Papers

We introduce a one-parameter family of notions between double ergodicity and metric ergodicity for measure-class preserving (i.e., nonsingular) actions of countable groups on standard probability spaces, providing infinitely many new invariants distinguishing weakly mixing actions. We call these properties (essentially) $k$-chaining, for $k \in \mathbb{N}$. We apply this framework to study boundary actions of free groups of finite rank $r \ge 1$, where the boundary is equipped with a stationary Markov measure. We prove that in this context, weak mixing is equivalent to $(2r-1)$-chaining, as well as to strict irreducibility of the transition matrix of the Markov measure.

Let Gamma be a convex-cocompact group of isometries of a CAT(-1) space X and let Y to X/Gamma be a Galois cover with a word-hyperbolic group of deck transformations. We show that, for almost every geodesic xi with respect to the Bowen-Margulis-Sullivan measure on the lift of X/Gamma, there exist m,sigma > 0 and a standard Brownian motion B(s) such that, for any lambda > 1/4, d(p(g(s,xi)), o) = ms + sigma B(s) + o(s^lambda), where g(s,xi) denotes the geodesic flow acting on the geodesics of Y and p the canonical projection to Y. The result is a consequence of an almost sure invariance principle for random walks on hyperbolic groups with dependent increments. Its proof makes use of a new Ruelle operator theorem for skew products and Martin boundary techniques for random walks with dependent increments.

We study the existence of periodic solutions and invariant tori bifurcating from a zero-Hopf equilibrium in a Hopf-Langford type system. Using second-order averaging theory, we establish explicit parameter conditions under which the system admits a periodic solution near the origin, together with its stability. We then apply recent results relating invariant tori to Neimark-Sacker bifurcations of the associated Poincaré map to obtain a smooth bifurcation curve along which a unique invariant torus surrounds the periodic solution. The relevant first Lyapunov coefficient is computed explicitly as a power series in the perturbation parameter, and we show that, under the conditions considered, this coefficient is always positive; consequently, whenever it exists, the bifurcating torus is unstable, regardless of the specific parameter values chosen within our hypotheses. This provides a structural obstruction to the existence of asymptotically stable tori in this family of systems. We illustrate our results with two numerical examples, one exhibiting an isolated asymptotically stable periodic solution with no bifurcating torus, and another exhibiting a periodic solution surrounded by an unstable invariant torus, the latter also illustrated through its Poincaré map.

Selective state space models (SSMs) have recently emerged as a compelling alternative to transformers, combining competitive performance with substantially improved inference efficiency. At each SSM layer, a sequence of hidden states are propagated by a recurrence, mixing information of different tokens. Despite using a different mechanism, this mixing plays a role analogous to attention in transformers. In fact, recent works have shown that the two architectures may be closer than they first appear, as this recurrence admits a formulation akin to linear attention. In transformers, attention is known to drive the tokens to cluster, i.e., to reach consensus, collapsing in the limit to a single direction. Thus, we ask: does the recurrence at the core of SSMs drive the tokens to consensus, as attention does in transformers? To answer this question, we take a dynamical systems perspective on SSMs, modeling the evolution of tokens across layers as an ordinary differential equation. By exploiting input-to-state stability arguments, we establish local exponential stability of the consensus equilibria and characterize their domain of attraction for time-varying weight matrices, a setting not addressed by previous results. We thereby show that the resemblance between SSMs and transformers does run deeper: the recurrence at the core of SSMs aggregates tokens just as attention does. Numerical experiments on a pretrained Mamba-2 model point to the output gate as the component that regulates the extent of this consensus, preventing the tokens from reaching it in full.

Let $\Gamma$ be the mapping class group of the plane minus a Cantor set, acting on the ray graph $\mathcal{R}$, and for $\gamma \in \Gamma$ let $\tau(\gamma)$ denote the translation length of $\gamma$ on $\mathcal{R}$. We prove that $\tau(\gamma) \le h(f)/\log(2)$ for every $C^\infty$ diffeomorphism $f$ of $S^2$ representing $\gamma$, where $h$ denotes topological entropy. The proof falls into two parts: a combinatorial argument bounding distance between two rays in $\mathcal{R}$ by the logarithm of the geometric intersection number; and a geometric argument that promotes geometric control (coarse length of iterates of a fixed ray) to combinatorial control (geometric intersection number). We conjecture that the $C^\infty$ hypothesis on $f$ can be removed.

