.\ 333, 1992) and of its recent extension to autonomous systems in $n$ variables by Aziz et al. (arXiv:2512.15522). Our results hold over any differential field of characteristic zero with an algebraically closed field of constants; in particular, they apply to non-autonomous systems. Finally, we exhibit a system that admits a first integral in a non-Liouvillian Picard-Vessiot extension but none in its Picard-Vessiot ring and thus establishing that the Liouvillian hypothesis cannot be dropped.

In this paper, we first introduce fiberwise supramenability for locally compact Hausdorff étale groupoids with compact unit space. Then, for such a minimal ample groupoid $\mathcal{G}$, we show that if $\mathcal{G}$ is fiberwise supramenable or topologically amenable, then the clopen type semigroup $S(\mathcal{G})$ is almost unperforated. Consequently, if $\mathcal{G}$ is $\sigma$-compact, then it has groupoid strict comparison. We then present several applications, including Matui's AH conjecture and pure infiniteness for groupoids and groupoid $C^*$-algebras.

In this note we present a dimension formula for certain randomized self-similar sets. The self-similar sets we are considering are $d$-dimensional generalizations of rational projections of inhomogeneous sponges: they have overlaps, but the overlapping structure is in some sense well-organized. The randomization happens according to a labelled Galton-Watson tree corresponding to the cylinders of the IFS, for example as in the case of the Mandelbrot percolation set. The dimension formula is given in terms of a pressure function corresponding to the expectation matrices, which describes the overlapping structure and the randomness of the system.

Although generalized hyperbolicity has proved to be a fundamental notion in linear dynamics, recent results show that it is too restrictive to characterize the shadowing property of invertible bounded linear operators on Banach spaces. This motivates the introduction of generalized pseudo-hyperbolicity, a weaker notion formulated through continuous homogeneous maps generating exponentially decaying Green families. We prove that, for arbitrary sequences of invertible bounded linear operators on Banach spaces, generalized pseudo-hyperbolicity is equivalent to both shadowing and topological stability. In the autonomous setting, this yields the equivalence between shadowing and topological stability, removing additional assumptions from previous results. We also prove robustness of shadowing and topological stability for sequences of invertible bounded linear operators.

We prove a local Hartman-Grobman theorem at zero for the unforced Navier-Stokes equation on the three-dimensional torus in the strong topology of $H^s$ for every $s>7/2$. More precisely, we construct a homeomorphism between open positively invariant neighborhoods of zero that conjugates the Navier-Stokes semiflow to the heat semigroup. We develop an infinite-dimensional boundary-time scheme built on a nested hierarchy of exact asymptotic dynamical factors. Their fiber geometry yields Sobolev block coordinates with autonomous finite-dimensional factor systems, on which boundary-time topological conjugacies can be constructed at every finite level and then assembled on a common neighborhood into the desired $H^s$ conjugacy. The infinite-dimensional assembly relies on new spectral estimates for the divergence-free transport structure, which control derivative loss and the blockwise tails of the finite-level conjugacies and their inverses without an additional spectral-gap assumption.

We extend the generalised Calderon Transfer Principle as presented in \cite{dBL} to the setting of trace preserving group actions of $\sigma$-compact locally compact Hausdorff groups on semifinite von Neumann algebras $\M$. At its essence the transfer principle consists of showing that a large class of regular operators in the group context may be translated to operators in the algebra context by means of this group action. The end result is a protocol for proving ergodic convergence results for group actions on noncommutative Orlicz space of semifinite von Neumann algebras. We start with proving the existence of the transferred operator in the noncommutative context. Two tools are developed for the purpose of actually proving convergence results: (1) a theory of what may be called 2-variable decreasing rearrangements for the algebra $L^\infty \overline{\otimes} \M$ (where ($\Gamma,\nu$) is a Radon measure space) and (2) a concepts of maximal operators suited to the present context. These tools are then used to lift the ergodic convergence results presented in \cite{dBL} to the noncommutative setting. In closing we present examples illustrating the application of the tools

