Papers

Non-degenerate periodic orbits in three-dimensional Reeb flows are classified into three types: positive hyperbolic, negative hyperbolic and elliptic. In the present paper, we consider a closed connected contact three-manifold with a non-degenerate contact form. We show that its Reeb flow has a simple positive hyperbolic orbit if it has at least three simple periodic orbits. We mainly study the case in which no elliptic orbit exists. We prove that there is no non-degenerate contact form on a closed connected three-manifold with $b_1=0$ such that all simple periodic orbits are negative hyperbolic. As a corollary, by combining the author's previous result in the presence of an elliptic orbit and the known result for $b_1>0$, we obtain the main result. The proof uses the Weyl law for ECH spectral invariants. We also use compactness for genus zero $J$-holomorphic curves counted by the $U$-map. Under the contrary assumption, the number of simple orbits in each action interval $[L,2L]$ is uniformly bounded with respect to $L$. We use this property to study ECH generators and genus zero $U$-curves.

We establish the dynamical Mordell–Lang conjecture over the complex numbers for a family of radially ramified polynomial skew products in arbitrary base dimension and fiber rank. We assume that one fixed iterate sends the vertical critical locus into the invariant zero section. Our result covers affine-linear bases and, under a degree-gap condition, nonlinear étale polynomial bases. We also establish a uniform Skolem–Mahler–Lech theorem for a fixed nonlinear orbit and a fixed polynomial-coefficient linear recurrence whose trailing coefficient is a nonzero constant. The theorem applies to joint polynomial and rational relations involving the orbit and finitely many successive terms of any solution. A single eventual period works for every such solution and relation, although the finite exceptional set may depend on the relation. Our proof combines controlled nonarchimedean realizations with an orbitwise transfer theorem for superattracting line-bundle dynamics.

We identify the sharp iterated-logarithmic thresholds at the critical exponent for quantitative measure equivalence and orbit equivalence between integer lattices. To be more precise, let $n>m$ be two positive integers, $\alpha$ be a positive number and $( \beta_j)_{j \geqslant 1}$ be a finitely supported sequence of non-negative numbers. We show that there is a quantitatively $t^{\alpha}\cdot \prod_{j \geqslant 1} ( \log^{(j)}{t} )^{-\beta_j} $-integrable measure equivalence from $\mathbb{Z}^n$ to $\mathbb{Z}^m$ if and only if either $\alpha<m/n$ or $\alpha=m/n$ and there is $j_* \geqslant 1$ such that $\beta_j= 1$ for all $1 \leqslant j < j_*$ and $\beta_{j_*} > 1$. Here $\log^{(j)}$ is the $j$-fold iterated logarithm. The same characterization holds for quantitative orbit equivalence. This characterization greatly strengthens the previous best-known result, due to the work of Delabie, Koivisto, Le Maître and Tessera (2022) and the work of Correia (2025), which asserts that there is a $t^{\alpha}$-integrable ($\alpha>0$) measure equivalence (or orbit equivalence) from $\mathbb{Z}^n$ to $\mathbb{Z}^m$ if and only if $\alpha<m/n$. In particular, our result solves, in much stronger forms, two open problems posed respectively by Delabie, Koivisto, Le Maître and Tessera, and by Naryshkin and Petrakos.

We solve the robust ten martini problem: Cantor spectrum with no arithmetic restriction on the irrational frequency for Type I analytic one-frequency quasiperiodic Schrodinger operators. We introduce Type I through a new acceleration-type quantity with stability under analytic perturbations. This yields a global open class of analytic potentials whose associated operators have Cantor spectrum for every irrational frequency.

Mather's $\beta$-function associates with each rotation number the least average action carried by an orbit of a twist map. For analytic standard maps in the KAM regime, we prove the following. For any family of $d \ge 1$ directions of perturbation satisfying a natural symmetry condition, the jet of the $\beta$-function at a single algebraic Diophantine rotation number locally determines the potential for an open and prevalent, hence dense, set of base potentials in the KAM domain. In particular, we obtain that for a residual and prevalent set of even potentials in the KAM domain, every real analytic deformation with finite Fourier support that preserves the full jet of the $\beta$-function at a fixed algebraic Diophantine rotation number is constant. These results are non perturbative within the KAM domain. As an intermediate result of independent interest, we establish a KAM theorem giving joint $C^\infty$–holomorphic dependence of the invariant curves, and hence of the $\beta$-function, on complex holomorphic potentials and on a domain of complexified rotation numbers whose boundary contains real Diophantine numbers.