We study linearizations of dynamical systems and some of its topological properties. Special attention is paid to the case of Lipschitz-free operators, and they are shown to model the dynamics of very general linearizations. We provide a new criterion, called the targeting property, for the (weakly) mixing property in a linear dynamical system through a (possibly) non-linear restriction of it. We then apply this criterion to the linearization $T_f$ of a Lipschitz map $f\colon M\to M$, where $M$ is a metric space and the operator $T_f$ is defined on the corresponding Lipschitz-free space $\mathcal F (M)$—the so-called Lipschitz-free operators. We show that, under some natural assumptions on the distance considered in $M$, the operator $T_f$ is hypercyclic if and only if the map $f$ has the targeting property.

We study Fourier decay of natural invariant measures for real analytic skew products $T(x,y)=(f(x),g(x,y))$ on $\mathbb{S}^1 \times [-1,1]$, where $f$ is expanding and $0<|\partial_y g(x,y)|<1$. Let $\mathcal S$ denote the associated solenoidal attractor. We prove a dichotomy: either $\mathcal S$ is a single horizontal graph, or both the SRB measure and the measure of maximal entropy have polynomial Fourier decay. No transversality assumption is imposed.

We introduce several notions of resonance for systems exhibiting weak or partial dissipation: $\dot u=Au+f(t)$, with $A$ the generator of a strongly stable semigroup $S(t)$ on a Hilbert space $H$. We refer to resonance as the phenomenon where a time-periodic $f$ may yield an unbounded $u$, classically occurring when $A$ has imaginary eigenvalues. In infinite dimensions, however, unbounded growth may depend on topological choices, and several “resonances" may occur. Classically, existence of $T$-periodic $u$ under $T$-periodic $f$ is equivalent to a range condition: $\mathcal{R}(I-S(T))=H$. Consequently, its failure permits unbounded solution growth and we elucidate that connection through properties of $S(t)$. We describe a hierarchy of range failures, and provide an associated resonance taxonomy. The growth rate of ~$||(A-\lambda I)^{-1}||_H$ on ~$i\R$ dictates a regularity gap between $u$ and $f$, and, for rapid growth in $\lambda$, that gap can be infinite. In this case, we demonstrate implications for periodic solvability, and a mechanism for constructing smooth resonant forces. The theory developed here is motivated by (and demonstrated for) a hyperbolic-parabolic system, where the latter component provides the only system dissipation. Such dynamics are a simplification of fluid-structure phenomena, which demonstrate strong stability but do not provide unconditional periodic well-posedness. Though point-spectral resonance is ruled out, we note that $\mathcal R(I-S(T))\neq H$. Our main result here shows that, for a carefully constructed geometry, a classical heat-wave system possesses periods $T$ which yield infinite derivative loss and, via supporting results, yield resonance with smooth forcing.

We present a framework for the modeling and analysis of geometric hybrid dynamical systems as hybrid inclusions on $C^1$-manifolds. Using tools from nonsmooth and set-valued analysis on manifolds, we derive coordinate-independent sufficient conditions for the existence of nontrivial solutions to this type of systems. We present geometric notions of uniform stability and attractivity of compact sets, and establish their equivalence to metric-based stability notions when the manifold is endowed with a Riemannian structure. We also introduce nonsmooth Lyapunov functions and a hybrid Lyapunov theorem providing sufficient conditions for uniform global asymptotic stability of compact sets. Finally, by characterizing $\omega$-limit sets of precompact solutions, we derive a hybrid invariance principle for geometric hybrid dynamical systems. The results are demonstrated through several running examples.

Let $\mu$ be a finitely supported probability measure on $\mathrm{SL}_2(\mathbb{R})$ whose support generates a Zariski-dense discrete subgroup. We prove that its stationary probability measure on $\mathbb{P}^1(\mathbb{R})$ is singular with respect to the $\mathrm{SO}(2)$-invariant probability measure. The proof compares winding variances on finite covers of a closed hyperbolic surface and uses a topological obstruction.

