CyberSec Research

We show positivity of the lower Fourier dimension for equilibrium states of nonlinear, area preserving, Axiom A diffeomorphisms on surfaces. To do so, we use the sum-product phenomenon to reduce Fourier decay to the study of some temporal distance function for a well chosen suspension flow, whose mixing properties reflects the nonlinearity of our base dynamics. We then generalize in an Axiom A setting the methods of Tsujii-Zhang, dealing with exponential mixing of three-dimensional Anosov flows arXiv:2006.04293. The nonlinearity condition is generic and can be checked in concrete contexts. As a corollary, we prove power Fourier decay for the density of states measure of the Fibonacci Hamiltonian, which is related to the measure of maximal entropy of the Fibonacci trace map. This proves positivity of the lower Fourier dimension for the spectrum of the Fibonacci Hamiltonian, and suggest strong phase-averaged dispersive estimates in quasicrystals.

We show that the winding of low-lying closed geodesics on the modular surface has a Gaussian limiting distribution when normalized by any natural notion of length.

In this paper, we proved that for every $C^1$ star vector fields on three-dimensional manifolds, every ergodic hyperbolic invariant measure which is not supported on singularities can be approximated by periodic measures, and the Lyapunov exponents of the ergodic hyperbolic invariant measure can also be approximated by the Lyapunov exponents of those periodic measures.

The Koopman operator has become a celebrated tool in modern dynamical systems theory for analyzing and interpreting both models and datasets. The linearity of the Koopman operator means that important characteristics about it, and in turn its associated nonlinear system, are captured by its eigenpairs and more generally its spectrum. Many studies point out that the spectrum of the Koopman operator has a multiplicative lattice structure by which eigenvalues and eigenfunctions can be multiplied to produce new eigenpairs. However, these observations fail to resolve whether the new eigenfunction remains in the domain of the Koopman operator. In this work, we prove that the spectrum of the Koopman operator associated to discrete-time dynamical systems has a multiplicative lattice structure. We further demonstrate that the Koopman operator associated to discrete-time stochastic process does not necessarily have such a structure, demonstrating an important nuance that lies at the heart of Koopman operator theory.

Using the wrapped Floer homology, we prove the existence of consecutive collisions at the primaries in the circular restricted three-body problem. We also prove the existence of a symmetric periodic orbit. These existence results are obtained for energy hypersurfaces slightly above the first critical value.

We consider the filtering problem with the partially observed Lorenz 96 model. Although the accuracy of the 3DVar filter applied to this problem has been established, that of the EnKF has not yet been. This study aims to establish the error bound of a variant of the EnKF, known as the PO method. By introducing the additive inflation and a projection of the background covariance to the observation space, we establish the error bound of the PO method. A numerical example validates theoretical findings and shows the potential to extend the analysis.

Jalali and Poor introduced an asymptotic framework for compressed sensing of stochastic processes, demonstrating that any rate strictly greater than the mean information dimension serves as an upper bound on the number of random linear measurements required for (universal) almost lossless recovery of $\psi^*$-mixing processes, as measured in the normalized $L^2$ norm. In this work, we show that if the normalized number of random linear measurements is strictly less than the mean information dimension, then almost lossless recovery of a $\psi^*$-mixing process is impossible by any sequence of decompressors. This establishes the mean information dimension as the fundamental limit for compressed sensing in this setting (and, in fact, the precise threshold for the problem). To this end, we introduce a new quantity, related to techniques from geometric measure theory: the correlation dimension rate, which is shown to be a lower bound for compressed sensing of arbitrary stationary stochastic processes.

We study the dynamics of generic volume-preserving automorphisms $f$ of a Stein manifold $X$ of dimension at least 2 with the volume density property. Among such $X$ are all connected linear algebraic groups (except $\mathbb{C}$ and $\mathbb{C}^*$) with a left- or right-invariant Haar form. We show that a generic $f$ is chaotic and of infinite topological entropy, and that the transverse homoclinic points of each of its saddle periodic points are dense in $X$. We present analogous results with similar proofs in the non-conservative case. We also prove the Kupka-Smale theorem in the conservative setting.

In this paper, we provide an upper bound on the number of maximal entropy ergodic measures with zero Lyapunov exponent for topologically transitive partially hyperbolic diffeomorphisms with compact one-dimensional center leaves on $\mathbb{T}^3$. Furthermore, we establish a comprehensive characterization of the support of these measures.

This paper considers a mean-field model of $n$ interacting particles whose state space is the unit circle, a generalization of the classical Kuramoto model. Global synchronization is said to occur if after starting from almost any initial state, all particles coalesce to a common point on the circle. We propose a general synchronization criterion in terms of $L_1$-norm of the third derivative of the particle interaction function. As an application we resolve a conjecture for the so-called self-attention dynamics (stylized model of transformers), by showing synchronization for all $\beta \ge -0.16$, which significantly extends the previous bound of $0\le \beta \le 1$ from Criscitiello, Rebjock, McRae, and Boumal (2024). We also show that global synchronization does not occur when $\beta < -2/3$.