We discover an exponent-dependent phase transition phenomenon for optimal hypercontractivity: for every prescribed $q_0>2$, there exists a reversible continuous-time Markov chain on three state space with normalized spectral gap whose $(2,q)$-optimal hypercontractivity time satisfies $$ \text{$t_{\mathrm{opt}}(2,q)=\frac12\log(q-1)$ if and only if $q\ge q_0$},$$ whereas the strict inequality $t_{\mathrm{opt}}(2,q)>\frac12\log(q-1)$ holds for $2<q<q_0$.

We developed a mathematical model to describe the inflammatory dynamics during arthritis development and resolution, focusing on the interaction between three key immune cell populations within the joint tissue, namely neutrophils (N), sublining synovial tissue-resident macrophages (SL-TRM, T), and circulating monocytes that are recruited into the inflamed joint and differentiate into monocyte-derived macrophages (MoMac, M). An inflammatory signal (S), representing the response to an external arthritogenic stimulus, drives the recruitment and activation of neutrophils and MoMac, while SL-TRM exert a regulatory and protective role. We propose a minimal ordinary differential equation model of acute inflammatory response into a joint that couples an inflammatory signal S to the three cell populations (N, T, M) and incorporates recruitment, stimulation, nonlinear self-limitation and mutual regulation. We show that the equilibrium structure depends on three adimensional parameters A, B, C and derive a cubic equation characterizing nontrivial steady states. We perform an analytical study of equilibria and their stability, and identify a bifurcation at A = 1 separating a disease-free regime from a sustained inflammatory state. Numerical simulations illustrate typical transient dynamics under various scenarios and show the ability of the model to capture the transition from acute to chronic inflammation, as well as the resolution of inflammation under certain conditions. In conclusion, our model provides a framework to explore mechanisms underlying acute versus sustained joint inflammation and offers a basis for parameter estimation from experimental data, as well as future extensions toward evaluation of therapeutic strategies.

Given a quadratic polynomial with an irrationally indifferent fixed point of bounded type, a bubble is an iterated preimage of its Siegel disk. Similar to pseudo-Siegel disks, one constructs pseudo-bubbles from bubbles by filling in parabolic fjords. We introduce the near-degenerate regime for the $\alpha'$-Dynamics controlling the geometry of these pseudo-bubbles in deep scale. Consequently, we obtain uniform bounds on the size and regularity of pseudo-bubbles independent of the rotation number. In [DLL], jointly with Lyubich, this result serves as a necessary ingredient for establishing uniform butterfly bounds and ultimately the combinatorial rigidity of the full attractor of neutral renormalization.

Recurrent Transformers reusing their weights rather than stacking $L$ distinct layers are becoming widely adopted due to their parameter efficiency [1,2,3]. However, the exact representational and dynamical differences between looped and stacked architectures remain uncharacterized. This paper presents a controlled study on the example of bViT model [1] applying one weight-tied block $L$ times. We train two models: bViT and standard ViT [4] on 12-lead electrocardiogram (ECG) classification tasks from the PTB-XL dataset under identical training protocols. Despite an $8.9\times$ parameter reduction, bViT achieves accuracy parity with ViT. Geometric similarity metrics demonstrate that both architectures construct comparable latent representations in an equivalent canonical order. Crucially, their dynamics differ: bViT exhibits smaller step sizes and inter-patient sensitivity, as well as near-neutral behavior away from the data manifold, whereas ViT exhibits collapsing dimensionality of representations and out-of-distribution feature expansion.

