Papers

We study foliations which force every transverse flow to have a closed orbit. We fully characterize this property among $C^2$ finite depth foliations on atoroidal 3-manifolds. We also show that for many foliations, this property can be broken by a $C^0$-small modification of the foliation. We produce closed orbits using the theory of endperiodic maps, and we construct $C^1$ flows without closed orbits by inserting dynamical plugs transverse to foliations along curves of attracting holonomy.

Nonlinear parabolic partial differential equations frequently occur in the modeling of applied phenomena. They often exhibit complicated dynamics which is organized by their associated equilibrium solutions. As a function of the involved model parameters, the equilibria can be detected using a bifurcation analysis, which in turn is centered around a deeper understanding of bifurcation points. In many applied systems establishing these organizing centers rigorously is difficult to achieve using traditional mathematical techniques. In this paper, we propose a theoretical framework for computer-assisted proofs of symmetry-induced transcritical and pitchfork bifurcations in nonlinear problems. This s accomplished by reformulating the existence question via a zero finding problem for an associated extended nonlinear system. The approach is general and allows for a wide variety of underlying symmetries. We demonstrate how these results can be applied in the setting of parabolic systems and higher-order equations, on both one- and two-dimensional domains.

This work provides a comprehensive characterization of the first dynamic transition of a general two-component reaction-diffusion system on a bounded interval in one spatial dimension. By rigorously addressing the difficulties arising from the non-linear interactions between the two components, this study significantly extends the scope of previous studies on dynamic transitions of reaction-diffusion systems, which focused on single-equation cases. The analysis is conducted by employing center manifold reduction within the standard classification of dynamic transition theory. A key contribution of this study is that the nonlinear terms are treated to involve arbitrary powers of the unknowns $u$ and $v$. Furthermore, the theoretical framework is complemented by the analysis of a representative and simplified system. As a result of this generalization, the dynamics of a broad class of fundamental reaction-diffusion systems can be systematically classified.

We construct an automorphism of a smooth projective K3 surface with positive topological entropy and a non-empty Fatou set. The Fatou set contains invariant domains biholomorphic to products of two annuli. On each domain, the automorphism is holomorphically conjugate to a fixed-point-free irrational rotation. These domains are two-dimensional analogues of Herman rings. The construction answers a question posed by McMullen in 2002 and several questions posed by Cantat in his 2018 ICM address.

A wide range of methods have been proposed, including physics-informed neural networks, which are powerful but do not guarantee identifiability of the dynamics, symbolic regression, which requires a set of precomputed operations, and causal discovery, which is more principled but usually relies on strong assumptions that physical systems may violate. In this work, we develop a theory-grounded method and prove that under a set of permissive assumptions, the structural drivers and drift of stochastic delayed differential equations are identifiable. Our method outperforms others on a benchmark for driver identifiability, and on a second benchmark to evaluate physical consistency of the learned dynamics.

In a continuous coordination game, under myopic best response, players change strategies, and thus strategic communities change their shapes in time. We show that the boundaries of these strategic communities evolve according to Free Boundary Mean Curvature Flow in the continuous time limit. The convergence relies on an approximation scheme for the Mean Curvature Flow, which has previously been used to describe other threshold dynamics. With this equivalence, we can characterize Nash equilibria of the continuous coordination game as minimal surfaces and explore related consequences in biased versions of the same game.

We discuss some contexts in which the topological dynamics of certain Anosov subgroups of higher-rank Lie groups on "bad" subsets of flag manifolds can be analyzed using results from homogeneous dynamics in the infinite-covolume rank-one setting. For example, we show that a group of projective transformations dividing a strictly convex domain in projective space and intersecting Zariski-densely the stabilizer of an ellipsoid has a dense orbit in the complement of the domain, and acts minimally on the space of full projective flags tangent to the domain. This accounts for all known examples of divisible strictly convex domains in sufficiently high dimensions. We also provide examples of Zariski-dense groups that are Anosov in a partial flag manifold but such that the equivariant projection from the Benoist-Guivarc'h limit set in the Furstenberg boundary to the Anosov limit set is not a fibration.

