Papers

We show that for any closed atoroidal 3-manifold $M$, there are only finitely many isotopy classes of links that can arise as singular orbits of pseudo-Anosov flows on $M$.

We study the landscape function on a discretized interval. The discrete landscape has an explicit closed form, and its spectral coefficients can be computed exactly. We show that these coefficients lie in explicit abelian extensions of $\mathbb{Q}$, and obtain the bound $[\mathbb{Q}(c_k(N)):\mathbb{Q}]\leq\varphi(N)$ for every odd $k$. For the first coefficient, exact computation for $3\leq N\leq 30$ gives the full degree $\varphi(N)$ in every case, motivating the Chandra–Jain conjecture that $[\mathbb{Q}(c_1(N)):\mathbb{Q}]=\varphi(N)$ for all $N\geq 3$. We then reduce the higher modes to the coprime case and discuss the remaining degree question. We also discuss connections with parity, the Arnold cat map, and Lefschetz numbers. The Chandra–Jain conjecture has since been proved by Q. Zhou (Zenodo, doi:10.5281/zenodo.21935814).

The computability of real numbers and functions using Turing Machines has been a central area of theoretical computer science since the mid-20th century. In the late 20th century, it was shown that chemical reactions can serve as a basis for computation using the Chemical Reaction Network (CRN) model. Recent advances in computing real numbers using Deterministic Chemical Reaction Networks (DCRNs) have identified numerous classes of DCRN-computable real numbers. In parallel, the works of R. Landauer and C. H. Bennett, spanning the 1960s to the early 2000s, showed that reversible computing offers significant advantages over irreversible methods, particularly in energy efficiency, motivating extensive research on reversible computation. In this work, we investigate the computability of real numbers using Reversible Chemical Reaction Networks (RevCRNs). The paper has two primary contributions: (1) establishing relationships among CRN-computable real number classes including Lyapunov CRN ($\mathbb{R}_{LCRN}$), Real-Time CRN ($\mathbb{R}_{RTCRN}$), rational numbers ($\mathbb{Q}$), and RevCRNs ($\mathbb{R}_{RevCRN}$), with key results: (i) $\mathbb{Q}$ is a strict subset of $\mathbb{R}_{RevCRN}$; (ii) the set of positive algebraic numbers ($ALG$), $\mathbb{R}_{LCRN}$, and real numbers computable by 1-species RevCRN ($\mathbb{R}_{RevCRN}^{1s}$) are equal; (iii) $\mathbb{R}_{RTCRN}$ and $\mathbb{R}_{RevCRN}$ exhibit non-empty overlap; and (iv) the set of real numbers computable by detailed-balanced RevCRNs ($\mathbb{R}^{DetBal}_{RevCRN}$) is a subset of $ALG$; and (2) exploring the existence of a hierarchy within $\mathbb{R}_{RevCRN}$. Finally, we leave open the exact relationship between $\mathbb{R}_{RevCRN}$ and $\mathbb{R}_{RTCRN}$ while conjecturing a general hierarchy of RevCRN-computable reals.

The class of Weierstrass functions $W_{a,b}$ is one of the first class of examples of continuous and nowhere differentiable real functions. A challenging line of research has been to determine the various dimensions of graphs $G(W_{a,b})$ of such functions. For instance, the Hausdorff dimension of $G(W_{a,b})$ was only recently determined by Shen in 2018, after a long series of partial results by many different authors. While the Assouad dimension of $G(W_{a,b})$ remains an open problem, also posed as a question by J. M. Fraser, there have been many indications that it might be equal to $2$. Such indications include the graph of Wiener processes, the graphs of almost all Hölder functions in the Baire category sense, and the graphs of Weierstrass functions after a series of countably many reflections all having Assouad dimension equal to $2$. In this paper we show that this is not the case, providing a quantitative upper bound on the Assouad dimension of $G(W_{a,b})$ that is strictly less than $2$. In particular, we show that such a bound is true for a class of generalized Weierstrass functions $W_{a,b}^\phi(x) = \sum_{j=0}^\infty a^j\phi(b^j x)$, which includes $W_{a,b}$ and the class of Takagi functions. The latter fact is used to also answer in the negative a conjecture of H. Yu on the Assouad dimension of graphs of Takagi functions for parameters $a\in (0,1)$, $b\in (1/a, \infty)\cap \mathbb{Z}$.