In this paper, we investigate finite-time blow-up solutions for distributed delay differential equations incorporating memory effects with a specific gamma distribution kernel. Focusing on typical nonlinear terms that cause finite-time singularities in ordinary differential equations (ODEs), we examine how memory effects change the existence, rate, and qualitative properties of blow-up. By comparing these behaviors with those of the corresponding non-delayed and memory-free ODEs, we show that time delay originating from memory not only essentially induces finite-time blow-up (“memory-induced blow-up”), but also drastically alters the blow-up profile, such as accelerating algebraic blow-up rates or driving a qualitative transition from ODE quenching to logarithmic blow-up. Furthermore, by incorporating a self-inhibitory term, we reveal a threshold phenomenon in the phase space that governs the occurrence and non-occurrence of blow-up depending on the initial conditions and parameters. These results are established by reducing the system to a two-dimensional ODE via the linear chain trick, and analyzing the dynamics at infinity using Poincaré-type compactification, blow-up techniques, and the center manifold theorem.

By exploiting complex coordinates and elementwise matrix products, a single and simple differential equation is derived that describes the motion of every multi-arm planar pendulum.

For an entire function $f$ and a non-zero complex number $\lambda$, let $f_\lambda(z)=f(\lambda z)$. We give conditions on $f$ which imply that the Hausdorff dimension of the set of non-escaping points in the Julia set of $f_\lambda$ tends to $1$ as $|\lambda|\to 0$. In fact, we give an upper bound for this dimension in terms of $\lambda$. This generalizes earlier results concerned with the case that $f(z)=\exp z$.

We introduce a fully–discrete variational approximation of Mather measures and sets for Tonelli Lagrangians on the flat torus, together with a numerical procedure for approximating the entire Mather set. The scheme is based on a fully–discrete Lax–Oleinik operator with integer winding labels. We prove an $O(\tau+h/\tau)$ error estimate for the critical value, convergence of critical solutions, and a finite-dimensional characterization of fully–discrete Mather measures. Accumulation points of the reconstructed minimizing measures are continuous Mather measures, while the supports satisfy complementary upper and lower convergence results involving the Mañé and Mather sets. To avoid the selection of only some minimizing components by exact discrete minimizers, we introduce a mass-threshold approximation based on almost-minimizing holonomic measures. This yields a finite-dimensional constrained optimization procedure designed to recover the whole Mather set.

We investigate the $k$-variation bounding property of the discrete-time Hankel operator, i.e., its invariance under the set of signals with a variation (number of sign changes) of at most $k$. Building on existing sign-consistency criteria, it is shown that this property is equivalent to the external positivity of $k+1$ explicitly realized linear discrete-time systems. Thus, making the property tractable via numerical and analytic certificates. We also derive dominant-pole restrictions for the case of $k=1$. A three-node thermal example illustrates the results and distinguishes variation bounding from variation diminishing.

The qualitative analysis of nonlinear dynamical systems relies on identifying geometric and topological structures that govern phase-space organization. We develop a computational framework based on novel topological and geometric descriptors of vector fields that enables robust comparison of dynamical regimes from both analytical models and sampled data. The framework comprises the Begin-End Point Embedding (BEPE), Density of Directions (DOD) and Euler characteristic-based summaries (Euler Characteristic Curve (ECC) and Euler Characteristic Profile (ECP)). BEPE captures local variations in vector field, while DoD characterizes the global organization of vector directions. ECCs and ECPs provide filtration-based topological summaries of vector-field structure, with ECPs extending these summaries to multiparameter settings. Together, these descriptors encode complementary geometric and topological information about flow organization. We formulate both continuous and sampled versions of the descriptors and establish their robustness with respect to perturbations, finite sampling and selected transformations. The effectiveness of the proposed approach is demonstrated on a collection of benchmark systems (the Hopf bifurcation, the FitzHugh-Nagumo model, the Lorenz system). Experiments on the discrete Hénon map demonstrate the applicability of the framework to displacement fields generated by discrete dynamical systems, while experiments on high-dimensional turbulent flows assess its scalability beyond low-dimensional phase spaces. Comparisons with classical approaches, including Conley-index-based methods and $L_p$-type metrics, show that the proposed descriptors provide complementary structural information while offering favorable robustness, interpretability and computational efficiency, establishing a practical connection between dynamical systems theory and topological data analysis.