We study the Arnold-Beltrami-Childress (ABC) flow in perturbative regimes near its three elementary integrable coordinate axes. Starting from a degenerate family of periodic streamlines of the limiting integrable system, we use first-order averaging to prove the existence of two isolated periodic solutions of the perturbed flow. The three perturbative regimes are related by the cyclic symmetry of the ABC vector field. In each case, the corresponding two-dimensional averaged system possesses two simple equilibria, one elliptic and the other hyperbolic, depending on the sign of the relevant parameter ratio. Poincaré sections illustrate how the degenerate periodic structure of the integrable limit breaks under perturbation and how the isolated periodic orbits predicted by averaging emerge within the surrounding dynamics. These periodic orbits provide the natural link between the perturbative analysis and the integrability problem. The eigenvalues of the linearized averaged system determine the leading-order behavior of the nontrivial characteristic multipliers of the bifurcating periodic solutions. Combining this relation with the Poincaré-Llibre-Valls criterion, we prove that, in a neighborhood of each of these periodic orbits, there exists no nonconstant first integral $H\in C^1$ that is regular along the orbit. The analytical results are further illustrated and confirmed by direct shooting, monodromy-matrix computations, and continuation of the elliptic and hyperbolic branches, demonstrating that the first-order averaging approximation remains quantitatively accurate over a substantially wider parameter range than the strict asymptotic regime.

We investigate the zero-Hopf bifurcation and the local $C^{1}$-non-integrability of the original Rössler system. First, we characterize the unique zero-Hopf equilibrium of the system and study its perturbation by means of the first-order averaging theory. We prove that, under suitable assumptions on the bifurcation parameters, three distinct families of periodic orbits bifurcate from the zero-Hopf equilibrium, substantially extending the previously known result in which only one bifurcating periodic orbit was obtained. In the second part of the paper, these bifurcating periodic solutions are used to investigate the integrability properties of the Rössler system. By combining the first-order averaging theory with the classical Poincaré criterion for first integrals, we prove that the bifurcating periodic orbits constitute a generic obstruction to the existence of local first integrals of class $C^{1}$. This demonstrates that averaging theory provides not only an effective tool for detecting periodic solutions but also a powerful method for the integrability analysis.

Ecological and evolutionary communities do not always settle into a single predictable configuration. Models such as Lotka-Volterra or replicator dynamics may predict several admissible states. The multiplicity of such states is central to understanding robustness, alternative community configurations, coexistence, and shifts in biodiversity. Yet, standard practices are often state-centered: they either focus on the properties of typical equilibria or pool states across systems, losing information about how many states can arise in a single system, including cases where no state exists. Here, we introduce a system-centered probabilistic formalism that captures this hidden structure and could be applied across a wide range of ecological and evolutionary dynamical models, drawing inspiration from hurdle and zero-inflated models. Each ecological system is viewed as generating a distribution over its possible outcomes. This allows us to jointly quantify the occurrence of a given state type, its multiplicity within each system, and the number of species co-existing within these states. The formalism is generic and can be applied to different families of dynamical models. To illustrate its scope, we consider two case studies: the generalized Lotka-Volterra and replicator dynamics with random and structured parameters. This unified perspective reveals patterns that remain invisible when equilibria are analyzed individually or pooled across systems. By turning the state space of a model into interpretable probabilistic quantities, our formalism offers a new way for studying the robustness of community structure, the likelihood of alternative outcomes under small changes in initial conditions, and the potential for shifts between states of low and high diversity.

Let $K$ be the non-singleton attractor of a finite iterated function system $\Phi=\{\phi_i(x)=a_i O_i x+t_i\}_{i=1}^m$ on $\mathbb{R}^d$, where $0<a_i<1$, $t_i\in\mathbb{R}^d$ and the $O_i$ are orthogonal transformations on $\mathbb{R}^d$. Suppose that $\Phi$ satisfies the open set condition, but not the strong separation condition. We show that $K$ cannot be generated by any finite iterated function system of similitudes satisfying the strong separation condition. This removes the homogeneity assumption from a result of Feng, Ruan and Xiong and gives a negative answer to a folklore question about the separation conditions on the generating iterated function systems of self-similar sets.