We prove that the upper box dimension of an inhomogeneous attractor of a $C^1$ iterated function system in $\mathbb{R}^d$ is bounded above by the maximum of the singularity dimension of the system and the upper box dimension of the condensation set. This extends the upper bound of Burrell and Fraser (2020) for inhomogeneous self-affine sets to the $C^1$ setting. The covering argument of Burrell and Fraser does not apply directly to nonlinear compositions. We overcome this difficulty by combining their approach with an extension of the recursive covering estimate of Feng and Simon (2023). We also obtain the corresponding bound for orbital sets driven by arbitrary compact subshifts.

Let $1,\alpha_1,\alpha_2$ be a basis for a cubic number field $K\subset\mathbb{R}$, and for $\tau_1,\tau_2\ge 1$ let $G(\mathbf{\alpha},\mathbf{\tau})$ denote the number of distinct gaps between the fractional parts of the numbers $m_1\alpha_1+m_2\alpha_2$, with $m_1,m_2\in\mathbb{Z}$, $0\le m_1<\tau_1$ and $0\le m_2<\tau_2$. We show that $G(\mathbf{\alpha},\mathbf{\tau})$ is unbounded as $\mathbf{\tau}$ varies. When $K$ is totally real this is deduced from a density theorem of Shapira for orbits of the diagonal group on the space of lattices. When $K$ has a complex embedding we give an explicit construction, which also shows that there are constants $c,\kappa>0$ for which $G(\mathbf{\alpha},\mathbf{\tau})\ge c(\log\max\{\tau_1,\tau_2\})^{\kappa}$ for infinitely many $\mathbf{\tau}\in\mathbb{N}^2$.

We study the extent to which periodic orbits determine a one-dimensional dynamical system. In particular, we show that they determine every topologically mixing interval map. More generally, for a non-mixing interval map $f$ with dense periodic points, the periodic orbits need not determine $f$, but they always do so for $f^2$. Finally, for maps on compact intervals, we obtain a local version of this rigidity: for each basic set $B$ arising in the spectral decomposition of an interval map $f$, there exists an iterate $f^q$ whose restriction to $B$ is determined by the periodic orbits of $f$.

We prove that the Boca–Cobeli–Zaharescu (BCZ) map is strongly mixing with respect to normalized Lebesgue measure on the BCZ triangle.

We prove that entropy alone, with no invariance hypothesis, forces a Borel probability measure on the circle toward Lebesgue measure under every sublacunary set of multiplicative endomorphisms $x \mapsto mx$. Here, we refer to an infinite set of multipliers $m$ as sublacunary if its consecutive ratios tend to $1$. Examples of such sets include the primes following perfect cubes, the range of the partition function, the integers $\lfloor n^{\log n} \rfloor$ and the semigroup generated by $2$ and $3$ which is the classical case of Furstenberg's conjecture. Specifically, we show that every measure $\mu$ has a weak-star limit of endomorphs which dominates Lebesgue measure scaled by the upper entropy dimension of $\mu$. These theorems are effective at finite entropy resolutions, and we provide the associated bounds for the primes and the perfect powers. The semigroup case may be interpreted as an invariance-free form of the Rudolph-Johnson theorem. We also show that an analog of Lyons' conjecture at full entropy dimension for every sublacunary multiplier set while failing by an arbitrarily large factor for the lacunary semigroup of powers of $3$. The proofs combine Fourier analysis with the algebraic structure of the multiplicative endomorphisms.

We construct a deterministic Farey-boundary algorithm converting a cyclic prefix table for an element of $\mathsf T$ into a word in addressed boundary flips. The reduction uses rational boundary pairs and cyclic order, with a marked edge determining the residual modular correction. Its unmarked stage uses at most twice the minimum number of diagonal exchanges. For input size $s$, the complete output has $O(s)$ factors, encoded length $O(s^2)$, and running time $O(s^3)$. Adjoining shuffles extends the construction to $\mathsf V$ with polynomial bounds. We also give explicit boundary generating sets and their local flip–shuffle relations. Changes of generators in the Lochak–Schneps and Bleak–Quick presentations yield a two-flip presentation of $\mathsf T$ and a presentation of $\mathsf V$ with two flips, one shuffle, and nine relations.