We prove that every codimension-one holomorphic Anosov diffeomorphism of a compact connected complex manifold is biholomorphically conjugate to a hyperbolic automorphism of a complex torus. This verifies a conjecture of Ghys in the codimension-one case and, in particular, in complex dimension three.

We supply the details of the subcritical dry Ten Martini statement announced in [6, Remark 1.3]: for every irrational frequency, every subcritical Type I energy with resonant rotation number is an endpoint of an open spectral gap. The proof combines the all-frequency Puig argument developed there with quantitative almost reducibility. A different proof was recently given by Li [7].

We prove combinatorial rigidity for the full renormalization attractor of neutral quadratic polynomials. More precisely, any two bi-infinite renormalization towers with the same combinatorics are conformally conjugate on neighborhoods of their Mother Hedgehogs. The rigidity theorem includes arbitrary irrational combinatorics and respective parabolic enrichments. The proof relies on a comprehensive analysis of neutral cascades, i.e. transcendental dynamical systems arising from the rescaled limits of the first return maps of neutral quadratic polynomials. We establish uniform butterfly bounds for neutral cascades and prove that any two combinatorially equivalent neutral cascades are affinely conjugate. We also prove rigidity for parabolic towers with equivalent backward combinatorics.

Rempe and Rippon gave a general result which enables the construction of a transcendental entire function with a Baker domain, in which the iterates tend to infinity at a prescribed arbitrarily slow rate. Here we construct a transcendental entire function with a Baker domain, in which the iterates are controlled from both above and below in a prescribed arbitrarily slow but regular manner. The proof uses distortion theorems for conformal mappings due to Ahlfors and Warschawski, and an approximation theorem due to Rempe and Rippon.

Motivated by the pioneer work of Viswanath \cite{VIS2000} on random Fibonacci sequences and the development of random Euclidean addition chains in cryptography, we introduce a new family of random Binomial-Fibonacci sequences. We study the asymptotic behavior of their finite-time Lyapunov exponents. In the special case of random Bernoulli-Fibonacci sequences, we show that they convergence almost surely to an explicit Lyapunov exponent and prove a central limit theorem with an explicit formula for the variance. We also show that they satisfy a large deviation principle.

In a referential that rotates with Earth, the dynamics of the configurations of satellites in a Delta Walker constellation can be analyzed as a dynamical system as a function of a translation on the torus. This paper extends this type of analysis to routing over a Walker constellation, leveraging fixed ground relays. It considers any deterministic, time-invariant routing rule on such a collection of satellites and relays. The main object of interest is the routing table associated with this rule, which specifies, for all source-destination pairs, a route made of satellites and relays between them, for instance the shortest, together with angular information on next hop at each step of the route, which is essential for beamforming in this context. The main result is that, for each such routing rule, the routing table inherits the dichotomy of the torus flow: when the ratio of the Earth-spin and satellite angular speeds is rational, the routing table process is periodic with the same period as the constellation. When it is irrational, the table is non periodic but admits ergodic long-run averages which can be computed as ensemble averages w.r.t. an explicit invariant measure. Simulations with mixed satellite–gateway greedy routing illustrate these findings in terms of both temporal and spectral properties of the Walker routing table time series. The paper also discusses how to define metrics that cope with such non periodic fluctuations when present. This is illustrated by a comparison of the round-trip times between two far away cities as offered by a Walker constellation using greedy routing on one side, and by the currently available terrestrial fiber on the other side. It is shown how ergodicity can be used to make this comparison between the time varying instantaneous round trip times of the constellation and the constant round trip time of the fiber network meaningful.

We study the Siegel–Veech transform on the Poincaé section for the horocycle flow consisting of lattice surfaces with a visible horizontal holonomy vector of length at most one. We compute its first moment and derive formulas for higher and factorial moments. We emphasize these formulas in the case of the square torus of unit area and, as an application, use its first moment formula to recover the Boca–Zaharescu pair-correlation density for Farey fractions.

