Set Of Periodic Points Research Articles

Pure Point Diffraction and Almost Periodicity

AbstractThis article deals with pure point diffraction and its connection to various notions of almost periodicity. We explain why the Fibonacci chain does not fit into the classical concept of Bohr almost periodicity and how it fits into the classes of mean, Besicovitch and Weyl almost periodic point sets. We report on recent results which characterize pure point diffraction as mean almost periodicity of the underlying structure, and discuss how the complex amplitudes fit into this picture.

Stable/unstable holonomies, density of periodic points, and transitivity for continuum-wise hyperbolic homeomorphisms

We discuss different regularities on stable/unstable holonomies of cw-hyperbolic homeomorphisms and prove that if a cw-hyperbolic homeomorphism has continuous joint stable/unstable holonomies, then it has a dense set of periodic points in its non-wandering set. For that, we prove that the hyperbolic cw-metric (introduced in Artigue et al (2024 J. Differ. Equ. 378 512–38)) can be adapted to be self-similar (as in Artigue (2018 Ergodic Theory Dyn. Syst. 38 2422–46)) and, in this case, continuous joint stable/unstable holonomies are pseudo-isometric. We also prove transitivity of cw-hyperbolic homeomorphisms assuming that the stable/unstable holonomies are isometric. In the case the ambient space is a surface, we prove that a cw F -hyperbolic homeomorphism has continuous joint stable/unstable holonomies when every bi-asymptotic sector is regular.

Chaotic dynamics of a continuous and discrete generalized Ziegler pendulum

We present analytical and numerical results on integrability and transition to chaotic motion for a generalized Ziegler pendulum, a double pendulum subject to an angular elastic potential and a follower force. Several variants of the original dynamical system, including the presence of gravity and friction, are considered, in order to analyze whether the integrable cases are preserved or not in presence of further external forces, both potential and non-potential. Particular attention is devoted to the presence of dissipative forces, that are analyzed in two different formulations. Furthermore, a study of the discrete version is performed. The analysis of periodic points, that is presented up to period 3, suggests that the discrete map associated to the dynamical system has not dense sets of periodic points, so that the map would not be chaotic in the sense of Devaney for a choice of the parameters that corresponds to a general case of chaotic motion for the original system.

Accelerating material property prediction using generically complete isometry invariants

Periodic material or crystal property prediction using machine learning has grown popular in recent years as it provides a computationally efficient replacement for classical simulation methods. A crucial first step for any of these algorithms is the representation used for a periodic crystal. While similar objects like molecules and proteins have a finite number of atoms and their representation can be built based upon a finite point cloud interpretation, periodic crystals are unbounded in size, making their representation more challenging. In the present work, we adapt the Pointwise Distance Distribution (PDD), a continuous and generically complete isometry invariant for periodic point sets, as a representation for our learning algorithm. The PDD distinguished all (more than 660 thousand) periodic crystals in the Cambridge Structural Database as purely periodic sets of points without atomic types. We develop a transformer model with a modified self-attention mechanism that combines PDD with compositional information via a spatial encoding method. This model is tested on the crystals of the Materials Project and Jarvis-DFT databases and shown to produce accuracy on par with state-of-the-art methods while being several times faster in both training and prediction time.

Synchronizing dynamical systems: Their groupoids and 𝐶*-algebras

Building on work of Ruelle and Putnam in the Smale space case, Thomsen defined the homoclinic and heteroclinic C ∗ C^\ast -algebras for an expansive dynamical system. In this paper we define a class of expansive dynamical systems, called synchronizing dynamical systems, that exhibit hyperbolic behavior almost everywhere. Synchronizing dynamical systems generalize Smale spaces (and even finitely presented systems). Yet they still have desirable dynamical properties such as having a dense set of periodic points. We study various C ∗ C^\ast -algebras associated with a synchronizing dynamical system. Among other results, we show that the homoclinic algebra of a synchronizing system contains an ideal which behaves like the homoclinic algebra of a Smale space.

On limit sets of simplest skew products defined on multidimensional cells

The purpose of this work is to describe two important types of limit sets of the most simple skew products of interval maps, the phase space of each of which is a compact n-dimensional cell (n ≥ 2): firstly, a non-wandering set and, secondly, ω-limit sets of trajectories. Methods. A method for investigating of a nonwandering set (new even for the two-dimensional case) is proposed, based on the use of the concept of C0-Ω-blow up in continuous closed interval maps, and the concept of C0-Ω-blow up introduced in the work in the family of continuous fibers maps. To describe the ω-limit sets, the technique of special series constructed for the trajectory and containing an information about its asymptotic behavior is used. Results. A complete description is given of the nonwandering set of the continuous simplest skew product of the interval maps, that is, a continuous skew product on a compact n-dimensional cell, the set of (least) periods of periodic points of which is bounded. The results obtained in the description of a nonwandering set are used in the study of ω-limit sets. The paper describes a topological structure of ω-limit sets of the maps under consideration. Sufficient conditions have been found under which the ω-limit set of the trajectory is a periodic orbit, as well as the necessary conditions for the existence of one-dimensional ω-limit sets. Conclusion. Further development of the C0-Ω-blow up technique in the family of maps in fibers will allow us to describe the structure of a nonwandering set of skew products of one-dimensional maps, in particular, with a closed set of periodic points defined on the simplest manifolds of arbitrary finite dimension. Further development of the theory of special divergent series constructed in the work will allow us to proceed to the description of ω-limit sets of arbitrary dimension d, where 2 ≤ d ≤ n - 1, n ≥ 3, in the simplest skew products.