A discrete group $\Gamma$ has the \emph{$C^*$–ISR property} if every unital $\Gamma$–invariant $C^*$–subalgebra $A\subseteq C_r^*(\Gamma)$ is of the form $C_r^*(N)$ for a normal subgroup $N\triangleleft\Gamma$. We show that $\text{PSL}_n(\mathbb{Z})$ satisfies the $C^*$–ISR property for every $n\geq3$.

Active-quiescent switching occurs in biological populations in which growth is confined to a proliferative active state, while cells may reversibly enter a nonproliferative quiescent state. Experiments often observe only part of this process, through active-state markers, aggregate population measurements, or aggregate data supplemented by sparse active-state observations. Such measurements arise when marker panels are limited, active-state assays require fixation or endpoint sampling, or only total cell number, optical density, tumor burden, or aggregate fluorescence is reported. These observation choices complicate sparse regression methods such as sparse identification of nonlinear dynamics (SINDy), because the measured variable may satisfy a different equation from the underlying active-quiescent system. A library chosen for the wrong observable may therefore fit a trajectory without preserving mechanistic interpretation or transferability. We study this issue using a two-compartment ordinary differential equation model. We define observable reduction as the elimination of hidden states to obtain the differential equation satisfied by the measured variable. For several biologically relevant growth laws, we derive observation-specific reductions and use them to construct sparse-regression libraries. Using synthetic data, we compare these structured libraries with standard polynomial SINDy. Polynomial libraries can match training trajectories while failing coefficient-relation and transfer tests, whereas reduction-guided libraries recover interpretable coefficient maps when the observed regime is informative. These results show that interpretable equation learning in hidden-compartment systems requires matching both the regression target and candidate library to the observation process.

Consider the two-dimensional semiclassical Schrödinger operator $P_\hbar=-\frac{\hbar^2}{2}\Delta+V$ on $\mathbb R^2$, where $V$ has a nondegenerate well at the origin, its harmonic frequencies are rationally independent, and $V$ is $\mathbb Z_2$-symmetric. We prove that, generically, the first three layers of the quantum Birkhoff normal form (QBNF) determine the full Taylor series of $V$ at the origin, up to the unavoidable spatial inversion $V(x)\mapsto V(-x)$. The generic condition depends only on the jet of $V$ through order ten. Consequently, for real analytic potentials with a unique isolated global well at the origin, the low-lying semiclassical spectrum generically determines $V$ up to spatial inversion.

We investigate local and global bifurcation of periodic solutions for two classes of nonlinear fractional differential equations with symmetry. For the one-sided periodic Weyl equation, the nonzero temporal Fourier modes give rise to complex characteristic functions, leading naturally to a two-parameter bifurcation problem. The associated local invariant is expressed in terms of winding numbers and twisted equivariant degree. In contrast, the periodic Riesz equation is governed by real spectral quantities, and its local bifurcation invariant is obtained from the jump of the equivariant degree across isolated critical values. In both settings, the decomposition into spatial isotypical components and temporal Fourier modes determines the critical representations and the possible symmetries of bifurcating solutions. A nonvanishing local invariant yields the existence of nearby nontrivial periodic solutions, while the corresponding global bifurcation theorems describe the continuation of connected solution components away from the trivial branch.

We construct examples of fibrewise Anosov flows over the Fenley–Mann–Potrie example of dimension 4. This is a family of non-algebraic Anosov flows with unstable dimension $1+4n$ and stable dimension $2+4n$, for every positive integer $n$.

