We obtain conditions, which when fulfilled, permit to transform the coordinates of a dynamical system into pairs of canonical ones for some Hamiltonian system. These conditions, restricted to the class of coordinate transformations which act on each coordinate independently, are greatly simplified. However, they are surprisingly successful in defining canonical coordinates and an associated Hamiltonian for several test examples. So, a method is proposed to exploit these simple transformations in a systematic manner.
We study the global behavior of the trajectories of the polynomial system $\dot x = x - x^2 y+p x y^2+ y^3, \ \dot y=y+p y^3 , \ \ p\in \mathbb{R}$. Our study is related to the paper {\it Alarcon B., Castro S.B.S.D., Labouriau I.S.} Glodal planar dynamics with star nodes: beyond Hilbert's 16th problem//arXiv:2106/07516v2 [math.DS].
We study the global behavior of the trajectories of the polynomial system $\dot x = x - x^2 y+p x y^2+ y^3, \ \dot y=y+p y^3 , \ \ p\in \mathbb{R}$. Our study is related to the paper {\it Alarcon B., Castro S.B.S.D., Labouriau I.S.} Glodal planar dynamics with star nodes: beyond Hilbert's 16th problem//arXiv:2106/07516v2 [math.DS].
We consider the propagation of equatorial waves of small amplitude, in a flow with an underlying non-uniform current. Without making the too restrictive rigid-lid approximation, by exploiting the available Hamiltonian structure of the problem, we derive the dispersion relation for the propagation of coupled long-waves: a surface wave and an internal wave. Also, we investigate the above-mentioned model of wave-current interactions in the general case with arbitrary vorticities.
Vivina L. Barutello, Irene De Blasi, Susanna Terracini
We prove the presence of topological chaos at high internal energies for a new class of mechanical refraction billiards coming from Celestial Mechanics. Given a smooth closed domain $D\in\mathbb{R}^2$, a central mass generates a Keplerian potential in it, while, in $\mathbb{R}^2\setminus \overline{D}$, a harmonic oscillator-type potential acts. At the interface, Snell's law of refraction holds. The chaoticity result is obtained by imposing progressive assumptions on the domain, arriving to geometric conditions which hold generically in $C^1$. The workflow starts with the existence of a symbolic dynamics and ends with the proof of topological chaos, passing through the analytic non-integrability and the presence of multiple heteroclinic connections between different equilibrium saddle points. This work can be considered as the final step of the investigation carried on in arXiv:2108.11159 [math.DS] and arXiv:2105.02108 [math.DS].
This paper presents some numerical experiments in relation with the theoretical study of the ergodic short-term behaviour of discretizations of expanding maps done in arXiv:2206.07991 [math.DS]. Our aim is to identify the phenomena driving the evolution of the Cramér distance between the $t$-th iterate of Lebesgue measure by the dynamics $f$ and the $t$-th iterate of the uniform measure on the grid of order $N$ by the discretization on this grid. Based on numerical simulations we propose some conjectures on the effects of numerical truncation from the ergodic viewpoint.
This paper is aimed to study the ergodic short-term behaviour of discretizations of circle expanding maps. More precisely, we prove some asymptotics of the distance between the $t$-th iterate of Lebesgue measure by the dynamics $f$ and the $t$-th iterate of the uniform measure on the grid of order $N$ by the discretization on this grid, when $t$ is fixed and the order $N$ goes to infinity. This is done under some explicit genericity hypotheses on the dynamics, and the distance between measures is measured by the mean of \emph{Cramér} distance. The proof is based on a study of the corresponding linearized problem, where the problem is translated into terms of equirepartition on tori of dimension exponential in $t$. A numerical study associated to this work is presented in arXiv:2206.08000 [math.DS].
AbstractWe give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice $${\mathbb{Z}^2}$$Z2 . We also determine the asymptotic spectral gap, asymptotic mixing time, and prove a cutoff phenomenon for the recurrent state abelian sandpile model on the torus $${\left(\mathbb{Z}/m\mathbb{Z}\right)^2}$$Z/mZ2 . The techniques use analysis of the space of functions on $${\mathbb{Z}^2}$$Z2 which are harmonic modulo 1. In the course of our arguments, we characterize the harmonic modulo 1 functions in $${\ell^p(\mathbb{Z}^2)}$$ℓp(Z2) as linear combinations of certain discrete derivatives of Green’s functions, extending a result of Schmidt and Verbitskiy (Commun Math Phys 292(3):721–759, 2009. arXiv:0901.3124 [math.DS]).
In [3] (Rend. Lincei Mat. Appl. 26 (2015), 1-10; see also arXiv:1503.08145 [math.DS]) the following result has been announced: Theorem. Consider a real-analytic nearly-integrable mechanical system with potential $f$, namely, a Hamiltonian system with real-analytic Hamiltonian $$H(y,x)=\frac12 \sum_{i=1}^n y_i^2 +\epsilon f(x)\ ,$$ $(y,x)\in{\mathbb R}^n\times{\mathbb T}^n$ being standard action--angle variables. For "general non-degenerate" potentials $f$'s there exists $\epsilon_0,a>0$ such that, if $0<\epsilon<\epsilon_0$, then the Liouville measure of the complementary of $H$-invariant tori is smaller than $\epsilon|\log \epsilon|^a$. In this paper we provide a proof of such result.
