Papers

Let $\Gamma\subset\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})$ be an irreducible, non-uniform lattice and define the space $X = \operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})/\Gamma$. Let $U$ be the standard horospherical subgroup in $\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})$. We show that for every $\delta>0$ and $x\in X$ with dense $U$-orbit, the $U$-orbit evaluated at almost squares, $\{(u_{(n^{2-\delta},m^{2-\delta})}~:~n,m\in\mathbb{N}\}\cdot x$, is dense in $X$. The main idea of the proof is to shadow large periodic $U$-orbits and reduce the statement to a density statement within the periodic $U$-orbit, i.e. a statement for the $2$-torus $\mathbb{T}^2$. This then follows from an effective version of Weyl's inequality to deduce the result. The effective shadowing of periodic $U$-orbits is achieved using tools from homogeneous dynamics, namely quantitative non-divergence of horospherical subgroups, recurrence under the diagonal flow and effective equidistribution of expanding horospherical subgroups. Our approach follows the strategy of the work \cite{KR25} of \citeauthor{KR25} who established density of almost squares of the horocycle flow for non-uniform lattices in $\operatorname{SL}_2(\mathbb{R})$.

Consider a number field $K$, a polarized endomorphism $f:X\to X$ on a normal projective variety, and a non-exceptional point $\alpha$ for $f$. We show that the extension $K(f^{-\infty}(\alpha))/K$ generated by the backward orbit of $\alpha$ is virtually abelian if and only if $f$ is an exceptional map with CM and $\alpha$ is $f$-preperiodic. This answers a conjecture of Andrews and Petsche and extends it to higher dimensions. To this end, we develop the theory of exceptional maps on normal projective varieties over $\mathbb{C}$ and prove that commuting polarized endomorphisms of multiplicatively independent degrees are exceptional.

In this article, we prove the existence of a sectorial transformation to a polynomial normal form a a non-degenerate 1-resonant biholomorphism in ${\mathbb C}^n$ . We also give their holomorphic classification.

We report on frequency bursting oscillations in a system of phase oscillators with adaptive and delayed coupling. Adaptation of the coupling strengths is considered slow and depends on the phase shift between the oscillators. We find due to the combined chain of adaptation, collective dynamics, and time delays, the system robustly achieves a state in which the oscillator's frequencies are nearly synchronized but detuned by an integer number of small adaptation frequencies. We demonstrate that this quantization of the detuning is caused by alternating slow and fast transitions. Moreover, the observed motions take the form of bursts of instantaneous frequency, and the number of spikes in each burst corresponds to the quantization level of the detuning. We provide a fast-slow analysis of this phenomenon and explain the mechanisms behind the emergence of bursts. Our findings indicate that these frequency bursting oscillations are robust and exist stably within finite parameter regions.

We study arithmetic operations and maps which preserve normality. Revisiting a recent argument of Dayan, Ganguly and Weiss, we give a streamlined, self-contained presentation of the probabilistic proof that the non-zero rationals are exactly the multiplicative normality preserving numbers in a fixed integer base. We also include the extension to all bases simultaneously, using the theorem of Hochman and Shmerkin. In the additive case, we extend Rauzy's characterization of deterministic numbers in a fixed integer base to absolutely normal numbers. Namely, a translation preserves absolute normality if and only if its parameter is deterministic in every integer base. We prove this extension by a sparse random perturbation argument. Finally, we show that all locally $C^2$ normality preserving maps need to be affine-linear. The regularity assumption cannot be weakened to $C^{1,1}$: we construct a non-affine $C^{1,1}$ diffeomorphism which preserves normality in every integer base, and hence absolute normality.