For $n\ge2$, we classify the affine diffeomorphisms $f_{A,v}(x)=Ax+v$ on $\mathbb R^n$ that admit a complete Riemannian metric with respect to which they are Anosov. Such a metric exists if and only if $A$ is hyperbolic or $f_{A,v}$ has no fixed point, equivalently $v\notin\operatorname{Im}(I-A)$. In the latter case, $f_{A,v}$ is smoothly conjugate to a translation when $\det A>0$ and to White's map times the identity when $\det A<0$. We also determine the possible stable indices. In the hyperbolic case, the index is determined by the stable spectrum of $A$, whereas a map with no fixed point admits complete Anosov metrics of every stable index from $1$ to $n-1$. Along the $1$-eigenspace, every such metric must exhibit exponential growth of vector norms along one of the two half orbits. We then determine the interior and boundary of the Anosov-realizable locus in the affine parameter space and describe the corresponding change of the index spectrum near regular drift parameters. Finally, we show that Anosov-realizability is not open in the two-sided weak $C^1_{\mathrm{loc}}$ topology but is open in the two-sided strong Whitney $C^1$ topology.

We establish a sharp prime orbit theorem for every homoclinic class of a $C^\infty$ diffeomorphism on a closed surface with positive topological entropy. Let $\mathcal{H}$ be a homoclinic class with topological entropy $h > 0$. Then there exists a constant $\chi_2 < 0$ such that for any $\chi_1 \in (0, h)$, \[ \lim_{\substack{l(\mathcal{H}) \mid n \\ n\to\infty}} \frac{\sharp P_{\chi_1,\chi_2}(n)}{e^{nh}} = l(\mathcal{H}). \] Here $P_{\chi_1,\chi_2}(n)$ stands for the set of period-$n$ saddle points in $\mathcal{H}$ with Lyapunov exponents lying outside the interval $[\chi_2,\chi_1]$, and $l(\mathcal{H})$ denotes the period associated with the homoclinic class $\mathcal{H}$.

We develop a general framework for studying generic homeomorphisms of compact metric spaces using combinatorial amalgamation methods inspired by Fraïssé theory. Our approach combines Rosendal's criterion for comeager conjugacy classes with combinatorial codings of compact spaces arising from the Bartoš-Bice-Vignati duality. We introduce a new amalgamation property, called weak minimal amalgamation, formulated in suitable paracategories encoding local dynamical data of homeomorphisms. This yields characterizations of the Rokhlin and the strong Rokhlin properties, i.e., the existence of dense and comeager conjugacy classes in homeomorphism groups of compact metric spaces. As applications, we reprove Hjorth's theorem that $\mbox{Homeo}_+([0,1])$ admits a generic homeomorphism, and establish the existence of generic homeomorphisms for the Cantor fan and the Lelek fan. Moreover, we show that generic homeomorphisms of these spaces have no Li–Yorke pairs, and therefore generically have zero topological entropy. More broadly, the paper develops new connections between combinatorial amalgamation properties and generic phenomena in topological group theory and topological dynamics.

We investigate equilibrium configurations and dissipative dynamics of simplified disclination–dislocation systems in planar elasticity. We consider an edge dislocation interacting with a fixed wedge disclination in a circular domain. We derive the reduced interaction energy, characterize the equilibrium states and their stability, and study the associated dissipative evolution. The analysis reveals characteristic length scales and collision times arising from the competition between the dislocation and disclination contributions. In particular, the presence of the disclination produces a nontrivial modification of the dislocation dynamics and can hinder its motion. This behavior is consistent with experimental observations and provides a simple mathematical description of the interaction between rotational and translational defects.

In this paper, we investigate a broad class of systems for which Katok's intermediate entropy conjecture holds. In particular, we show that a dynamical system with an amenable group action satisfies Katok's intermediate entropy conjecture provided that it has the specification property and is asymptotically entropy expansive.

We study long-branched unimodal maps for which the sequence of values of the kneading map form a Sturmian word. We identify these kneading classes with those arising from the stunted Lorenz construction of Anušić, Bruin, and Činč, and their quadratic representatives with the maps constructed by Blé from rigid rotations. We compute the cutting and co-cutting times explicitly and establish criteria for the existence and nonexistence of absolutely continuous invariant probability measures. We also show that, for Lebesgue-almost every rotation angle, no finite-order $S$-unimodal realization satisfies the Collet-Eckmann condition.

