Papers

We establish that for a $C^r$-generic surface repeller ($1 < r \leq \infty$), its Hausdorff and box dimensions are exactly the unique zero of the sub-additive topological pressure function. As an application of our framework, we show that the attractor of a uniformly non-conformal and weakly irreducible planar non-linear iterated function system (IFS) satisfying the strong separation condition (SSC) attains its expected Hausdorff and box dimensions. Furthermore, as a direct consequence of our generic repeller theorem, we deduce that this dimension formula also holds for a generic $C^r$ planar IFS satisfying the SSC. Using this approach, we also extend the dichotomy result for graphs of Weierstrass-type functions of Ren and Shen (2021) by weakening their real-analytic requirement to arbitrary $C^r$ regularity for $r > 1$.

The Lagrange spectrum is a classical object in number theory, defined as the set of values of $\liminf_{q\to\infty} q\, \mathrm{dist}(q\alpha,\mathbb{Z})$ where $\alpha$ runs through irrational numbers. It has a complicated structure, with the discrete part, Hall's ray, and a transitional part in between. One can similarly define Lagrange spectrum in the multidimensional set-up, and until now not much has been understood about it. In this paper we prove that, unlike in the one-dimensional case, the closure of the multidimensional Lagrange spectrum is equal to the interval between $0$ and its supremum. The proof relies on a correspondence between Diophantine approximation and dynamics on the space of unimodular lattices and proceeds by studying a dynamical counterpart of the Lagrange spectrum that we call dynamical Lagrange spectrum. The latter is shown to be equal to the interval between $0$ and its maximum by means of an argument utilizing the higher rank nature of the set-up. A passage from full dynamical spectrum to the density of the Diophantine spectrum is achieved by applying equidistribution of expanding translates of horospheres in the space of lattices.

For every $1\le d<n$, we prove joint equidistribution, as the height tends to infinity, of $d$-dimensional $K$-subspaces in a rigid adelic space over a number field $K$, together with their archimedean Grassmannian images and the $K$-linear isometry classes of the normalized subspace and quotient. We identify the leading constant, prove no escape of mass from either normalized factor, and obtain global equidistribution against bounded continuous test functions. Over a general number field, the two normalized shapes are coupled by a determinant-class relation; conditional on the determinant class, their limiting law is the product of the Haar-induced probability measures on the corresponding fibers.

In this paper, we study how the classification complexity of natural orbit equivalence relations changes when the full homeomorphism group of a compact metrizable space is replaced by a dense non-closed subgroup. For a compact space $X$ and a subgroup $G \leq \mathcal{H}(X)$, we consider three canonical actions: the left shift action on $\mathcal{H}(X)$, the induced hyperspace action on $\mathcal{F}(X)$, and the conjugation action on $G$ We first analyze subgroups of the group $\mathcal{H}^+([0,1])$ of increasing interval homeomorphisms, focusing on bi-Lipschitz homeomorphisms, diffeomorphisms, and bi-absolutely continuous homeomorphisms. We show that, in contrast to the behavior of closed subgroups, passing to these subgroups strictly increases the complexities of the associated classification problems or makes them incomparable with the corresponding full-group relations. In the second part, we investigate hyperspace actions of bi-absolutely continuous homeomorphisms on the Cantor space and the Hilbert cube with respect to some Borel probability measure and show that a similar behavior occurs on these spaces as well.

We examine pursuit and evasion problems in the space $ \mathbb{R}^{n} $ involving a lone pursuer and evader. Motion of each player is governed by a first-order linear differential equation. The players' control functions are subject to the Gronwall-type inequality. The game's duration is fixed and represented by a positive number $\vartheta $. When the state of a pursuer coincides with that of the evader, we then say the pursuit is completed. On the contrary, evasion is possible when throughout the game there is no agreement between the states of pursuer and evader. The result obtained related to the pursuit problem depends on the attainability domains of the two players. On the other hand, one sufficient condition is given for evasion to be possible.