Let $\Gamma$ be a countable discrete $C^*$-simple group. Let $\mu$ be a $\Gamma$-invariant Borel probability measure on $\text{Sub}(\Gamma)$, i.e., an IRS. We show that $\mu$-almost every subgroup $H$ is $C^*$-simple.

Lyapunov constants are a crucial tool for analyzing bifurcation behavior in switching systems. This paper proposes an efficient algebraic symbolic algorithm for computing Lyapunov constants in planar switching systems. By mapping the vector fields into the complex domain, we construct a normal form algorithm based on Laurent poly?nomials. The classical Poincaré map method requires repeated symbolic integration, which may lead to substantial computational difficulties at high orders. Our ap?proach improves computational efficiency by replacing continuous integration with direct algebraic operations. We apply this approach to investigate two switching models. For a generalized switching quartic Liénard system, the algebraic method extracts the center conditions and proves the existence of 10 small-amplitude limit cycles, establishing a new lower bound for its cyclicity. Furthermore, a physically motivated circuit-level modification of the Alpazur oscillator is investigated. We show that at most one small-amplitude limit cycle can bifurcate from the center and that this upper bound is attainable, illustrating the applicability of the proposed algorithm to a practical switching circuit. This application shows that our method provides a systematic framework for analyzing and interpreting complex dynamical phenomena arising in practical piecewise smooth systems.

Determining the momentum missing on a Poincaré section of a two-degree-of-freedom Hamiltonian is usually posed as the root-finding problem $H(0,q_2,p_1,p_2)=E$, which degenerates at the boundary of the energetically allowed region. We argue that the degeneration is the visible symptom of a geometric object and reorganize the problem around it. The section is a coisotropic hypersurface, the missing momentum is its characteristic direction, and recovering it is a reconstruction through the associated coisotropic reduction. Its carrier is not the fibrewise least-energy function, which we show is informationally incomplete, but the correspondence $\Sigma_E^{\mathrm{sec}}=\Sigma_E\cap C$ together with the reducing projection $\pi_E$, a branched reconstruction over the admissible region whose discriminant is identified with the Hill boundary under a base-side regularity condition. We prove, isolating for each conclusion its minimal hypothesis: that $\pi_E$ is proper and a covering away from the discriminant (coercivity); that the fibres have exactly two points over the interior (unimodality); that along the discriminant $\pi_E$ is a fold (fibrewise nondegeneracy and transversality); and that the discriminant coincides with the topological boundary of the admissible region precisely under a regularity of the least-energy function on the base, which no fibrewise hypothesis supplies. Strict convexity of $H$ in the momenta is shown to be a sufficient but non-fundamental bundle of these conditions. The boundary is a discriminant, not a Lagrangian caustic. Only afterwards does the squared fibrewise energy residual $F_b=(h_b-E)^2$ appear, as the analytic resolution of the fold; inside the admissible region it is a double well with two global minimizers, so branch selection is a separate rule.

We study the dynamics of a single point vortex in an unbounded planar channel whose interfaces are quasi-periodic in the longitudinal direction. The motion is governed by the Robin function. We introduce a hull formulation that lifts the quasi-periodic geometry to a periodic problem on a finite-dimensional torus and yields an exact quasi-periodic representation of the Robin function. Under a natural transversality condition, we construct global quasi-periodic invariant graphs describing transporting vortex trajectories and show that a full neighborhood of each boundary component is foliated by such graphs. Under a Diophantine condition on the spatial frequencies, the dynamics along each graph can be straightened to a constant drift, so that the vortex motion is quasi-periodic modulo translation. A central contribution concerns the distribution of vortex equilibria. In the quasi-periodic setting, where no fundamental spatial cell exists, we identify critical points of the Robin function with crossings of a hypersurface by a Kronecker flow on the hull torus. Exploiting unique ergodicity, we establish a general zero-counting theorem, allowing finite-order tangencies, which yields an explicit geometric flux formula for the asymptotic density of critical points and their limiting phase distribution. The result is nonperturbative once the critical hull is constructed. Explicit periodic and quasi-periodic models reveal bifurcations and phase transitions in the critical-point distribution, while numerical computations provide quantitative validation of the analytical results.