We investigate the pathwise asymptotic dynamics of a nonlocal lattice equation with fixed delay on $\ell^2(\mathbb Z)$ driven by an infinite-dimensional stationary weakly geometric rough path. Using a rough-flow Doss-Sussmann transformation and the method of steps, we establish global well-posedness in the controlled-path framework and construct a continuous random dynamical system. We further construct a tempered pullback absorbing family and use a spatially localized Lyapunov-Krasovskii functional, together with a sewing estimate for the spatial tails of the driving rough path, to prove that the lattice tails vanish uniformly. The existence of a unique random pullback attractor is then established by combining this lattice-tail estimate with an Arzelà–Ascoli compactness argument. Finally, for the delay-compatible compensated Euler schemes and finite-dimensional lattice truncations, we establish uniform absorbing and lattice-tail estimates, which yield the existence of the approximate random pullback attractors and their joint upper semicontinuity, without a coupling condition between the time step and the spatial truncation level.

In this paper, we study the weak limit of the invariant measures attached to an analytic family $\{f_t\}_{t\in\mathbb{D}^*}$ of regular polynomial automorphisms over $\mathbb{C}^N$ parametrized by the unit punctured disk which may degenerate at the origin, by using the techniques called hybrid spaces developed by Boucksom–Favre–Jonsson. For each $t$, the invariant measure $\mu_t$ is constructed by Sibony, and we show that the weak limit of $\{\mu_t\}_t$ over the hybrid space is given by the invariant measure of the regular polynomial automorphism $f_{\mathbb{C}((t))}^{\operatorname{an}}$ over $\mathbb{C}((t))$ induced from the original family. This is a generalization of the author's previous result for degenerating families of dynamics of Hénon mappings.

Machine-learning emulators of chaotic and stochastic systems are usually validated on forecast skill and long-run statistics. Neither certifies that an emulator responds correctly to forcing, the property that projection and attribution studies rely on. Linear response theory makes this testable: the forced response follows from unperturbed correlations through a generalized fluctuation-dissipation relation, and decomposes over the stochastic Ruelle-Pollicott resonances of the Koopman generator. Building on the Koopmanism Response framework, we turn this into a calibrated, mode-resolved test for learned surrogates: each surrogate rollout passes or fails each check, and failure rates are compared with those of independent realizations of the true system. On stochastic Lorenz-63, a three-variable toy model, we evaluate SINDy, an MLP, a reservoir computer, a neural ODE and a neural SDE with learned diffusion, over up to 80 rollouts each. A sparse-regression model with the correct library passes every check at rates consistent with the true system. Invariant-statistics fidelity and response fidelity dissociate in both directions: a quarter of reservoir-computer rollouts pass every invariant-statistics check and match the static susceptibility $\chi(0)$, yet misrepresent the slow relaxation modes, while the neural ODE and SDE rarely meet the invariant-statistics floor but recover those modes in three quarters of rollouts. As expected of a time-integrated quantity dominated here by fast relaxation, $\chi(0)$ does not separate these cases. For a fixed network, the training formulation (one-step drift, flow map, or multi-step through the integrator) decides which of these properties it gets right.

We answer, in the principal setting, a question of Matui on the almost finite–purely infinite dichotomy for minimal amenable ample groupoids. We prove that a second-countable Hausdorff minimal principal topologically amenable ample groupoid with Cantor unit space is strongly almost finite if and only if it admits an invariant probability measure; if no such measure exists, then it is purely infinite. We also prove that every bounded-degree uniformly Borel amenable Følner graph is Borel almost finite. The proofs combine the recent three-to-two comparison method of Glasner and Liu with the randomized Følner packing methods of Elek and Timár.