Given a discrete group $\Gamma$ acting on a compact space $X$, and a stabilizer subgroup $\Lambda\leq \Gamma$ of $X$, we use the Rieffel induction of covariant $(\Lambda, X)$-representations to study classes of representations induced by certain characters on $\Lambda$ and the $\text{C}^*$-algebras they generate. When $X$ is a $\Gamma$-boundary, we obtain new classes of simple, traceless group $\text{C}^*$-algebras. When the $\Gamma$-boundary $X$ is an extreme boundary, we show that the associated group $\text{C}^*$-algebras are also purely infinite, answering part of a question posed by Kalantar and Scarparo on $\text{C}^*$-algebras associated with Thompson's groups. The latter $\text{C}^*$-algebras are shown to be selfless in the sense of Robert.

We prove that on any closed manifold there exists a C1 open and dense subset of star flows for which the periodic orbits grow at most expo- nentially. We also explain how our methods apply to the sectional-hyperbolic attractors including the Lorenz polynomial equation.

Multistable mechanical systems can store and release elastic energy through snap-through instabilities, but controlling transition pathways between stable states remains challenging when multiple routes are accessible. Here, we introduce a two-mass von Mises truss as a general model for studying pathway selection governed by coupled saddle-node bifurcations. The system consists of two coupled snap-through units with geometric imperfections, giving rise to four stable configurations: a fully inverted state, a fully natural state, and two intermediate mixed states. We show that the coupling stiffness reorganizes the quasi-static bifurcation structure and selects among three transition pathways under release: sequential snapping through one mixed state, direct cooperative snapping, or sequential snapping through the other mixed state. Using pseudo-arclength continuation, we track the relevant saddle-node bifurcations and identify the parameter regimes associated with each quasi-static pathway. We then demonstrate that dynamic bifurcation delay provides an additional rate-dependent mechanism for pathway selection. Even when the quasi-static bifurcation structure favours a unique sequential pathway, finite-rate loading delays snap-through beyond the corresponding static saddle-node points and can reorder the snapping sequence of the two units. A local reduction of the coupled dynamics near each saddle-node yields normal forms with coupling-dependent critical points and coefficients. The resulting theory identifies distinct rate-dependent delay laws in the inertia-dominated and overdamped regimes and predicts the critical rate at which the snapping order reverses. These results establish a general mechanics framework for tuning transition pathways in multistable systems through elastic coupling and loading-rate control.

Relative-frequency trajectories are often used to infer selective advantage in competing biological populations. A common empirical approach is to fit a linear function to the logit-transformed frequency of an invading type and interpret the slope as the relative advantage. Here we test how this estimator is affected when the competing types are observed after type-specific delays. We use SARS-CoV-2 variant replacement in the United Kingdom as empirical motivation and study the mechanism with simple two-type models. In an ideal exponential replacement model, fixed or randomly distributed observation delays change the intercept of the observed log-odds trajectory but not its slope, provided that delayed counts are aggregated before frequencies are formed and boundary effects are absent. In nonlinear SIR-type models, where the relative growth is time-dependent, delays can compare the competing variants at different dynamical phases and substantially bias the fitted logit slope. This bias can occur even when the delayed replacement trajectory remains nearly linear on the logit scale. Thus, a good logistic fit to the observed data is not sufficient to guarantee that the inferred logit slope approximates well the relative growth advantage. Observation delays should therefore be accounted for when logit slopes are used as proxies for relative fitness in nonlinear evolutionary competition.

We study the geography of taut foliations on compact $3$-manifolds with toroidal boundary components. The central object of our work is the set of boundary multislopes realized by foliations transverse to a fixed flow on such a manifold. We prove that these sets exhibit remarkable structural properties (rationality, rigidity, and convexity) which motivate the name ziggurats. Our main tool, of independent interest, is a two-way correspondence between foliations and contact structures on $3$-manifolds with boundary: we generalize the Eliashberg-Thurston theorem, which produces pairs of positive and negative contact structures from foliations, and a construction of the first author, which builds foliations from such contact pairs. Using both directions of this correspondence, we bring contact-geometric methods to bear on the architecture of ziggurats.