We consider the parametrised family of sets $$ E(x,y)=\left\{\sum_{n=1}^{\infty}\frac{\varepsilon_n}{4^n}: (\varepsilon_n) \in \{0,x,y,x+y\}^{\mathbb{N}}\right\} $$ for $(x,y) \in \mathbb{N}^2$. This family can be viewed through three lenses: (a) as homogeneous self-similar sets; (b) as achievement sets of bi-geometric series; or (c) as the set of `rational' orthogonal projections of the four-corner Cantor set. We synthesise these three perspectives to obtain a complete topological classification of $E(x,y)$ for $(x,y) \in \mathbb{N}^2$. Next, we collapse this topological classification to a binary one according to whether or not $E(x,y)$ has interior. When this binary classification is visualised, it reveals a two-colour tiling $T$ of the lattice $\mathbb N^2$, which, despite being visibly structured, turns out to be aperiodic; indeed, we prove it has no non-trivial translational symmetries. Due to the rigidity of our model, this same binary classification simultaneously captures several dichotomies. Most notably, when the family $\{E(x,y)\}_{(x,y) \in \mathbb{N}^2}$ is viewed through the theory of self-similar sets, $T$ can be seen to describe the emergence of dimension drop within the family. Finally we examine the mechanism underlying the tiling's aperiodic order. By considering the number-theoretic properties of the tiling, we characterise its substitution structure, and discover that $T$ is a factor of a substitution tiling on four “hidden” arithmetically defined states.

We give a complete classification of algebraically primitive Teichmüller curves in $\Omega\mathcal M_g(g-1,g-1)^{\mathrm{hyp}}$ for $g>2$. These curves are precisely those generated by the Veech $2p$-gons, where $p=2g+1$ is prime.

We prove the Global Attractor Conjecture for complex balanced mass-action reaction networks whose reachable siphons satisfy two structural conditions, allowing multiple linkage classes. The proof proceeds in two steps. A structural condition relating the stoichiometric space to the reactions active within a boundary face ensures that a stoichiometric compatibility class contains at most one boundary equilibrium with any prescribed zero set. Finiteness of the possible zero sets and connectedness of the $\omega$-limit set then imply that any boundary limit set consists of a single equilibrium. To exclude convergence to such an equilibrium, we consider the embedded reaction network obtained by projecting onto the vanishing species and freezing the concentrations of the surviving species at a positive limit. The structural condition guarantees complex balance of this embedded network, while a further condition on its minimal active linkage classes ensures that an explicit Chetaev function is strictly increasing near the boundary. This excludes the boundary point as an accumulation point whenever the surviving concentrations converge. Consequently, every positive trajectory converges to the unique positive equilibrium in its stoichiometric compatibility class. Examples illustrate the hypotheses and their relation to strong endotacticity.

In environmental feedback systems, self-interested behaviors of rational agents often undermine cooperation and environmental sustainability. Although punitive incentive mechanisms are widely recognized as effective in addressing such social dilemmas, the timing of their implementation under different environmental conditions remains insufficiently understood. In this paper, we develop a multi-agent environmental feedback game framework with coupled incentive intensities across resource states. We investigate how system dynamics vary with incentive intensity under both resource-abundant and resource-scarce conditions. Theoretical analysis shows that imposing penalties in resource-abundant states can transform the system from a tragedy of the commons into a bistable regime that supports full cooperation. In contrast, imposing penalties in resource-scarce states leads to complex dynamical behaviors, including interior stability, heteroclinic cycles, and Hopf bifurcations, which hinder the emergence of stable cooperation. These results highlight the importance of timing in incentive regulation. Proactive intervention under favorable environmental conditions is more effective in sustaining cooperation than delayed enforcement after environmental degradation.

This paper investigates some properties of the compound marker endomorphisms appearing in the study of the monodromy action of Hénon maps. The main result is to present a computable criterion for checking the bijectivity of a single marker endomorphism. Some direct corollaries are also developed, keeping in mind the Hénon dynamical aspect. Another point of the paper is the reformulation of compound marker endomorphisms in terms of permutive functions, which have been introduced by Hedlund. This reformulation allows to directly apply some of his results.

We establish topological transitivity for a class of analytic partially hyperbolic diffeomorphisms on thickened nilmanifolds. These systems take the form of skew products, constructed as interval extensions of Anosov diffeomorphisms on nilmanifolds. Specifically, we develop a mechanism for topological transitivity allowing arbitrary finite-order parabolic degeneracies in the central direction.