We count maximal cylinders on zero holonomy $\mathbb{Z}$-covers of genus $2$ square-tiled surfaces, up to $\mathbb{Z}$-action, obtaining quadratic asymptotics. We also show that the leading term of the asymptotic, called the Siegel-Veech constant, can be recovered via a large-genus approximation by intermediate finite covers. Our work applies to the infinite staircases introduced by P. Hubert and G. Weitze-Schmithüsen. For many members of this family, we explicitly compute the associated Siegel-Veech constants. In particular, we exhibit the first infinite family of examples of zero holonomy $\mathbb{Z}$-cover in which the number of cylinders grows sub-quadratically.

This survey studies C*-algebraic crossed products arising from topological dynamical systems with an eye toward their classifiability in the sense of the Elliott program. We introduce the crossed product construction for actions by discrete groups in full generality, and then focus on commutative systems to establish some of the fundamental structural results: nuclearity of the crossed product is equivalent to amenability of the action, and in this setting simplicity is equivalent to the combination of minimality and topological freeness. With these properties in place, the only condition left to translate into dynamical terms is tensorial absorption of the Jiang-Su algebra $\mathcal{Z}$. We present the main dynamical tools known to obtain $\mathcal{Z}$-stability, both in the settings of amenable and nonamenable groups, and highlight the two conjectures that are believed to capture the full picture. The aim of the survey is to provide both an accessible entry point and a comprehensive reference on the classification of crossed products, describing the state of the art and a number of central open problems in the field.

For residually finite groups, we study the automorphism and normalizer groups of odometers and their symbolic extensions: Toeplitz subshifts. In relation to orbit equivalence theory, we show that two continuous orbit equivalent $G$-odometers have commensurable normalizers when $G=\mathbb{Z}^d$, but that this result fails for other nilpotent groups $G$. The automorphism and normalizer groups of any Toeplitz subshift embed into those of its underlying odometer; despite this constraint, they exhibit considerable flexibility. For instance, we exhibit examples of low-complexity $G$-Toeplitz subshifts whose centralizer is isomorphic to $G$ (or, at the other extreme, restricted to its center) yet whose normalizer group is as large as possible.

We prove a noncommutative maximal ergodic inequality for averages along polynomial sequences. Let $\gamma$ be a trace-preserving automorphism of a semifinite von Neumann algebra $(\mathcal N,\tau)$. We show that the associated polynomial averages \begin{equation*} A_Nf:=\frac1N\sum_{n=1}^N\gamma^{P(n)}(f), \qquad N\in\mathbb N, \end{equation*} satisfy a strong maximal inequality on $L_p(\mathcal N)$ for every $1<p<\infty$, extending the previously known restricted range of $p$. The proof follows Bourgain's major-arc strategy but requires substantially new ideas and tools for operators that may have further applications in noncommutative analysis. More precisely, we obtain a localized maximal inequality by developing a noncommutative version of Stein's extrapolation, novel combinatorial methods, and a surprising multilinear version of Doob's maximal inequality. For the required decaying $L_2$-approximation, we combine two of Bourgain's constructions in a way that avoids the multi-frequency maximal inequality used in the scalar proof.

Let $\mathcal H$ be the space of support functions of centrally symmetric, strictly convex, compact planar bodies whose boundary is a $C^1$ curve, endowed with the $C^1$ topology. We prove that for a residual subset of $\mathcal H$ the outer billiard maps about the associated bodies admit unbounded orbits escaping to infinity. In particular, this residual set includes support functions arbitrarily $C^1$-close to those of disks or ellipses. This provides an affirmative answer, in the strictly convex $C^1$ setting, to a famous question of Moser and Neumann concerning the existence of unbounded outer-billiard orbits. The key ingredient is a rigidity theorem: if $h\in\mathcal H$ has a purely singular radius-of-curvature measure, then every continuous invariant tangent graph consists entirely of periodic points. We then show that for a generic table in $\mathcal H$ these periodic invariant graphs are absent, allowing us to construct escaping orbits via Mather's diffusing mechanism in a Birkhoff region of instability.