This survey explores the rich interplay between (taut) foliations and (tight) contact structures in dimension three, highlighting recent work of the author. We outline a new method for constructing taut foliations from suitable pairs of contact structures, and discuss some applications and future research directions. In particular, we give a brief account of work in progress with Jonathan Zung on ziggurats for taut foliations transverse to pseudo-Anosov flows, and we propose a (mostly speculative) contact perspective on the $L$-space conjecture.

Let $M$ be a finite-volume hyperbolic $d$-manifold with $d\ge3$. We prove that the number of primitive nonsimple closed geodesics has exponential growth rate strictly smaller than $d-1$. Consequently, asymptotically almost every primitive closed geodesic in $M$ is simple. In contrast, we show that on an arithmetic hyperbolic manifold of type~I, the unit vectors tangent to nonsimple closed geodesics are dense in the unit tangent bundle.

The interconnection of port-Hamiltonian systems is usually understood as structure-preserving due to the geometric fact that the composition of the underlying Dirac structures produces again a Dirac structure. Pointwise, this is doubtless true, but when comparing dynamical systems, their trajectories are usually compared, and trajectories usually obey some regularity assumptions. It is shown that both classical and weak solutions do not necessarily exhibit a one-to-one correspondence between solutions of the interconnected port-Hamiltonian system, and the dynamical system realising the interconnection. In the mostly studied cases of constant Dirac structures or port-Hamiltonian ODE systems, both systems are behaviourally equivalent; for non-constant Dirac structures, a sufficient condition is given.

We answer two questions of Kozlov about polynomial-in-momentum first integrals of billiard systems. For billiards in bounded polytopes, such integrals are independent of position. For hard-ball gases in a rectangular box, every polynomial-in-momentum first integral is a polynomial in the total kinetic energy whenever the collision graph is connected.

We study nonzero-frequency phase-locked motions in asymmetrically coupled Kuramoto networks. Such motions are relative equilibria with fixed phase differences and a nonzero common angular velocity, and we call them dancing equilibria. Their existence requires all coupling sums to have the same nonzero value. We show that neither symmetric coupling nor an acyclic associated digraph can support a dancing equilibrium. We introduce structurally equitable and $q$-twisted state equitable partitions and prove a partition-based criterion for the resulting class-constant profiles, with standard labeled $q$-twisted profiles recovered from singleton partitions. For the forward $m$-neighbor model, we characterize existence by an exact indivisibility criterion. Stability is studied modulo the common phase-shift direction. For general directed networks, strong connectivity and edgewise phase differences in $\left(-\pi/2,\pi/2\right)$ imply local orbital exponential stability and yield an explicit positively invariant set contained in the local basin of attraction. For arbitrary twisted indices, this contraction argument gives a low-winding stability regime with explicit positively invariant neighborhoods. For each existing $q$-twisted branch of the forward model, a discrete Fourier transform criterion yields local orbital exponential stability when all nonzero Fourier-mode factors are positive and nonlinear instability when at least one is negative. In the unstable case, the proof constructs explicit escaping real Fourier perturbations. We further derive additional explicit stability and instability ranges for arbitrary twisted indices in terms of constants $N$, $m$, and $q$. For the first two twisted branches, sharper arguments yield a first-mode transition criterion for $q=1$ and a complete finite-size classification for $q=2$, with the degenerate case in each branch handled separately.

We prove that every Poisson action of positive intensity of a noncompact locally compact second countable sofic group has completely positive measure sofic entropy with respect to every locally compact sofic approximation. The proof constructs random microstates for arbitrary measurable factors by completing local Poisson configurations with an independent exterior Poisson process. Probability estimates and a packing argument then show that every nontrivial factor has positive entropy. We also give a compact-group example showing that the noncompactness assumption is essential.