A subcritical age-structured branching population dies out almost surely. Conditioned on survival, it converges to its Yaglom limit, a quasi-stationary equilibrium that we take as the operating point for control. We model preventive removal (culling) as an age-dependent actuator that raises the mortality rate and leaves the offspring law untouched, and we show that its authority over this equilibrium is bounded for structural reasons. Two facts drive the result. First, the input is matched to the killing rate but unmatched with respect to the Foster–Lyapunov drift, so the transmission barrier $\Lbar$ set by reproduction alone is invariant under such actuation. Second, and this does not follow from invariance alone,} the supremum of the reachable decay rates is $\Lbar+\nu^\star$, where $\nu^\star\le0$ is the Malthusian parameter of the lineage conditioned never to die childless; the gap $|\nu^\star|$ is given in closed form and vanishes exactly when no individual has two or more offspring. Consequently no removal law of this class reaches the barrier, and along the admissibility boundary the achievable decay rate is governed by the shape of the actuator rather than by its size. We illustrate these results on a model calibrated to the 2001 Cumbrian foot-and-mouth outbreak.

We investigate planar piecewise-analytic vector fields, focusing on the merged focus and other monodromic tangential singularities when hysteresis or time-delay is incorporated into the switching condition. If the singularity is an asymptotically stable solution of the system with an instantaneous switch, then the introduction of hysteresis or time-delay causes an attracting limit cycle to be formed locally. We derive asymptotic expressions for the size and period of the limit cycle, allowing any degrees of tangency and any order for the smallest non-zero Lyapunov coefficient. We find that the growth rate of the limit cycle differs for hysteresis and time-delay, and differs to that of the related pseudo-Hopf bifurcation.

We prove Arnol'd's chord conjecture for a large new class of contact manifolds: for every contact form and every closed Legendrian submanifold there exists a non-constant Reeb chord with endpoints on the Legendrian. This class is characterized by contact non-orderability and rigidity of symplectizations. This proves the chord conjecture for Brieskorn manifolds, many prequantization spaces, and for all prequantization spaces under a mild topological condition on the Legendrians. Moreover, it provides a uniform upper bound on the length of the minimal chord. Our approach involves a new link between Mohnke's construction and contact Hofer geometry.

Motivated by Helfter's notion of scaling, we define Bowen, upper capacity, and local measure-theoretic entropy scales. Under a controlled decay condition, we establish a variational principle on compact subsets by combining a Billingsley-type theorem with a Frostman-type construction. We also prove factor inequalities for upper capacity entropy scales on compact sets and for Bowen entropy scales on arbitrary subsets. For induced systems on spaces of probability measures, we give sufficient conditions for the preservation of zero upper capacity entropy scales and show, under an additional comparison condition, that positivity for the original system forces the induced entropy scale to be infinite. Finally, we define upper capacity entropy scales along prescribed observation times and establish the corresponding zero-level equivalence for induced systems.

Elliptical Lunar Frozen Orbits (ELFOs) are attractive candidates for lunar satellite constellations supporting lunar south pole exploration. Building on prior frequency-based orbit analysis methods, this investigation develops a torus-based, frequency-domain framework for constellation-level ELFO design and analysis. Within a doubly-averaged dynamical model, each ELFO is characterized by three frequency-angle pairs; a systematic comparison across progressively higher-fidelity models identifies the additional frequency components introduced at each level. The angular parametrization admits multiple symmetries that substantially reduce the design space, enabling a rapid, epoch-free global survey of constellation configurations. A higher-fidelity refinement and analysis layer then combines a (1) Frequency-Domain Differential Corrector (FDDC) that targets desired frequency properties and suppresses undesired long-period oscillations, (2) Fourier surrogate re-optimization for inter-satellite phasing without propagation in the loop, and (3) spectral attribution that separates design-refinable coverage losses from those inherent to the dynamics. Together, these elements supply a fidelity-bridging and diagnostic framework for ELFO constellation design and analysis.