This paper investigates analytical solutions of a continuous analogue of a class of Volterra quadratic stochastic operators of a two-sex population using the homotopy analysis method. The positive invariance of the state space of the system is established. By means of Lyapunov functions, inclusions for the $\omega$-limit sets are obtained, together with exponential decay estimates for individual components of the trajectories under the corresponding conditions on the coefficients of the system. Sufficient conditions are obtained for the coincidence of the $\omega$-limit sets of the discrete and continuous models. For the diagonal invariant subset, sufficient conditions are established for the periodicity of all interior non-equilibrium trajectories. By means of a majorant series, a sufficient condition for the absolute and uniform convergence of the homotopy series on $[0,+\infty)$ is obtained under the Hurwitz stability of the reduced Jacobian of the system. In addition, explicit analytical solutions of the Cauchy problem are found for special relations between the heredity coefficients. Computational experiments show a rapid decay of the terms of the homotopy series and good approximation accuracy already when using the first few terms of the expansion.

We prove a Denjoy-Wolff theorem for uniformly quasiregular self-maps of the unit ball in $\mathbb{R}^n$, $n\geq 2$: either all orbits are relatively compact, or the iterates converge locally uniformly to a single point of the sphere. No properness, surjectivity, or boundary extension is assumed. The key tool is a Schwarz-Pick lemma valid in all dimensions: every quasiregular map preserving a bounded measurable conformal structure is non-expanding for a modulus metric built from that structure, and uniformly quasiregular maps preserve such a structure by a theorem of Iwaniec and Martin. Since this metric is not geodesic, we pass to an associated chain metric, which we show is Gromov hyperbolic with boundary the sphere, and apply a theorem of Karlsson. As an application, a stable Fatou component of a uniformly quasiregular map of $\overline{\mathbb{R}^n}$ that is a uniform domain with uniformly perfect boundary is an attracting basin, a rotation domain, or a parabolic basin with a unique boundary fixed point. Non-injective examples of both escaping types are constructed in every dimension.

Let $G$ be a group acting continuously on a compact metric space $X$. This naturally induces an action of $G$ on the space $\mathcal{K}(X)$ of nonempty compact subsets of $X$. In [KS26], Kra and Schmieding define a system $(X,G)$ to be IC-rigid if the union of the supports of the $G$-invariant Borel probability measures on $\mathcal{K}(X)$ is as small as possible. They leave open the question of existence of IC-rigid abelian systems. We show the existence of IC-rigid actions for three classes of abelian groups: mixing rank-one $\mathbb{Z}$-actions, horocycle flows, and certain mixing actions on a Cantor space of $G=\bigoplus_{n=1}^{\infty}G_n$, where $(G_n)_n$ is any sequence of nontrivial finite abelian groups. In the course of proving these results, we introduce the notion of weakly asymptotic pairs and establish its close connection to topological minimal self-joinings.

We present Suêtes, a fully differentiable, non-hydrostatic, limited-area atmospheric dynamical core implemented in JAX. Suêtes includes both a semi-implicit semi-Lagrangian scheme and a Split-Explicit Runge-Kutta scheme with acoustic substepping, enabling applications from coarse regional simulations to convection-permitting dynamics. Its terrain-following geometry, spatial discretization, time integration, initial conditions, lateral boundary treatment, and included physical parameterizations are all compatible with reverse-mode automatic differentiation. A single reverse-mode pass can therefore compute sensitivities of forecast diagnostics to selected initial and boundary fields, physical parameters, topography, and computational geometry. We provide a series of numerical experiments—including a tracer-source inverse problem, a sensitivity analysis of a three-dimensional squall-line, adversarial perturbation construction an ERA5-driven downslope hurricane force wind event, topography optimization, and terrain-following coordinate optimization—demonstrating the versatility of Suêtes as a framework for gradient-based diagnosis and model development at regional and convection-permitting scale.

It is well-known that Douady and Hubbard proved that the Mandelbrot set is connected by constructing a uniformizing holomorphic map for its complement from the Böttcher coordinates. This strategy has a clear analogue in the setting of quasiregular Mandelbrot sets. In this paper, we show that the natural generalization of the Douady-Hubbard uniformizing map is quasiconformal in a neighbourhood of infinity, as well as, in fact, outside a certain neighbourhood of the quasiregular Mandelbrot set. On the other hand, we also show that this uniformizing map can reverse orientation at some parameters, which forces it to have folds and, in particular, prevents it from being injective. This feature cannot happen in the holomorphic setting. The connectivity of the quasiregular Mandelbrot sets therefore cannot be established along the Douady-Hubbard route, and remains open.