Fix an integer $p\geq 2$ and $0<s<1$. We construct an ergodic $\times p$-invariant measure $\mu$ on $\mathbb T^2$ of dimension $s$, such that every line projection preserves dimension, including when the corresponding projected IFS has exact overlaps. In fact, our measure assigns mass $O(w^s)$ to every planar tube of width $w$. The result is sharp at the endpoint $s=1$ as shown by Pyörälä, Shmerkin, Suomala and Wu (2025). In contrast, we show that every quasi-Bernoulli $T_p$-invariant measure with positive dimension admits a dimension-dropping projection.

Motivated by a theorem of A. Schmidt on the part of the complex Lagrange spectrum below $2$, we study the restricted Lagrange spectrum $L_{\frac{1}{2}+i\mathbb{R}}$ arising from the approximation of complex numbers of the form $\frac{1}{2}+i\alpha$, $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ by Gaussian rationals $p/q$ with $p,q\in\mathbb{Z}[i]$, $q\neq 0$. We show that this spectrum admits a description in terms of a dynamical spectrum associated with a real horseshoe. As a consequence, we obtain several fractal properties of $L_{\frac{1}{2}+i\mathbb{R}}$, such as continuity of the dimension function $t\mapsto\dim_H(L_{\frac{1}{2}+i\mathbb{R}}\cap(-\infty,t))$. We also prove that the set of complex numbers $z$ satisfying \begin{equation*} \left\lvert z-\frac{p}{q}\right\rvert\geq\frac{1}{2|q|^2}, \quad\text{for all } p,q\in\mathbb{Z}[i], q\neq 0, \end{equation*} is uncountable. In fact, we show that this inequality holds for every complex number of the form $z=\frac{1}{2}(1+i\theta)$ where $\theta\in\mathbb{R}\setminus\mathbb{Q}$ is a root of one of Schmidt's $C$-minimal forms.

We study the self-intersections of billiard trajectories in a square with small pockets of size $\varepsilon$ removed from its four corners. In particular, we establish an asymptotic formula for the $r$-th moment of the number of self-intersection points for each positive integer $r$, as well as the distribution of such points.

Let $\Lambda=\operatorname{diag}(m_1,\ldots,m_d)$ be a diagonal expanding integer matrix with $m_1\ge\cdots\ge m_d\ge2$ and at least two distinct diagonal entries, and let $X\subset\mathcal A^{\mathbb N}$ be a subshift with weak specification over $\mathcal A=\prod_{r=1}^d\{0,1,\ldots,m_r-1\}$. Let $K=R(X)$ be the corresponding self-affine set. Writing $\gamma=\dim_{\mathrm H} K$, we prove a sharp critical-measure dichotomy: \begin{align*} \dim_{\mathrm H} K=\dim_{\mathrm B} K \Longleftrightarrow 0<\mathcal H^\gamma(K)<\infty,\quad \dim_{\mathrm H} K<\dim_{\mathrm B} K \Longleftrightarrow \mathcal H^\gamma(K)=\infty. \end{align*} In the second case, $\mathcal H^\gamma$ is not $\sigma$-finite on $K$. This resolves Peres' 1994 conjecture for planar systems and extends to higher dimensions. For sofic subshifts, we prove the existence of the box dimension and show that dimension equality implies that the critical Hausdorff measure is positive and $\sigma$-finite, although it may be infinite. We also construct two examples with $\dim_{\mathrm H} K<\dim_{\mathrm B} K$: in one example the critical Hausdorff measure is positive and finite, whereas in the other it is infinite but $\sigma$-finite.

Let $M_d(X)$ denote the average number of rational preperiodic points among degree-$d$ polynomials over $\mathbb{Q}$ with vanishing $z^{d-1}$ coefficient, constant term $1$, and height at most $X$. We prove that, for every integer $d\ge2$, $M_d(X) \sim \gamma_d X^{-1}$ for an explicit constant $\gamma_d>0$. For $d\ge4$, the error term is $O_{d,\varepsilon}(X^{-2+\varepsilon})$ for every $\varepsilon>0$; for $d=3$ the error term is $O(X^{-3/2})$; and for $d=2$ the error term is $O(X^{-3/2}\log X)$.