Reinfection and relapse are known to induce backward bifurcation (a subcritical pitchfork bifurcation) in many infectious disease models, particularly tuberculosis. This phenomenon implies the coexistence of multiple equilibria when the basic reproduction number is below one: a stable disease-free equilibrium and two endemic equilibria, one stable and one unstable. As a result, hysteresis may arise, meaning that reducing the reproduction number below one may not eliminate the disease given that the system remains in the basin of attraction of the endemic state. Another relevant phenomenon, often associated with intervention delays, is the occurrence of supercritical Hopf bifurcations. In this case, the endemic equilibrium loses stability for reproduction numbers above one, leading to sustained oscillations in the form of stable limit cycles. Although less explored in the tuberculosis literature, this behavior has important implications for disease control. In this work, we analyze a tuberculosis model to investigate these dynamical effects and show that Hopf bifurcation can arise even without delays or saturation effects. The bifurcation analysis focuses on two key public health interventions -case finding and case holding- and their combined effect on disease dynamics. Numerical simulations, based on epidemiological data from Salta, Argentina -a province with high TB burden- illustrate the challenges these dynamics pose for the design and implementation of public health policies.

We introduce a new amenability property for étale groupoids, termed fiberwise amenability, along with a stronger variant termed ubiquitous fiberwise amenability. (Ubiquitous) fiberwise amenability emerges naturally from a coarse-geometric perspective on étale groupoids and, in the special case of transformation groupoids, it coincides precisely with the amenability of the acting group (rather than topological amenability of the action). It is also tightly linked to the existence of invariant measures on the unit space of the groupoid. The coarse-geometric framework for étale groupoids that we develop systematically in this work allows us to establish several foundational properties of (ubiquitous) fiberwise amenability. As an application, we prove a Følner–paradoxical dichotomy for minimal étale groupoids, which will serve as a key tool in a sequel on almost elementariness of étale groupoids.

We formulate gene regulatory network reconstruction from RNA velocity as a sparse dynamical inverse problem. We show that control-only data can be structurally nonidentifying and characterize excitation conditions under which controlled perturbations restore identifiability by generating complementary regulator trajectories. To enable stable reconstruction from noisy data, we develop an integral sparse estimator that avoids numerical differentiation and derive recovery bounds separating stochastic error from systematic contributions due to latent-time uncertainty, kinetic-parameter error, numerical quadrature, and model misspecification. Synthetic experiments illustrate perturbation-assisted identifiability, improved conditioning, and the robustness of integral reconstruction. Applied to perturbation-resolved RPE1 RNA-velocity data, the framework yields an empirically full-rank design whose conditioning improves with perturbational diversity and a reconstructed network core stable under perturbation subsampling. Held-out evaluation further shows that identifiability and reconstruction stability do not imply uniform predictive improvement. These results connect perturbational excitation, identifiability, and stable sparse recovery in regulatory dynamical systems.

Let $(X,T)$ be a topological dynamical system and $(\mathcal M(X),T_*)$ be its induced system. For a non-empty compact subset $K\subset X$, we define $\mathcal M(K)$ as the set of Borel probability measures supported on $K$. In this paper, we systematically study the relationship between various entropies of $(T,K)$ and of $(T_*,\mathcal M(K))$. We show that: \begin{equation*} \begin{aligned} & h_{\mathrm{top}}^{\mathrm{UC}}(T,K)>0 \iff h_{\mathrm{top}}^{\mathrm{UC}}(T_*,\mathcal{M}(K))=\infty, \qquad &h_{\mathrm{top}}^{P}(T,K)>0\iff h_{\mathrm{top}}^{P}(T_*,\mathcal{M}(K))>0, \qquad &h_{\mathrm{top}}^{B}(T,K)>0 \implies h_{\mathrm{top}}^{B}(T_*,\mathcal{M}(K))>0 , \end{aligned} \end{equation*} where $h_{\mathrm{top}}^{\mathrm{UC}}(T,K)$, $h_{\mathrm{top}}^{P}(T,K)$, and $h_{\mathrm{top}}^{B}(T,K)$ denote the upper capacity topological entropy, the packing topological entropy, and the Bowen topological entropy of $K$, respectively. Additionally, we present a counterexample involving a non-invariant set, demonstrating that the converse of the third assertion is not valid in general.