Let $d\geq2$ be an integer, and let $\operatorname{MPoly}^d$ be the moduli space of degree-$d$ polynomials. For every number field $K$, we prove that there exists $\epsilon_{K,d}>0$ such that \[ \left\lbrace\alpha\in\operatorname{MPoly}^d(K^{\mathrm{ab}})\colon h_{\mathrm{crit}}(\alpha)<\epsilon_{K,d}\right\rbrace=\left\lbrace\alpha\in\operatorname{MPoly}^d(K^{\mathrm{ab}})\colon h_{\mathrm{crit}}(\alpha)=0\right\rbrace \] is finite, where $h_{\mathrm{crit}}$ is the critical height. In particular, only finitely many $K^{\mathrm{ab}}$-rational points of $\operatorname{MPoly}^d$ are postcritically finite. The proof combines the correspondence method of Ji–Song–Xie with local ramification estimates and the uniform finiteness of intertwined classes in $\operatorname{MPoly}^d$. We also prove an analogous finiteness theorem for non-preperiodic marked points in non-isotrivial one-parameter polynomial families.

We study strong completeness and the existence of a set attractor for the stochastic flows induced by a class of two-dimensional stochastic differential equations resembling a planar Ornstein-Uhlenbeck process with an additional radius-dependent rotational drift term. Our main results give both sufficient and necessary conditions for the stochastic flows to be strongly complete and for the existence of set attractors. In particular, we demonstrate that if the derivative of the angular velocity $\rho(r)$ with respect to the radius $r$ satisfies $|\rho'(r)| \geq K_2\, r^3$, for large $r$ and some universal constant $K_2$, the diameter of a compact set can grow exponentially fast with positive probability. Furthermore, we show that for $|\rho'(r)|\geq r^{3+\varepsilon}$, $\varepsilon>0$, with probability one, exceptional initial conditions diverge to infinity in finite time, ruling out the existence of a global stochastic flow.

The dynamics of chemical reaction systems with two chemical species can be described (under mass-action kinetics) by planar autonomous systems of ordinary differential equations (ODEs) with right-hand sides containing polynomials. While their behaviour close to hyperbolic critical points is well understood, their dynamics has not been fully characterized for strongly non-hyperbolic critical points, where the linearization close to the equilibrium is given by the zero matrix. In this paper, the behaviour of such chemical reaction systems is investigated. Considering second-order chemical reaction networks with a positive equilibrium point, we show that only nine geometric equivalence classes are possible out of the sixteen classes for general ODE systems. Examples of chemical reaction networks in each class are presented. All sixteen geometric equivalence classes can be realized by chemical systems if the non-hyperbolic equilibrium point is located at the origin.

Regional weighted phase-volume contraction for nonlinear systems is studied using positive densities selected from a finite-dimensional log-affine family. The affine dependence of the density divergence on the design parameters turns density selection into a semi-infinite convex optimization problem that maximizes a certified uniform contraction margin on a compact region. This provides a constructive alternative to testing a preselected density and ensures a well-posed design problem under compact parameter constraints. Finite active-point and minimax characterizations identify the worst-case states governing the optimum, while rigorous sampling bounds allow continuum contraction guarantees to be recovered from finitely many inequalities. For scalar power densities, feasibility is characterized by an exact interval condition. The same structure is preserved in discrete time and admits distance-weighted and conformal-geometric interpretations. Overall, the framework provides computable certificates for detecting regional contraction that may remain hidden from ordinary divergence. A two-parameter limit-cycle example demonstrates this effect when both ordinary phase volume and a natural single-basis density fail.