We prove a local stability criterion for the no focal points property on compact Riemannian surfaces. More precisely, if a smooth metric $g$ has non-positive Gaussian curvature and its zero-curvature set is finite, then $g$ belongs to the $C^2$-interior of the set of metrics without focal points. Thus, every sufficiently small $C^2$-perturbation of $g$ still has no focal points, although arbitrarily small perturbations may create regions of positive Gaussian curvature. By Ruggiero's characterization of the $C^2$-interior of the set of metrics without conjugate points, all metrics in the resulting neighborhood are Anosov. We also show that, starting from any hyperbolic metric on a compact surface, one can prescribe an arbitrary finite set as the zero set of the Gaussian curvature of a smooth conformal non-positively curved metric. These metrics can be approximated smoothly by negatively curved metrics, so they lie on the boundary of the negatively curved regime while remaining interior points of the no-focal-points regime. The proof of the stability theorem combines uniform local convexity, Gulliver's bound on the length of geodesic segments contained in small balls, and an inductive argument on the Riccati equation that controls successive passages through the regions where positive curvature may appear.

We establish the dynamical Mordell–Lang conjecture over \(\mathbb C\) for endomorphisms that are relatively étale over arbitrary endomorphisms of finite products of smooth projective curves. In particular, we establish the conjecture for arbitrary endomorphisms of product of smooth projective curves and relatively étale skew products. The arithmetic input is our unconditional theorem for split endomorphisms over \(\overline{\mathbb Q}\), proved by reducing to a simultaneous Hasse principle for the local periods of critical points and analyzing local inertia in joint arboreal towers. We combine this arithmetic input with relative étaleness, simultaneous algebraic specialization preserving wandering base coordinates, prescribed-reduction \(p\)-adic embeddings, and \(p\)-adic interpolation to obtain the result in the complex case.

This paper addresses the existence of pullback attractors for a class of nonlinear non-autonomous strongly damped plate equations with delay. The model contains the biharmonic operator, the strong damping term, a nonlinear source term and a delay operator acting on the history of the solution. The interaction between the strong damping mechanism and the hereditary delay effect gives rise to several analytical difficulties in deriving uniform estimates and proving pullback asymptotic compactness. To overcome these difficulties, we construct a modified energy functional adapted to the strong damping structure and employ the contractive function method to handle the delay term in the history phase space. By establishing the existence and uniqueness of weak solutions, uniform pullback estimates and the pullback asymptotic compactness of the associated non-autonomous process, we prove the existence of pullback attractors for the strongly damped plate equation with delay.

We find and classify every stationary configuration of the system of four identical point vortices in the plane. Up to the natural symmetries of the system, there are exactly three configurations, which are relative equilibria: a square, a collinear configuration, and an equilateral triangle with a central vortex. Among them, the square configuration is the global minimum of the Hamiltonian, whereas the other two configurations are saddle points. In particular, the square is the unique stable configuration among the stationary configurations. Using the special algebraic structure of four points, we introduce a change of variables based on the Hadamard matrix of order four, which transforms the original four-vortex problem into an equivalent three-point problem.

This paper studies the existence of pullback $\mathcal{D}$-attractors for non-autonomous Kirchhoff wave equations with strong damping and delay effects in the phase space $\mathcal{E} = C_{H^1_0(\Omega)} \times C_{L^2(\Omega)}$. The model contains a strong damping term $-\Delta \partial_t u$, a nonlocal Kirchhoff term $\Psi(\|\nabla u\|^2)$, a state-dependent delay term $\phi(t, u_t)$, and a time-dependent external force $h(x, t)$. There appear to be no results on the pullback attractors of Kirchhoff wave equations involving both strong damping and delay effects in the literature. To this end, we establish delicate uniform estimates and employ the contraction function method to prove the pullback asymptotic compactness of the associated process, which yields the existence of pullback $\mathcal{D}$-attractors.