The Kantowski–Sachs interior of Grumiller's two-dimensional dilaton gravity model reduces to $\dot H=\tfrac12(\Lambda-3H^2-u^2)+Qu$, $\dot u=-Hu$, where $H$ is the expansion rate of the orbit two-spheres, $u$ their inverse areal radius, and $Q$ is minus twice the Rindler acceleration. For $Q\neq0$ it has two distinct rank-one nilpotent equilibria with complementary Bogdanov–Takens (BT) degeneracies: $(a,b)=(0,\neq0)$ with multiplicity $\mu=3$, and $(a,b)=(\neq0,0)$ with $\mu=2$. We show that this complementarity is forced by a single divisor-organized structure. The system is Darboux integrable, $X=\tfrac12u^4X_I$, where $I$ is the mass function and $u^4$ is an inverse integrating factor whose zero divisor is the invariant axis $\{u=0\}$. Off the divisor, every nilpotent equilibrium of $RX_I$ with $R(p)\neq0$ has $b=0$, from $b=D_{q_0}\operatorname{tr}DX$. On the divisor, invariance of the coordinate axis forces $a=0$ because $\det DX|_{u=0}=\Phi_xG$ and both factors vanish at the corner. Hence $ab=0$ throughout the class, so a versal two-parameter BT unfolding is impossible. The Bernstein–Kushnirenko bound fails at both points; instead, the exact local identity $\mu_0=m_1\ell_y+m_2\ell_x$ gives $\mu=3,2$ without a nondegeneracy hypothesis. Both multiplicities have a mixed-covolume interpretation, with $\operatorname{Covol}=\sum_{k,i}\min(p_kq_i',p_i'q_k)$ on convenient diagrams; the corresponding general theorem for the non-convenient diagrams arising here remains open. For $\dot H=\tfrac{1-m}{2}H^2+\psi(u)$, $\dot u=-Hu$, with $\psi(0)=0$, $\psi'(0)\neq0$, $m\neq1$, the vacuum always has $\mu=3$, $a=0$, $b=-m$. Its Dumortier–Llibre–Artés discriminant is $b_1^2+8a_3=(m-2)^2$, so for every $m>1$ it has one hyperbolic and one elliptic sector, independently of $\psi$.

Factor complexity counts the distinct factors of each length in an infinite word, while palindromic complexity counts those invariant under reversal. For recurrent aperiodic words with reversal-closed language, the total number of palindromic factors in two consecutive lengths is at most the one-step growth of factor complexity plus two. We prove that equality is forced whenever the factor complexity at a given length does not exceed three halves of that length plus one. We construct examples showing that the bound is optimal. We also obtain an adaptive local criterion and consequences for reversal-closed quasi-Sturmian words and palindromic defect.

A folding rational map is a postcritically finite rational map $f$ admitting an essential Jordan curve $\beta$, disjoint from the postcritical set $P_f$, whose preimage is disconnected and consists of Jordan curves homotopic to $\beta$ relative to $P_f$. Such $\beta$ is called a characteristic curve of $f$. We construct a real postcritically finite rational map $f$ of degree $6$ with $\#P_f=4$, which admits two non-homotopic characteristic curves $\beta$ and $\beta'$. It is hyperbolic with orbifold signature $(2,2,3,\infty)$. This non-uniqueness is not a Lattès phenomenon, nor does it arise from a mating decomposition. The map is constructed piecewise, and a numerical model is provided to illustrate its dynamics. Thurston obstructions are excluded by arc-lifting and intersection-number arguments.

We introduce new techniques for determining combinatorial properties of sets of multiple recurrence by considering weighted averages with quickly growing weights. Our main result is a far-reaching generalization of Szemerédi's Theorem which additionally confirms a conjecture of Bergelson-Moreira-Richter and contains as special cases both the Polynomial Szemerédi Theorem due to Bergelson-Leibman-Lesigne and the fact that if $f$ belongs to a broad class of smooth functions and satisfies $x^{d-1}\prec f(x)\prec x^d$ for some $d\in \mathbb{N}$ then for any $\ell\in \mathbb{N}$, any invertible measure preserving system $(X,\mathscr{B},\mu,T)$, and any $A\in \mathscr{B}$ with $\mu(A)>0$, the set $\{n\in \mathbb{N}: \mu(A\cap T^{-[f(n)]}A\cap T^{-2[f(n)]}A\cap \cdots\cap T^{-\ell[f(n)]}A )>0\}$ is thick, meaning that it contains arbitrarily long intervals of natural numbers. Additionally, we formulate and prove a generalization to weighted averages of Boshernitzan's criterion for uniform distribution which we use in the proof of our main result.