Dynamical localization requires all position moments of a quantum wavepacket to remain bounded in time, but for quasiperiodic Schrödinger operators such bounds are generally not uniform in phase. In the positive Lyapunov exponent regime, the best known phase-uniform estimates instead grow on a logarithmic scale. We prove that both the logarithmic scale and the dependence on the moment order are sharp for a class of one-frequency quasiperiodic Schrödinger operators with even potentials. Our main ingredient is a reflective version of semi-uniformly localized eigenfunctions, adapted to the two localization centers forced by a completely resonant phase, from which we obtain matching logarithmic lower bounds along sequences of times.

In this paper, we study typical periodic optimization (TPO) for almost additive potentials in two perturbation spaces. Our main setting is the Banach quotient $\mathcal E_{\rm orb}(X,T)$ of orbit-Lipschitz almost additive potentials, where we extend the theory of maximizable sets and countable maximizable families developed by W. Huang, O. Jenkinson, L. Xu and Y. Zhang [Typical periodic optimization for dynamical systems: symbolic dynamics, Invent. Math. 245 (2026), 1–63], and establish a global structural theorem. For a countable maximizable family, global TPO holds if every non boundary member has $X$-extendable TPO and the boundary region has empty interior. As an application, we construct a compact system with global TPO for which $\mathcal E_{\rm orb}(X,T)$ is infinite-dimensional and the maximizing periods in open locking regions are unbounded. For a fixed almost additive potential $\Phi$, we also develop relative TPO theory on its Lipschitz leaf. When $\Phi=0$, this framework reduces to classical Lipschitz TPO. We prove the corresponding leafwise structural theorem and give a non additive rank-one matrix example on a full shift.

In this paper, we show that for a shift dynamical system $(X_{\mathcal{L}}, \sigma)$ with a weakened form of the specification property, there exists a Xiong chaotic set $C$ of $X_{\mathcal{L}}$ with full topological entropy everywhere (i.e. the intersection of $C$ and arbitrary non-empty open subset of $X_{\mathcal{L}}$ has full topological entropy). Moreover, $C$ satisfies that for any non-empty subset $A$ and any continuous map $F: A\rightarrow X_{\mathcal{L}},$ there exists an increasing sequence $\{p_{k}\}_{k=1}^{+\infty}$ of positive integers such that $\lim\limits_{k\rightarrow +\infty}\sigma^{p_{k}}(x)=F(x)$ holds for any $x\in A.$

For a stochastic autoparametric block-and-pendulum system, the long-time dynamics exhibit two fundamental features: the almost-sure stability of the single mode solution, characterized by its Lyapunov exponent, and the global asymptotic dynamics when this single mode solution loses stability. This naturally raises the question of whether these dynamical features are preserved under discretization, since such preservation is essential for the resulting discrete system to faithfully capture the qualitative behavior of the continuous system. To address this question, we first establish the existence of a random attractor for the continuous system subject to multiplicative stochastic excitation, providing a rigorous characterization of the global asymptotic dynamics. We then propose a numerical discretization that induces a discrete random dynamical system and prove the convergence of its random attractor to the continuous one as the step size tends to zero. In addition, we show that the numerical Lyapunov exponent of the single mode solution has the same sign as its continuous counterpart for sufficiently small step sizes, thus preserving the corresponding almost-sure stability or instability classification. These results demonstrate that the proposed discretization captures both the global asymptotic dynamics and the stability characteristics of the underlying stochastic autoparametric system.

We study statistical properties of nonstationary compositions of a sequence of heterochaos baker maps. For maps whose central direction is mostly expanding, we establish three main results: exponential rate of memory loss for a broad class of measures (Theorem 1.1), a functional correlation bound with stretched exponential decay (Theorem 1.2), and an error bound for the multivariate central limit theorem (CLT) in the Wasserstein-1 distance (Theorem 1.3). The proofs are based on the construction of suitable Gibbs–Markov induced maps for the nonstationary compositions. In a certain special case we obtain a concrete estimate on the rate of convergence in the multivariate CLT (Theorem 1.4).