This paper forms the second part of a series of three papers by the authors concerning the structure of nilspaces of Antolín Camarena and Szegedy. A nilspace is a compact space X X together with closed collections of cubes C n ( X ) ⊆ X 2 n C_n(X)\subseteq X^{2^n} , n = 1 , 2 , … n=1,2,\ldots satisfying some natural axioms. From these axioms it follows that a nilspace can be built as a finite tower of extensions where each of the successive fibers is a compact abelian group. Our main result is a new proof of a result due to Antolín Camarena and Szegedy [Nilspaces, nilmanifolds and their morphisms, arXiv:1009.3825v3 (2012)] stating that if each of these groups is a torus, then X X is isomorphic (in a strong sense) to a nilmanifold G / Γ G/\Gamma . We also extend the theorem to a setting where the nilspace arises from a dynamical system ( X , T ) (X,T) . These theorems are a key stepping stone towards the general structure theorem in [The structure theory of nilspaces III: Inverse limit representations and topological dynamics, arXiv:1605.08950v1 [math.DS] (2016)] (which again closely resembles the main theorem of Antolín Camarena and Szegedy). The main technical tool, enabling us to deduce algebraic information from topological data, consists of existence and uniqueness results for solutions of certain natural functional equations, again modelled on the theory in Antolín Camarena and Szegedy’s paper.
In a pioneering classic, Warren McCulloch and Walter Pitts proposed a model of the central nervous system. Motivated by EEG recordings of normal brain activity, Chvátal and Goldsmith asked whether or not these dynamical systems can be engineered to produce trajectories which are irregular, disorderly, apparently unpredictable. We show that they cannot build weak pseudorandom functions.
We investigate the existence of wandering Fatou components for polynomial skew-products in two complex variables. In 2004, the non-existence of wandering domains near a super-attracting invariant fiber was shown in Lilov (Fatou theory in two dimensions, PhD thesis, University of Michigan, 2004). In 2014, it was shown in Astorg et al. (Ann Math, arXiv:1411.1188 [math.DS], 2014) that wandering domains can exist near a parabolic invariant fiber. In Peters and Vivas (Math Z, arXiv:1408.0498, 2014), the geometrically attracting case was studied, and we continue this study here. We prove the non-existence of wandering domains for subhyperbolic attracting skew-products; this class contains the maps studied in Peters and Vivas (Math Z, arXiv:1408.0498, 2014). Using expansion properties on the Julia set in the invariant fiber, we prove bounds on the rate of escape of critical orbits in almost all fibers. Our main tool in describing these critical orbits is a possibly singular linearization map of unstable manifolds.
We prove the results in [1] using Theorem 1 of the recent paper [2] by Crovisier and Yang. References: [1] Arbieto, A., Rojas, C., Santiago, B., Existence of attractors, homoclinic tangencies and singular-hyperbolicity for flows, arXiv:1308.1734v1 [math.DS] 8 Aug 2013. [2] Crovisier, S., Yang, D., On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis, arXiv:1404.5130v1 [math.DS] 21 Apr 2014.
Let $K/\mathbf Q$ be a degree $d$ extension. Inside the ring of integers $\mathcal O_K$ we define the set of $k$-free integers $\mathcal F_k$ and a natural $\mathcal O_K$-action on the space of binary $\mathcal O_K$-indexed sequences, equipped with an $\mathcal O_K$-invariant probability measure associated to $\mathcal F_k$. We prove that this action is ergodic, has pure point spectrum and is isomorphic to a $\mathbf Z^d$-action on a compact abelian group. In particular, it is not weakly mixing and has zero measure-theoretical entropy. This work generalizes the paper by the first author and Sinai arXiv:1112.4691 [math.DS] where $K=\mathbf Q$ and $k=2$.
Let $K/\mathbf Q$ be a degree $d$ extension. Inside the ring of integers $\mathcal O_K$ we define the set of $k$-free integers $\mathcal F_k$ and a natural $\mathcal O_K$-action on the space of binary $\mathcal O_K$-indexed sequences, equipped with an $\mathcal O_K$-invariant probability measure associated to $\mathcal F_k$. We prove that this action is ergodic, has pure point spectrum and is isomorphic to a $\mathbf Z^d$-action on a compact abelian group. In particular, it is not weakly mixing and has zero measure-theoretical entropy. This work generalizes the paper by the first author and Sinai arXiv:1112.4691 [math.DS] where $K=\mathbf Q$ and $k=2$.
This is an expanded version of [arXiv:1107.4836v1 [math.DS]]. Using techniques from [Chapter XI, The Selberg Trace Formula, in Eigenvalues in Riemannian Geometry, by Isaac Chavel], in which a differential-geometrically intrinsic treatment of counterparts of classical electrostatics was introduced, it is shown that on some compact manifolds, certain stable configurations of points which mutually repel along all interconnecting geodesics become equidistributed as the number of points increases.