In 1970, Klingenberg proved a fundamental result on closed Riemannian manifolds with Anosov geodesic flows, asserting that such manifolds are without conjugate points. The proof, based on the Morse theory of the energy functional, had crucial gaps pointed out by Anosov, who saved the theorem by providing a completely independent proof. In this article, we fill the gaps in the original argument, and thus provide a full Morse theoretic proof of Klingenberg's theorem.

We study minimal models and flops for foliations. We show that if $\mathcal F$ is a rank one foliation with canonical singularities on a normal projective $\mathbb Q$-factorial variety and $K_{\mathcal F}$ is pseudo-effective, then any two outputs of the $K_{\mathcal F}$-MMP are isomorphic. For co-rank one foliations on threefolds, we prove existence results for $D$-flops in the klt setting and, under additional hypotheses, in the F-dlt setting. By contrast, we construct examples showing that rank one foliations display pathologies absent from the classical MMP: flopping contractions need not admit $D$-flops, and nef and big canonical divisors need not give rise to canonical models, even in the category of algebraic spaces.

Let $(M^n,g)$ be a closed, connected, oriented Riemannian manifold with negative sectional curvature and $n\ge 2$. It is known, due to Brin and Gromov, that the oriented frame flow is ergodic in every odd dimension $n\ge 3$, $n\neq 7$. We prove that if a closed, connected, oriented negatively curved Riemannian manifold is strictly quarter-pinched, then its oriented frame flow is ergodic.

We establish three averaging principles for distribution-dependent stochastic reaction–diffusion equations with rapidly oscillating coefficients on the torus $\mathbb T^d$, $d\le3$. First, solutions converge in mean square, uniformly on finite time intervals, to solutions of the averaged equation. Under a contraction condition, both the original and averaged equations admit unique bounded entire solutions whose mean-square distance vanishes uniformly for all $t\in\mathbb R$. At the level of probability laws, the original nonautonomous equation possesses a family of pullback attractors, whereas the averaged equation has a global attractor; the former converge upper-semicontinuously to the latter, uniformly over the coefficient hull. As an application, we present a class of stochastic reaction–diffusion models motivated by large-scale interacting systems.

We study the isomorphism groupoid $\mathcal{T}(M)$ of $\theta$-almost twisted Poisson ($\theta$-atP) structures on a smooth manifold $M$, focusing on the internal structure of its morphisms. A morphism in $\mathcal{T}(M)$ is a $C^\infty(M)$-linear isomorphism $\Phi:\gO^1(M)\to\gO^1(M)$ that simultaneously intertwines the anchor maps and the $(H,\theta)$-twisted Koszul brackets associated with two $\theta$-atP structures. Every such morphism induces a canonical isomorphism in $\theta$-atP cohomology. We define a classifying functor $$ \Delta : \mathrm{Mor}(\mathcal{T}(M)) \longrightarrow (Z^1_{\mathrm{dR}}(M) ,+), \qquad \Delta(\Phi)=\theta' - \theta, $$ which is additive under composition and partitions the morphisms into two complementary families: the sub-groupoid $\mathcal{T}_{\mathrm{fix}}=\ker\Delta$ of isomorphisms preserving $\theta$, and the family $\mathcal{T}_{\mathrm{mod}}$ of isomorphisms shifting $\theta$. We describe each element in this partition.

Twisted rational maps arise naturally in relative algebraic dynamics: if a rational self-map preserves a fibration, then the induced map on the generic fiber is usually not an ordinary rational self-map over the function field, but a twisted one. This suggests that twisted rational maps form a natural framework for studying relative dynamics. In this framework we extend the theory of dynamical degrees, the numerical invariants measuring the asymptotic complexity of a dynamical system: the defining limits exist, are independent of the choice of polarization, and are birational invariants. We also identify the relative dynamical degrees of a semi-conjugacy with the dynamical degrees of the induced twisted rational map on the generic fiber, and prove the corresponding mixed degree formula. Finally, using the spectral interpretation of dynamical degrees and valuative methods at infinity, we prove an algebraicity result for the first dynamical degree of twisted endomorphisms of affine varieties.