We establish new pointwise convergence results for weighted ergodic averages along sequences of the form \( (\lfloor a(n) \rfloor)_{n \in \mathbb{N}}, \) where $a(x)$ is a subpolynomial function in a Hardy field. For example, we establish pointwise convergence of logarithmic averages along sequences of the form $(\lfloor n^k + \log^{c} n \rfloor)_{n \in \mathbb{N}}$, where $k \in \mathbb{N} \cup \{0\}$ and $c > 0$. This result should be juxtaposed with the fact that either for $k=0$ or for $k \geq 2$ and for sufficiently small $c>0$ (depending on $k$), the standard ergodic averages along these sequences fail to converge pointwise. We also obtain pointwise joint ergodicity results for multiple weighted ergodic averages along slow Hardy field functions. For example, it follows from our results that for $c> 0$ and for any $f, g \in L^{\infty} (\lambda)$, \begin{equation*} \lim_{N \rightarrow \infty} \frac{1}{\log N } \sum_{n=1}^{N} \frac{1}{n} f(T_b^{\lfloor \log^c n \rfloor}x) \, g(T_G^{\lfloor \log^c n \rfloor} x) = \int f \, d \lambda \cdot \int g \, d \mu_G \quad \text{for almost every } x \in [0,1], \end{equation*} where $T_b:[0,1] \rightarrow [0,1]$ is the times-$b$ map defined by $T_b x = bx \, \bmod \, 1 $ and $T_G:[0,1] \rightarrow [0,1]$ is the Gauss map defined by $T_G(x) = \frac{1}{x} \bmod \, 1$ for $x \ne 0$ and $T_G (0) =0$. Here $\lambda$ is the Lebesgue measure on $[0,1]$ and $\mu_G$ is the Gauss measure on $[0,1]$ given by $\mu_G (A) = \frac{1}{ \log 2} \int_A \frac{1}{1+x} dx$ for any measurable set $A \subset [0,1]$.

A result of Futer, Kalfagianni, and Purcell implies that an upper volume bound for all link complements in the 3-sphere cannot depend solely on the braid index. In this paper, we introduce the Lorenz braid index and generalise the bunch algorithm to provide a general upper volume bound for all link complements in the 3-sphere. Such an upper bound is a quadratic polynomial in the Lorenz braid index. In addition, we construct an explicit family of hyperbolic Lorenz knots for which the classical braid index and the Seifert genus both tend to infinity, while the Lorenz braid index remains bounded.

We consider smooth symmetric bimodal maps with Fibonacci combinatorics. We prove that, when the critical order is sufficiently large, the common $\omega$-limit set of the two critical points is a wild Cantor attractor. Using a full-family argument, we first prove the existence of smooth symmetric Fibonacci bimodal maps with any prescribed critical order $\ell>3$. The proof of the existence of the wild attractor is based on a two-to-one semi-conjugacy which reduces the generalized renormalization of the bimodal map to a Fibonacci box mapping with one critical point.

We initiate a joining rigidity theory for rational maps on the Riemann sphere $\mathbb P^1=\mathbb P^1(\mathbb C)$. Let $f_1,f_2\colon\mathbb P^1\to\mathbb P^1$ be rational maps of degree at least $2$, and $\mu_1,\mu_2$ their respective measures of maximal entropy, whose supports are the Julia sets $J(f_1)$ and $J(f_2)$. We study ergodic joinings of the systems $(J(f_1),f_1,\mu_1)$ and $(J(f_2),f_2,\mu_2)$, namely ergodic probability measures on $J(f_1)\times J(f_2)$ which are invariant under $f_1\times f_2$ and whose marginals are $\mu_1$ and $\mu_2$. Our main theorem shows that a positive-mass local holomorphic relation forces algebraic rigidity. More precisely, if the joining charges the graph of a local biholomorphism, then that local relation globalizes to an invariant algebraic curve and yields either a finite cycle of rational graph or transpose-graph relations, or a genuinely multi-valued invariant algebraic correspondence. If no local biholomorphic graph has positive joining measure, then the joining generates a compact non-discrete family of local holomorphic relations. The proof introduces normalized inverse branch transfer maps and studies their cluster limits. Starting from a local biholomorphic graph of positive joining measure, recurrence and contraction of inverse branches produce recurrent local intertwining relations. These are promoted to an algebraic relation by a local-to-global rigidity argument in the non-Lattès case and by affine uniformization in the Lattès case. In the absence of any positive-mass local biholomorphic graph, the cluster family must be infinite, and its non-discrete closure gives the second alternative.