We study two-dimensional networks of FitzHugh-Nagumo oscillators in regimes that mimic cardiac tissue. The networks include small layers of cells that have fast-recovery characteristics and adjacent layers of cells that have slow-recovery. Numerical simulations show that the boundary between the fast- and slow-recovery cells can initiate fibrillatory-like behavior when activated by a single, healthy wave in the network. We discuss possible connections between these structures and the onset of atrial fibrillation, with the goal of improving our understanding and treatment of this costly condition. Our main conclusion is that a single, normal excitation of small areas of heterogeneity embedded in healthy tissue can initiate arrhythmias and the two dimensional network can then sustain irregular firing patterns in the form of double swirling-back curves and spiral waves. This occurs with a very simple model of excitable cells and we give a clear mechanistic description of the arrhythmia. The model illustrates how increasing heterogeneity in heart tissue can progressively cause first intermittent and then persistent arrhythmic behavior. Another conclusion is that the center of problematic firings can drift away from the heterogeneity into healthy tissue, thus presenting phantom targets for treatment.

The parasitic mite Varroa destructor poses a severe existential threat to global honey bee Apis mellifera populations. In this paper, we present a dynamical systems model of hive-mite interactions incorporating a eusocial Allee effect to evaluate the efficacy of chemical interventions. We partition treatments into “soft” miticides (targeting phoretic mites) and “harsh” miticides (penetrating the capped brood to target reproductive mites). Our bifurcation analysis reveals a fundamental trade-off: while harsh treatments effectively eradicate the protected mite reservoir by shifting the transcritical bifurcation boundary, they impose sublethal toxicity on the bees. We analytically show that exceeding a critical dosage threshold culminates in a catastrophic saddle-node bifurcation that guarantees colony collapse. To navigate this toxicity limit, we formulate an optimal control problem using Pontryagin's Maximum Principle. Numerical solutions reveal that a dynamic “shock and maintain” cocktail strategy optimally balances reservoir clearance with hive viability. Expanding the model to include treatment-resistant strains demonstrates that single-chemical reliance forces the hive onto a highly toxic, nearly unsustainable chemical treadmill. Finally, our mathematical framework yields a striking ecological prediction: because the eradication boundary scales intimately with the colony's carrying capacity, unchecked mite pressure will likely exert evolutionary forces that select for smaller, naturally resistant hives over the massive colonies favored by commercial agriculture.

This paper is concerned with multivariable stochastic Hamiltonian systems governed by an ordinary differential equation for the position and an Ito stochastic differential equation for the momentum. The momentum dynamics involve both conservative and nonconservative forcing including Langevin viscous damping and the Wiener process as a random force. The setting allows for rotational degrees of freedom, a non-quadratic potential energy function and position-dependent mass, diffusion and damping matrices. In addition to modelling nonlinear physical dynamics subject to random forcing, the system is motivated by a stochastic optimisation viewpoint, where the search for global minima of the potential energy (for example, using the gradient descent) is carried out through convergence to a Boltzmann equilibrium measure which favours smaller values of this function. Under certain regularity conditions, we show that the $\chi^2$-divergence or, equivalently, Renyi's second-order relative entropy with respect to the invariant measure has a nonpositive time derivative and thus provides a Lyapunov functional candidate for this convergence. However, this derivative vanishes at some moments of time, which is related to the behaviour of the probability distribution of the momentum conditioned on the position and has a link to a family of auxiliary quantum harmonic oscillator Hamiltonians parameterised by the position space of the underlying classical system. This position-momentum conditioning is used in order to prove that such pre-equilibrium break times in the entropy dissipation are isolated, thus making the $\chi^2$-divergence a strictly decreasing Lyapunov functional for the Fokker-Planck-Kolmogorov equation which governs the joint position-momentum distribution in such systems.