Koopman autoencoders (KAEs) seek a higher-dimensional latent representation in which nonlinear dynamics evolve linearly. However, many interesting systems have multiple basins of attraction, and both theoretical and empirical work has shown these multibasin systems cannot generally admit a single finite-dimensional global Koopman embedding under standard assumptions. We posit that encoders with a sparsity-inducing objective encouraging few active latent coefficients will provide latent supports as an inspectable basin-modeling principle for Koopman autoencoders. We use these encoders producing sparse latents in training Sparse Koopman Autoencoders (SKAEs) without basin labels or other regime annotations, and treat the learned latent supports as model-produced regime variables after training. Across a range of procedurally generated multibasin systems and chaotic flows, we show that SKAEs have superior forecasting performance compared to dense-latent KAEs. We also perform a mechanistic study that shows latent supports produced by SKAEs are both essential for the quality of the representation and useful for identifying basins on basin-interior states, whereas dense-latent KAEs collapse to an uninformative single support family. These results identify sparse latents and their corresponding supports as label-free, interpretable regime variables for Koopman learning in nonlinear systems with multiple local dynamical laws.

In this paper, we focus on the asymptotic behavior of random iterated functions systems, in which both the family of transformations and the distribution of selecting them vary at each step. We consider two types of dynamics: the time-inhomogeneous Markov chain arising from forward iterates and the non-Markovian process generated by backward iterates. For both settings, we provide certain criteria for the existence of a probability measure that is geometrically attracting in the bounded Lipschitz distance, independently of the initial distribution. Finally, we illustrate our results through applications to specific models involving affine transformations.

Many open systems are observed only through aggregate snapshots, while the rule that moves objects between states stays hidden. This paper studies the inverse problem of recovering the transfer operator of a multicluster open system from two adjacent snapshots. The consistent operators fill an affine manifold: the data fix the aggregate flows and leave the internal routing free, so every recovery method commits to a prior that selects one operator. Tikhonov regularisation with a zero reference gives the uniform outflow on each source simplex, which underestimates retention in an inertial system. We propose least action on the operator trajectory, under which the rule changes as little as the data permit. In discrete time this is exact as a probability statement: the action prices a trajectory by its likelihood under a Gaussian reference drawn toward full retention, so least action selects the most probable trajectory consistent with the balance. A causal construction keeps the balance exact by moving the operator only inside the null space of the constraints, and we relate it to optimal transport, to Perron-Frobenius coarse-graining, and to the Schrödinger bridge. Two instantiations carry it: the Russian labour market and a radiation-damped relativistic electron ensemble. The construction holds the balance to machine precision where temporal smoothing corrupts it by twelve percent on the labour series and sixty percent in the physical one. The spectral gap is non-identifiable from aggregate snapshots but available from micro-trajectories, where Ulam's method reproduces the Landau-Lifshitz rate to within six percent. On the short labour series no method beats the naive forecast in point accuracy at conventional significance levels. The value is structural: an interpretable operator, an exactly preserved balance, and an explicit information limit of aggregate observation.

Pseudorandom structures are structures that behave like random ones, without necessarily being random themselves. It is a prominent, and evidently very difficult, conjecture that $$ V(n) = \lambda \cos \Big( 2 \pi \Big( x_1 + n x_2 + \frac{n(n-1)}{2} \alpha \Big) \Big) $$ is pseudorandom from a Schrödinger operator perspective in that the Schrödinger operator in $\ell^2(\mathbb{Z})$ with potential $V$ displays Anderson localization, that is, pure point spectrum with exponentially decaying eigenfunctions for almost all parameter values — the same spectral features as those produced by random potentials. We show that $V$ is pseudorandom in terms of its diffraction properties, that is, the associated diffraction measure is purely absolutely continuous for all $\lambda \not= 0$, all irrational $\alpha$, and all $x_1,x_2$ — which is the case as well for the random case. Our result gives further evidence for the conjecture in the Schrödinger case and it elucidates the apparent dual behavior of Schrödinger spectral measures and diffraction measures.