We give geometric characterizations of analytically linearizable nondegenerate Siegel foliations in $\C^2$ using complex tangencies with shrinking strictly pseudoconvex hypersurfaces. Our principal result treats a fixed bounded smoothly bounded strictly pseudoconvex Reinhardt domain: in coordinates compatible with the eigendirections of the linear part, two-dimensional tangency loci along a shrinking sequence characterize analytic Siegel linearizability. For round spheres we prove a stronger statement: no a priori alignment of the Euclidean coordinate axes with the eigendirections is required; the tangency hypothesis itself forces the linear part to be unitarily diagonalizable with negative real eigenvalue ratio. We also establish a polynomial Shilov-boundary criterion on a round sphere and a logarithmic one-boundary criterion valid for arbitrary smooth strictly pseudoconvex domains, provided the embedded closed bidisc lies on the pseudoconvex side. Finally, we study the transverse-holomorphic dynamics of smooth tangency tori, including a rigidity theorem in the periodic case.

Any objects in space may encounter singularity due to the inverse-square law of the gravitational force, hindering the analysis and propagation of particle motion in space. Regularization in celestial mechanics removes such gravitational singularities in equations of motion by transforming the time and coordinates. This paper addresses a class of regularization methods known as global regularization for restricted three-body problems (R3BPs), which simultaneously removes the two singularities (e.g., Earth and Moon in cislunar space). It enables analysis, propagation, and prediction of particle motion throughout the domain of R3BPs, which is in stark contrast to regularizing each singularity individually. In particular, this paper develops a formulation for global regularization of the spatial elliptic R3BP (ER3BP) under external perturbations. The presented formulation leverages mathematical tools from quaternionic (hyper-complex) algebra and Hamiltonian mechanics. This approach enables us to (i) elegantly handle the four-dimensional coordinates arising from the dimension-lifting operation for spatial regularization, (ii) systematically derive the coordinate-momentum transformation while preserving the canonical structure and incorporating perturbing forces, and (iii) provide geometric insights for analyzing the regularity of the transformation, resulting Hamiltonian, and dynamical flow. Numerical examples demonstrate the validity of the mathematical results as well as significant numerical benefits of the regularization formulation in perturbed Earth-Moon ER3BP.

We construct hyperbolic renormalization horseshoes for analytic generalized interval exchange transformations (GIETs) of bounded type and with bounded geometry, provided their boundaries are small (but not necessarily zero). As a consequence, we describe the manifold structure of smooth conjugacy classes of GIETs.

We construct hyperbolic horseshoes for piecewise-analytic homeomorphisms of the circle with multiple break-type singularities, under the assumptions of bounded type rotation numbers and bounded geometry – provided the sizes of the breaks are uniformly small. As a consequence, we obtain a $C^{1+\alpha}$-rigidity result for such maps and prove that rigidity classes are analytic submanifolds.

We prove that topological conjugacy and flip conjugacy of minimal homeomorphisms of a Cantor space are complete analytic relations and hence are not Borel. We also prove that mutual reducibility by injective continuous graph homomorphisms and topological graph isomorphism are complete analytic relations on the space of nonempty compact graphs on a fixed Cantor vertex space with continuous chromatic number two. Both relations remain complete analytic on the larger space of nonempty compact graphs on the same Cantor vertex space with continuous chromatic number two or three.

We study distributional chaos in over-saturated sets under the non-uniform specification property. We prove that for any dynamical system with the non-uniform specification property, every non-empty over-saturated set exhibits distributional chaos of type 1. We further provide sufficient conditions for certain differences of over-saturated sets to exhibit distributional chaos of type 1. We also construct examples showing that some of these conditions are optimal.