We study rolls and homoclinic snaking in a variation of the one-dimensional Swift-Hohenberg equation, whose standard forms are prototypical order-parameter models for pattern formation in the sciences. Motivated by classes of differential equation models that involve continuous non-smooth low order nonlinear terms, we replace the standard quadratic-cubic nonlinearity by $\nu |u|^\alpha-u^3$, $\alpha \in [1,2]$ with $\nu > 0$. In the vicinity of zero, for $\alpha<2$ this nonlinearity falls outside the scope of classical Taylor expansion and bifurcation analysis. Our partially analytical and partially numerical results highlight that the non-smooth term modifies the criticality of pattern-forming bifurcations and alters the associated branches of solutions. In particular, $\alpha\in(1,2)$ implies subcriticality of roll bifurcations for any $\nu>0$. At $\alpha=1$ differentiability is lost, which has a strong impact on the bifurcations of sign-changing rolls including the disappearance of homoclinic snaking. Homoclinic snaking thus emerges non-smoothly as $\alpha$ increases from $\alpha=1$, and persists when retaining an additional, e.g., quadratic term.

We consider discrete-time dynamical systems generated by the iteration of asymptotically autonomous maps, whose asymptotic behavior may be either conservative or dissipative. These systems are of particular interest in the modeling of complex phenomena with time-dependent parameter variation. With an appropriate time compactification, the past and future infinity become normally hyperbolic invariant hyperplanes, and thus the plausible asymptotic states of the system can be determined by considering the appropriate fibers of the stable and unstable foliations or suitable sections of the asymptotic manifolds to their compact invariant subsets. In this context, we explore how tipping induced by a change of phase–namely, how a phase shift affects the evolution of the system–is encoded in the unstable manifolds of compact invariant subsets of the past equation. We further connect the tipping phenomenon to the splitting of invariant manifolds and identify certain system reversibilities able to prevent tipping. The theoretical framework we use clarifies the relationships among the various notions of tipping induced by phase-change introduced in the literature.

We consider exponential polynomials of the form $F_l(z)=P_1(z)\exp(Q_1(z))+P_2(z)\exp(Q_2(z))+\cdots+P_{l-1}(z)\exp(Q_{l-1}(z))+P_l(z)$, where $l\geq2$, each $Q_i$ is a non-constant polynomial and each $P_i$, $i=1,2,3\ldots,l-1$, is a polynomial (possibly constant), while $P_l$ is a non-constant polynomial. This article studies the topology of the preimages under $F_l$ of the neighborhoods of the essential singularity at infinity. We prove that for each disk $D$ (with respect to the spherical metric) centered at infinity, the set $F_l ^{-1} (D) $ is connected and, in fact, is an infinitely connected domain with all its boundary components bounded. In such a situation, the point at $\infty$ is called the Baker omitted value of the function. Our methods extend those developed by Das and Nayak for the function $\exp(z)+P(z)$ for any non-constant polynomial $P$ (Complex Var. Elliptic Equ. 70 (2025), 1831-1847), providing a broader framework for studying this phenomenon. We further show that these functions do not admit any Baker wandering domain. Finally, we conclude by posing some problems arising out of this work.

We prove quantitative stability and instability estimates for reversible periodic solutions of strictly Legendre-convex Lagrangian systems. If the configuration dimension is $n$, the fixed-period Morse index is $k$, and $k_D,\nu_D$ are the Dirichlet index and nullity on a half-period, set \[ \widetilde k=k-2k_D-\nu_D. \] Then the algebraic number $\nu_+$ of positive real Floquet multipliers satisfies \[ \nu_+\ge 2n-2\widetilde k, \] and we obtain corresponding bounds for the positive real spectrum away from $1$, allowing arbitrary Jordan chains at the unit multiplier. Hence at most $2\widetilde k$ Floquet multipliers can lie outside the positive real axis. In particular, when $k=0$ we recover the positive-real-spectrum theorem for reversible minimizers, while for nonconstant autonomous brake orbits the bounds improve by two. If, moreover, $n\ge2$ and the algebraic multiplicity of $1$ is two, an index-one brake orbit is transversely hyperbolic, with $n-1$ stable and $n-1$ unstable directions. Decoupled oscillators show that the refined bounds can be sharp even when the estimate based on $k$ alone is vacuous, and the Hénon–Heiles straight-line orbit provides a nonlinear illustration of the half-period mechanism.