For a totally bounded metric dynamical system $(X, d, T)$ we introduce a new critical value $\mathfrak s(X, d, T)$ which quantifies the asymptotic separation of periodic orbits. More precisely, $\mathfrak s(X,d,T)$ is defined to be the supremum of all $s\ge 0$ for which there exists a sequence of periodic orbits $\{\mathcal O_k\}_{k=1}^\infty$ in $(X,d,T)$ such that \[ \lim_{k\to \infty} \#\mathcal O_k = +\infty \quad \text{and} \quad \liminf_{k\to\infty}\#\mathcal O_k\cdot\eta(\mathcal O_k)^s >0, \] where $\eta(\mathcal O_k)$ denotes the smallest distance between distinct points in $\mathcal O_k$. When the dynamical system is induced by a self-similar iterated function system $\mathcal F$ satisfying the strong separation condition, we prove that this critical value is equal to the Hausdorff dimension $s_{\mathcal F}$ of its self-similar attractor. Furthermore, under a quantitative separation condition on periodic orbits we show that the associated empirical periodic measures converge weakly to the normalized $s_{\mathcal F}$-dimensional Hausdorff measure on the self-similar attractor.

We prove the infinitesimal dynamical spectral rigidity of the circular Bunimovich stadium with sufficiently long flat edges, in the class of domains obtained by replacing its two circular arcs with strictly convex $C^{4}$ arcs and keeping its reflection symmetry across the axis parallel to the flat edges. That is, if a $C^{1}$ one-parameter family in this class passing through the stadium preserves the length spectrum, then its deformation function at the stadium vanishes identically. For flat edges of arbitrary length, the deformation functions of such families still span a finite-dimensional space.

Rotating detonation engines (RDEs) exhibit strongly nonlinear, multiscale wave dynamics that set the observed thermal field. High-fidelity simulations (DNS/LES) resolve these structures but remain computationally prohibitive, while low-order models such as the one-dimensional Koch-Kutz model capture circumferential wave motion yet lack the expressivity for high-frequency content. We use continuous data assimilation (nudging) to synchronize the Koch-Kutz solver with processed high-fidelity temperature data, introducing the prediction-observation mismatch as a relaxation source in the conserved energy equation; where observations are temporally sparse, interpolation supplies a target at every source update. As the nudging strength increases, the reduced model is progressively drawn onto the high-fidelity trajectory, and the forcing recorded along it provides an explicit, state-dependent estimate of the correction the model requires. We then train a Jacobian-regularized closure a priori on this recorded source. With the observation term removed, the corrected model advances autonomously, remains bounded, and recovers the temperature spectrum and the marginal statistics of the conserved variables relative to the baseline.

Consider a compact leaf $\mathcal{L}$ of a holomorphic foliation $\mathcal{F}$ such that all the leaves of the restriction of $\mathcal{F}$ to some neighborhood $U$ of $\mathcal{L}$ are closed in $U$. We show that there exists an invariant germ of analytic set $V$ in a neighborhood of $\mathcal{L}$, of dimension higher than $\dim (\mathcal{F})$, consisting of compact leaves, such that the volume of the leaves in $V$ is uniformly bounded by above. We also provide a globalization of this result if the ambient manifold is projective. In order to study closed leaves foliations, and via the holonomy representation, we introduce a new concept, of independent interest, for subgroups $G$ of germs of holomorphic diffeomorphisms, the so called torsion locus. We show that it is non-trivial if $G$ has finite orbits.

We establish level-1 large deviation principles for continuous time suspension flows over countable Markov shifts, a class of non compact dynamical systems. Our results both quantify deviations of Birkhoff averages from their typical values and, importantly, provide a large deviation principle for orbits that at a finite time are close to escaping to infinity. To overcome the challenges posed by non compactness, we introduce the notion of topological pressure at infinity and prove a variational principle linking it to its measure theoretic counterpart. This framework allows the study of a natural class of potentials, known as strongly positive recurrent, which exhibit robust thermodynamic properties. Moreover, we show that compactly supported ergodic measures are pressure dense in this setting. Applications to pressure gaps and examples are given.