The evolution of weighted phase volume under a fixed dynamical system is investigated using a positive weight raised to an arbitrary real exponent. Sufficient conditions for uniform exponential contraction and expansion of transported weighted volume are obtained in terms of the corresponding weighted divergence. These conditions make it possible to reveal dissipative properties that may remain undetected by the ordinary divergence. Consequences for invariant sets are established, and, under the assumption of a compact absorbing set, the weighted measure of a global attractor is characterized. The approach is extended to invariant submanifolds through a distance-weighted ambient-volume estimate. A differential-geometric interpretation is also provided, in which the weighted divergence is identified with the ordinary divergence associated with a conformally transformed metric. The dependence on the exponent is used to reveal a duality between mutually inverse weights. In addition, sufficient local conditions for exponential attraction to an invariant hypersurface are derived using a sign-changing weight. Weighted-volume decay is interpreted as an integral property of transported sets and, by itself, is not identified with pointwise Lyapunov stability.

Let $f$ be a regular endomorphism of $\mathbb{A}_{\mathbb{C}}^N$ of algebraic degree $d$ and let $f_{\infty}$ be the induced endomorphism of the infinity hyperplane $H_{\infty}$. Suppose that for every periodic point $x_0\in H_{\infty}(\mathbb{C})$ of period $n_0$, the geometric mean of the multiplicities $e_{f_{\infty}}(x_0),\dots,e_{f_{\infty}}(f_{\infty}^{n_0-1}(x_0))$ is strictly less than $d$. Then we show that the dynamical Mordell–Lang conjecture for $f$ holds for curves in $\mathbb{A}_{\mathbb{C}}^N$.

We introduce temporal metric spaces, a framework for formalizing metric and topological structures on continuously evolving domains. Rather than defining a single metric on a fixed space, we construct a family of interval-dependent metrics on an associated trajectory space, enabling distances to be measured between continuous trajectories over finite time intervals. We establish the fundamental topology of the trajectory space, prove that it inherits completeness from the underlying metric space, and develop the corresponding temporal hyperspace together with its completeness. Finally, we develop a theory of iterated function systems acting on these evolving spaces, proving that their attractors evolve coherently with the underlying dynamics and remain stable even when the evolving geometry causes the generating mappings to lose their classical contraction property.

Mean-field games provide a continuum framework for modeling the dynamics of large, interacting populations of non-cooperative agents. This paper studies symmetry-breaking in a finite-horizon, two-choice min-LQG mean-field game in which identical agents with linear stochastic dynamics choose one of two equally desirable terminal destinations, while trading off control effort against social pressure to conform. The model has an odd symmetry between the two destinations and therefore always admits a symmetric, dynamic Nash equilibrium in which the population splits evenly between them, producing a deadlock collective state at final time. Numerical studies have suggested that, as the penalty for social nonconformity increases, this symmetric equilibrium loses stability as a fixed point of an associated scalar, self-consistency map, and asymmetric consensus Nash equilibria emerge, where most agents select the same destination. By analyzing the linearized forward-backward PDE system through the scalar map representation, we provide a proof of this loss of stability of the symmetric equilibrium. Together with the odd symmetry of the map, this implies the existence of symmetry-broken mean-field game equilibria corresponding to consensus on either destination.

Lyapunov functions have been a cornerstone of dynamical systems theory ever since the early 1900s. Moreover and akin to Morse theory, Lyapunov functions have been the key in linking topology to dynamical systems theory. In this note we continue along these lines. Suppose that some vector field on Euclidean space has a unique equilibrium point that is globally asymptotically stable (GAS). Under these conditions, it is known that there is always a smooth Lyapunov function to certify stability. One may wonder if this function can always be chosen to be Morse. We show that if the equilibrium point is hyperbolic, this is indeed true. Moreover, we show that if this would be true in general, then the generalized 4-dimensional smooth Poincaré conjecture (SPC4) must be true. Even so, we provide an example that is GAS, but does not admit a Morse Lyapunov function, closing this route of proving SPC4.