We study a generalized vanishing discount problem for Hamilton–Jacobi equations without assuming monotonicity either pointwise or in the averaged sense with respect to all Mather measures. Specifically, we consider \[ \lambda a(x)u(x)+H\big(x,Du(x)\big)=c_0, \] and assume that, for every sufficiently small $\lambda>0$, this equation admits a strict subsolution bounded from below uniformly with respect to $\lambda$. We prove that, under a smallness assumption on the Mather quotient, the maximal viscosity solution converges uniformly as $\lambda\to 0^+$ under a sign condition on the discount coefficient $a(x)$ determined by a prescribed part of the Aubry set. The limit is characterized explicitly in terms of the Peierls barrier and the selected Mather measures. As an application, the elementary solution of the critical Hamilton–Jacobi equation \[ H\big(x,Du(x)\big)=c_0 \] associated with any isolated static class can be realized as the vanishing discount limit. This provides the first mechanism for selecting multiple critical solutions in vanishing discount problems beyond the classical monotonicity framework, showing that different families of Mather measures yield different limiting critical solutions. We finally propose a variant of the discounted problems for which the strict subsolution hypothesis is automatically verified and hence applies to more general Hamiltonians $H$ and functions $a$.

Let $\varrho\neq -1$ be a complex number with modulus $|\varrho|=1$ and $K(x,y)=P_1(x)+P_2(y)$ for two polynomials $P_1$ and $P_2$. We consider the non-oscillatory solutions such that $\lambda(f)<\infty$ of the generalized Hill equation $f''-K(e^z,e^{-\varrho z})f=0$ ($\sharp$). When $\varrho=1$, the Hill equation $f''-K(e^z,e^{-z})f=0$ ($\dag$) is also written as $x^2u''-[K(x,x^{-1})-1/4]u=0$ ($\ddag$). We point out that there is a full correspondence between the non-oscillatory solutions of equation ($\dag$) and the Liouvillian solutions of equation ($\ddag$). Then this paper has two purposes. First, we show that if equation ($\sharp$) has a nonzero non-oscillatory solution, then $\varrho=1$. To this end, we solve entire solutions of a general Tumura–Clunie type differential equation. Second, for the particular Hill equation $f''-(e^{\mathbf{k}z}+b_{\mathbf{s}}e^{\mathbf{s}z}+b_0)f=0$, where $\mathbf{k}>\mathbf{s}\geq 1$ are integers and $b_{\mathbf{s}}\not=0$, we use Kovacic's algorithms to determine the non-oscillatory solutions with relatively few zeros.

We study minimal sets of torus homeomorphisms admitting an irrational circle factor whose fibres are thin essential annular continua. For totally irrational pseudo-rotations and homeomorphisms in a nontrivial Dehn-twist class, we prove uniqueness of the minimal set when the fibres are Jordan curves on a residual set of base parameters. The same conclusion holds if the fibre cores are Jordan curves, or if the fibres are locally connected, on a nonmeagre set of parameters. The residual hypothesis cannot be replaced by a full-measure hypothesis, even under area preservation and topological transitivity. For every totally irrational rotation vector, we construct such a map with Jordan-curve fibres almost everywhere and uncountably many pairwise disjoint uniquely ergodic minimal Cantor sets. We also construct examples in every nontrivial Dehn-twist class, with prescribed irrational vertical rotation number and bounded deviations. In both families, the Jordan-curve parameters form a meagre set of full Lebesgue measure.