We study the dynamical behavior of tokens in transformers from a control-theoretic perspective. Our model includes the feed-forward layer present after the self-attention mechanism, with the self-attention mechanism interpreted as an interacting particle system and the feed-forward layer as an independent control. Our main theoretical result establishes that the feed-forward network can steer the tokens arbitrarily close to consensus regardless of the key, query, and value matrices. Our result are easily extended to convergence to many clusters and to multi-head attention. We conduct numerical experiments to verify our results, and compare thresholding behavior from our theory to real-world LLMs.

Let $f$ be a Hénon map of $\mathbb C^2$. We provide quantitative versions of several results involving dynamical objects associated to $f$. Our results can be interpreted as a quantitative version of Pesin theory with geometric control. As an application we show that the perdiodic points of period $n$ of $f$ equidistribute towards its equilibrium measure exponentially fast as $n$ tends to infinity.

We develop a general criterion for establishing absolute continuity of convolutions of fractal measures on the line, and more generally of prescribed line projections of fractal measures in the plane. The crucial new ingredient is exact scaling covariance of the \(L^1\)-norm of Littlewood-Paley pieces of the projected measure, under affine renormalization. We apply this criterion in three key settings. First, we show that the convolution of two measures on the line is absolutely continuous whenever their dimensions sum to more than one, provided one is a self-conformal measure whose defining IFS is not \(C^2\)-conjugate to linear, and the other is either self-conformal or Ahlfors–David regular. Second, we show that every line projection of a planar complex-analytic self-conformal measure of dimension greater than one is absolutely continuous, under natural nonlinearity and nondegeneracy assumptions. Finally, we show that every line projection of a planar self-affine measure is absolutely continuous, under natural irreducibility and proximality assumptions, whenever the correlation dimension of the measure and the Frostman dimension of its Furstenberg measure sum to more than two.

While structure-preserving methods for the forward simulation of Hamiltonian systems are well established, the mathematical foundations of their data assimilation and filtering counterparts remain comparatively less developed, particularly regarding rigorous accuracy guarantees. To address this gap, we propose an online filtering algorithm to reconstruct an unknown function $u^\dagger$ from finitely many measurements, assuming that $u^\dagger$ solves a parametric Hamiltonian PDE with unknown inputs. The method uses low-dimensional symplectic approximation spaces that evolve in time according to the PDE model and informed by the measurement data. The method can also be coupled with a dynamical sensor-placement strategy designed to optimize reconstruction stability. We derive error bounds in terms of stability and best approximation, and show that the reconstruction remains in a symplectic space and preserves energy up to the approximation error. Numerical experiments demonstrate the effectiveness of the method and the benefits of dynamic over static sensor placement.

We prove that for every choice of four positive masses and every cyclic ordering of the bodies there is exactly one strictly convex planar central configuration with that ordering, up to similarity. This answers Problem 10 in the list of Albouy, Cabral and Santos, which Santoprete calls the Simó-Yoccoz conjecture. The main step is a uniform lower bound for the Hessian at convex central configurations: it is at least one quarter of its radial part, which vanishes only on translations and rotations. Dziobek's relations turn the indefinite part of the Hessian into a negative multiple of a square, and a Cauchy-Schwarz argument bounds this term by the radial part times the trace of an explicit $2\times 2$ matrix in which the masses do not appear. We prove that this trace is less than 3/4 by interval arithmetic on the three-dimensional set of normalized convex central configurations, in the coordinates of Corbera, Cors and Roberts. Uniqueness then follows by a covering argument from the case of four equal masses. The computation has been repeated with a second, independently written program. The Hessian bound and the uniqueness theorem, including the computation, have been formalized in the Lean proof assistant and checked by its kernel, using only the standard axioms. As consequences, every degenerate four-body central configuration is concave, the convex central configuration depends analytically on the masses, and the known symmetry theorems for kites, isosceles trapezoids and rhombi follow in a few lines. The analytic dependence and the symmetry theorems are also formalized.