This study focuses on the topological pressure of nonautonomous iterated function systems defined on a compact metric space. We establish an inequality relating two topological pressures associated with a factor map of nonautonomous iterated function systems.

In this paper we study the dynamics of Halley's and Traub's root-finding algorithms applied to a symmetric family of polynomials of degree $d+1\geq 3$. We discuss the (un)boundedness and simple connectivity of the immediate basins of attraction of the fixed points associated to the roots of the polynomials. In particular, we show the existence of polynomials for which the immediate basin of attraction of a root is bounded under Halley's method

The complement of the union of a collection of disjoint open disks in the $2$-sphere is called a Schottky set. We prove that a subset $S$ of the $2$-sphere is quasiconformally equivalent to a Schottky set if and only if every pair of complementary components of $S$ can be mapped to a pair of open disks with a uniformly quasiconformal homeomorphism of the sphere. Our theorem applies to Sierpi\'nski carpets and gaskets, yielding for the first time a general quasiconformal uniformization result for gaskets. Moreover, it contains Bonk's uniformization result for carpets as a special case and does not rely on the condition of uniform relative separation that is used in relevant works.

We introduce a new methodology for the analysis of the phenomenon of chaotic itinerancy in a dynamical system using the notion of entropy and a clustering algorithm. We determine systems likely to experience chaotic itinerancy by means of local Shannon entropy and local permutation entropy. In such systems, we find quasi-stable states (attractor ruins) and chaotic transition states using a density-based clustering algorithm. Our approach then focuses on examining the chaotic itinerancy dynamics through the characterization of residence times within these states and chaotic transitions between them with the help of some statistical tests. The effectiveness of these methods is demonstrated on two systems that serve as well-known models exhibiting chaotic itinerancy: globally coupled logistic maps (GCM) and mutually coupled Gaussian maps. In particular, we conduct comprehensive computations for a large number of parameters in the GCM system and algorithmically identify itinerant dynamics observed previously by Kaneko in numerical simulations as the coherent and intermittent phases.

In this work, we introduce a quadratically convergent and dynamically consistent integrator specifically designed for the replicator dynamics. The proposed scheme combines a two-stage rational approximation with a normalization step to ensure confinement to the probability simplex and unconditional preservation of non-negativity, invariant sets and equilibria. A rigorous convergence analysis is provided to establish the scheme's second-order accuracy, and an embedded auxiliary method is devised for adaptive time-stepping based on local error estimation. Furthermore, a discrete analogue of the quotient rule, which governs the evolution of component ratios, is shown to hold. Numerical experiments validate the theoretical results, illustrating the method's ability to reproduce complex dynamics and to outperform well-established solvers in particularly challenging scenarios.

We investigate the interplay between monomial first integrals, polynomial invariants of certain group action, and the Poincar\'{e}-Dulac normal forms for autonomous systems of ODEs with diagonal matrix of the linear part. Using tools from computational algebra, we develop an algorithmic approach for identifying generators of the algebras of monomial and polynomial first integrals, which works in the general case where the matrix of the linear part includes algebraic complex eigenvalues. Our method also provides a practical tool for exploring the algebraic structure of polynomial invariants and their relation to the Poincar\'{e}-Dulac normal forms of the underlying vector fields.

In this paper, we study the local behaviour of solutions near the fixed points of a reaction-diffusion equation with discontinuous nonlinearity. By employing an appropriate linearization around the fixed points, which involves the Dirac delta distribution, we analyze the stability of the stationary solutions and demonstrate that they exhibit a saddle-point structure. As a result, we establish the hyperbolicity of the fixed points.

We present a mechanism for Arnold diffusion in energy in a model of the elliptic Hill four-body problem. Our model is expressed as a small perturbation of the circular Hill four-body problem, with the small parameter being the eccentricity of the orbits of the primaries. The mechanism relies on the existence of two normally hyperbolic invariant manifolds (NHIMs), and on the corresponding homoclinic and heteroclinic connections. The dynamics along homoclinic/heteroclinic orbits is encoded via scattering maps, which we compute numerically. We provide two arguments for diffusion. In the first argument, we successively apply a single scattering map and use Birkhoff's Ergodic Theorem to obtain pseudo-orbits that, on average, gain energy. In the second argument, we use two scattering maps and, at each step, select one that increases energy. Either argument yields pseudo-orbits of scattering maps along which the energy grows by an amount independent of the small parameter. A shadowing lemma concludes the existence of diffusing orbits.