We introduce several Ramsey-theoretic properties of 0-dimensional ambits and obtain their dynamical characterizations. In consequence, we obtain Ramsey-theoretic criteria for triviality and for profiniteness of some important invariants (in particular, of the Ellis groups) of the ambits in question. This yields a criterion for the structural property that each distal minimal factor of the given ambit is a profinite flow. Another result is a criterion for metrizability of the minimal left ideals in the Ellis semigroup. Then we study three specializations of the above abstract context: to first order theories, to definable groups, and to the classical Kechris-Pestov-Todorčević theory, recovering known and obtaining new results in each of these contexts.

This paper establishes time-uniform accuracy guarantees for deterministic square-root ensemble Kalman filters in linear–Gaussian state-space models under perfect-model dynamics. For hyperbolic, detectable systems, we prove that the ensemble means and covariances approximate their Kalman filter counterparts uniformly in time, with high probability and without inflation or resampling. The required ensemble size depends on the effective rank of the initial covariance and the dimension of the unstable subspace, rather than on the ambient state dimension. We also analyze a localized square-root ensemble Kalman filter for weakly coupled spatial systems. We show that the stabilizing Kalman covariance inherits spatial decay from the dynamics and derive accuracy bounds that separate sampling error from localization bias. For the localized filter, the required ensemble size depends on the local intrinsic dimension and only logarithmically on the number of spatial blocks, while the localization bias scales with the interaction strength. Our analysis combines stability and forgetting for possibly singular Riccati recursions, nonasymptotic covariance concentration, and perturbation estimates for localized covariance dynamics.

This paper presents a nonlinear control law for a quadrotor unmanned aerial vehicle (UAV) under single-rotor failure that guarantees set stabilization via quasi-static feedback (QSF). Given a geometric curve in three-dimensional space, we characterize and stabilize the zero-dynamics manifold, also known as the path-following manifold, which represents all feasible motions along the path. Stabilizing this manifold ensures path-invariance: a UAV with a failed rotor initialized on the path with an appropriate orientation remains on the path for all future time. Furthermore, local exponential convergence to the manifold is guaranteed under certain conditions, implying that, under the stated assumptions, rotor failure during flight does not cause transverse deviation from the path. The proposed controller thus provides theoretical safety guarantees, which are validated through numerical experiments in the Drake physics-based simulation engine. The code is publicly available at https://gradslab.github.io/quasistatic-ftc/.

We study the stability of nonlinear systems subject to both slow and fast time variations. Both can have discontinuities, covering switched systems as a special case. The fast variation is assumed to be periodic; thus, we rely on averaging to construct an average system. Importantly, the equilibrium of the average system depends on the slow variation and is assumed to be exponentially stable when this slow input is frozen. Using perturbation and Lyapunov analyses, we establish a practical stability result showing that the system state remains within a neighborhood of the moving equilibrium when the total variation (flows and jumps) of the slow variation is appropriately bounded and the fast input varies sufficiently fast. The result is illustrated via a nonlinear switched system with slow-fast switching and a mode-dependent equilibrium.

We give a new elementary proof of Sullivan's no-wandering-domain theorem for rational maps of degree at least two using hyperbolic area. We then develop this method further to prove that a transcendental entire function has no wandering Fatou components if its set of singular values is compact and all accumulation points of that set lie in the Fatou set. Neither proof uses quasiconformal techniques.

We show that for closed negatively curved Riemannian manifolds whose length spectrum is exponentially separated, the length spectrum is a local isometry invariant. In particular, this holds for a dense set of negatively curved metrics. The main tool used in the proof is a finitary version of the Livshits theorem with an exponential tail.