We show that the degree $d$ Jouanolou foliation $\mathcal J_d$ on $\mathbb P^2$ is structurally stable for every $d \geq 2$. We prove the existence of a neighbourhood $\mathcal U_d$ of $\mathcal J_d$, such that every foliation $\mathcal F \in \mathcal U_d$ satisfies: the Fatou set is a connected disc bundle over a compact Riemann surface of genus $d(d+1)/2$, no leaf is dense in $\mathbb P^2$, and all but countably many regular leaves are biholomorphic to the unit disc. Finally, we prove transverse perfectness of the Julia set on the regular locus for $d \geq 2$.

Let $\Gamma$ be a countable discrete group and let $A$ be a unital $\Gamma$-simple $\Gamma$-$C^*$-algebra. We prove that $A\rtimes_r\Gamma$ has the generalized Powers averaging property if and only if the canonical map $A\rtimes_r\Gamma\to I_\Gamma(A)$ is the unique $\Gamma$-equivariant pseudo-expectation. We also show that this averaging property lifts to $I_\Gamma(A)\rtimes_r\Gamma$, with the averaging coefficients still belonging to $A$.

We prove that the postcritical set of every quadratic polynomial with a neutral fixed point has zero Lebesgue measure. In particular, no arithmetic condition on the rotation number is required. Our proof uses sector renormalization and the pseudo-Siegel bounds of Dudko-Lyubich to derive uniform distortion estimates for the renormalization change of variables. For sector renormalization towers, we also prove the same zero area statement for the associated postcritical set and the optimal Brjuno criterion for the interior of the associated Mother Hedgehog.

We study the problem of extracting reliable and interpretable linear models from nonlinear dynamical systems with periodic behavior. Dynamic Mode Decomposition (DMD) is a widely used data-driven method for approximating dynamics by a linear system and is commonly interpreted through Koopman operator theory. However, because standard DMD assumes linear evolution in the chosen observables, it can produce misleading results on nonlinear systems. Recent extensions address this limitation by modeling the dynamics as a linear system driven by a nonlinear forcing term, but their relationship to Koopman theory and their reliability remain unclear. We observe that the Koopman theoretical framweork allows for explicit, finite-dimensional, forced linear representations of dynamical systems, independently of the chosen observable space. The main question is therefore how to choose a useful, computable, and interpretable model for the forcing term, since forced linear representations are generally nonunique. In the framework considered here, this modeling question is related to the choice of a defect space: a complementary space that closes the action of the Koopman operator on the chosen observables. Existing forced-DMD approaches can be related to different modeling choices for this defect space, where the structure of the learned linear operator depends on the choice of the defect space. As an example of defect-space modeling for algorithm design, we introduce Correlation-Basis enhanced DMD (CB-DMD), a data-driven method whose goal is to learn an effective autonomous linear operator for periodic nonlinear dynamics. The forcing term is used to absorb nonlinear components that cannot be represented by the learned linear operator. Experiments on three nonlinear periodic systems demonstrate that CB-DMD yields more reliable linear models than standard DMD and related variants in the tested examples.

We study statistical limit theorems for time-series generated by multidimensional random fields that are finitary factors of iid random processes. Based on the decay of the tail of the coding radius, we establish the Central Limit Theorem with explicit rates of convergence, as well as the weak convergence to a Brownian sheet, for suitably regular functions. Our results can be applied to a variety of classical models in statistical mechanics, such as the Ising model, the hard-core model, and proper colourings.

The Katok examples on $S^2$ are induced by Randers metrics obtained by perturbing the round metric by the standard rotational Killing field: allowing a scaling factor of the round metric gives a two-parameter family of Randers metrics $F_{\alpha,\beta}$, parametrised by positive real numbers $(\alpha,\beta)$. We compute the set of all $(\alpha,\beta)\in (0,2+\sqrt{3})^2$ for which $D^*(S^2,F^*_{\alpha,\beta})$, the unit codisc bundle with respect to $F^*_{\alpha,\beta}$, symplectically embeds into the round codisc bundle $D^*S^2$. This set has the structure of an infinite staircase. We also establish a dictionary between embeddings of these codisc bundles, singular $A_1$-ellipsoid embeddings, and $\mathbb{Z} _2$-equivariant ellipsoid embeddings, showing that these embedding problems are equivalent. Furthermore, we prove the analogous results for the $\mathbb{R} P^2$ case and discuss a broader family of examples for which this dictionary applies.