On the real line, Bárány, Kolossváry, and Troscheit [5] made exponential separation of analytic self-conformal systems checkable through a dual iterated function system on a space of analytic functions. The quantity their condition separates is the pre-Schwarzian derivative, and this observation lets us carry the construction to conformal iterated function systems in every dimension, where the pre-Schwarzian is $T_f=\nabla\log\|Df\|$. A pointwise separation of the pre-Schwarzian cocycle implies the strong exponential separation condition, and in dimensions one and two separation of the Schwarzian cocycle implies the same condition modulo Möbius maps. Each hypothesis is equivalent to a uniform gap condition, and these are $\mathcal{C}^2$-open. On the line they are also dense, so both separation conditions are $\mathcal{C}^2$-generic there; this sharpens the genericity theorem of Bárány, Kolossváry, and Troscheit, whose open and dense set carries only the plain condition. In the plane they are dense on Jordan domains with simply connected extension domains, and both separation conditions are $\mathcal{C}^2$-generic there too. In dimensions at least three every conformal map is Möbius, so no system satisfies the condition modulo Möbius maps, and the pre-Schwarzian hypothesis becomes a pole-separation condition, dense when the generators contract strongly enough and stay away from the similarities, but not dense in general. On Jordan domains, the planar genericity and a theorem of Feng and Rapaport [13] settle the dimension drop conjecture for $\mathcal{C}^2$-generic planar systems with injective generators.

In this paper, we investigate the dynamics and long-time stability of hierarchical configurations in an $N$-point vortex system on $\mathbb{R}^2$ and in simply connected domains, without the Cantor-set restriction inherent in KAM theory. Since the Hamiltonian perturbation lacks a standard power-series expansion in the usual normalized variables, we develop a modified Birkhoff normal form based on successive ratios of adjacent variables, tailored to the hierarchical structure. A key feature is the preservation of the resulting natural exponent structure throughout the normal form procedure. Consequently, for $n$ sufficiently large, the hierarchical configuration persists for times of order at least $\varepsilon^{-\left(n-2(N-1)\right)}$.

We prove that every action of a countably infinite discrete amenable group on a nonempty compact Hausdorff zero-dimensional space has dynamical comparison, without assuming minimality, freeness, or metrizability. More precisely, strict inequalities under all invariant probability measures imply comparison in the clopen type semigroup with an order-unit remainder. For minimal Cantor actions, the clopen type semigroup is cancellative and almost unperforated, and its canonical map onto the positive cone of the coinvariant group is an isomorphism of ordered monoids. These results answer Melleray's cancellation question and the amenable case of his comparison question. For minimal Cantor actions, we also prove goodness of the associated affine evaluation map and obtain a minimal homeomorphism of the same Cantor space with the same invariant probability measures. We identify the precise obstruction to recovering clopen equidecomposability from affine evaluation alone. This obstruction is given by the infinitesimal subgroup: for every nonempty proper clopen set $A$, it parametrizes the topological-full-group orbits of clopen sets having the same affine evaluation as $A$. Combining comparison with the theorems of Kerr and Szabó, we prove that every free action of a countably infinite discrete amenable group on a nonempty compact metrizable space with the topological small boundary property is almost finite. This settles Naryshkin's almost-finiteness conjecture for finite-dimensional compact metrizable spaces. If the action is also minimal, its reduced crossed product is $\mathcal Z$-stable, has nuclear dimension at most one, and satisfies all the regularity conditions in the Toms–Winter conjecture.

We study the functional equation $ f \circ p = g \circ q,$ where $ f,$ $g,$ $p,$ $q$ are branched covering maps between closed surfaces. In particular, we extend results on semiconjugate holomorphic maps between compact Riemann surfaces to this broader setting. As an application, we present an alternative proof of the fact that every quotient of a torus endomorphism has a parabolic orbifold, along with some extensions of this result.

We provide a complete algebraic characterization of homogeneous self-affine measures whose Fourier transform does not tend to $0$ at infinity. Moreover, we prove that an integral self-affine measure has power Fourier decay if and only if it is absolutely continuous. For a given integral affine IFS, we also provide an algorithm for determining those probability vectors for which the associated self-affine measure is absolutely continuous.

We establish a resolvent factorization of the hyperbolic projection associated with the finite-range dual operators, by a kernel with exponential off-diagonal decay. As an application, we prove that, for every irrational frequency and every analytic potential, each subcritical Type I energy satisfying the gap-labelling condition is an endpoint of an open spectral gap. This confirms the conjecture of Ge–Jitomirskaya–You [GJY,You] in the subcritical regime.