Relatively pointwise recurrence on local dendrites

Let X be a local dendrite and let f : X → X be a continuous map. Denote by P ( f ) and R ( f ) the sets of periodic and recurrent points of f, respectively. We say that f is relatively pointwise recurrent if R ( f ) ¯ = X . We show that, if X is almost meshed (i.e the union of the interior of all free arcs is dense in X), then f is relatively pointwise recurrent if and only if either f is topologically conjugate to an irrational rotation on the circle or it has a dense set of periodic points.

A mechanism for ejecting a horseshoe from a partially hyperbolic chain recurrence class

AbstractWe give a $C^1$ -perturbation technique for ejecting an a priori given finite set of periodic points preserving a given finite set of homo/heteroclinic intersections from a chain recurrence class of a periodic point. The technique is first stated under a simpler setting called a Markov iterated function system, a two-dimensional iterated function system in which the compositions are chosen in a Markovian way. Then we apply the result to the setting of three-dimensional partially hyperbolic diffeomorphisms.

Density Functions of Periodic Sequences of Continuous Events

Periodic Geometry studies isometry invariants of periodic point sets that are also continuous under perturbations. The motivations come from periodic crystals whose structures are determined in a rigid form, but any minimal cells can discontinuously change due to small noise in measurements. For any integer kge 0, the density function of a periodic set S was previously defined as the fractional volume of all k-fold intersections (within a minimal cell) of balls that have a variable radius t and centers at all points of S. This paper introduces the density functions for periodic sets of points with different initial radii motivated by atomic radii of chemical elements and by continuous events occupying disjoint intervals in time series. The contributions are explicit descriptions of the densities for periodic sequences of intervals. The new densities are strictly stronger and distinguish periodic sequences that have identical densities in the case of zero radii.

Remarks on minimal sets on dendrites and finite graphs

ABSTRACT Let X be a dendrite and let be a continuous map that has an infinite minimal set M in X. In the article we obtain conditions on the structure of a dendrite X and the set M, under which f has an arc horseshoe, and, hence, f has a positive topological entropy. We show that this result is not correct both for continuous maps with an empty set of periodic points, and for continuous maps with a non-empty set of periodic points on finite graphs.

Decidability of CPC-irreducibility of subshifts of finite type over free groups

This paper attempts to study the irreducibility on complete prefix code (CPC-irreducibility) of a Markov shift over a free group, a topological mixing property first considered for shift spaces over free semigroups that induces chaotic behavior such as the existence of a dense set of periodic points. An example shows that the ({textsf{d}},{textsf{c}})-reduction, an effective algorithm of determination of CPC-irreducibility of Markov shifts over free semigroups (Ban et al. in J Stat Phys 177:1043–1062, 2019), fails for general Markov shifts over free groups. This paper reveals an algorithm for determining the CPC-irreducibility of Markov shifts over both free semigroups and groups. Furthermore, such an examination is finitely checkable, and an upper bound for the complexity of the algorithm is provided.

Computing periodic points on Veech surfaces

A non-square-tiled Veech surface has finitely many periodic points, i.e. points with finite orbit under the affine automorphism group. We present an algorithm that inputs a non-square-tiled Veech surface and outputs its set of periodic points. Our algorithm serves as a new proof of the finiteness of periodic points for non-square-tiled Veech surfaces. We apply our algorithm to Prym eigenforms in the minimal stratum in genus 3, proving that in low discriminant these surfaces do not have periodic points, except for the fixed points of the Prym involution.

Topological sequence entropy and topological dynamics of tree maps

We prove that a zero topological entropy continuous tree map always displays zero topological sequence entropy when it is restricted to its non-wandering and chain recurrent sets. In addition, we show that a similar result is not possible when the phase space is a dendrite even when we consider only the restriction on the set of periodic points.

A fixed-point curve theorem for finite-orbits local diffeomorphisms

AbstractWe study local biholomorphisms with finite orbits in some neighborhood of the origin since they are intimately related to holomorphic foliations with closed leaves. We describe the structure of the set of periodic points in dimension 2. As a consequence we show that given a finite-orbits local biholomorphism F, in dimension 2, there exists an analytic curve passing through the origin and contained in the fixed-point set of some non-trivial iterate of $F.$ As an application we obtain that at least one eigenvalue of the linear part of F at the origin is a root of unity. Moreover, we show that such a result is sharp by exhibiting examples of finite-orbits local biholomorphisms such that exactly one of the eigenvalues is a root of unity. These examples are subtle since we show they cannot be embedded in one-parameter groups.

Topological equicontinuity and topological uniform rigidity for dynamical system

In this paper, we study topological equicontinuity, topological uniform rigidity and their properties. For a dynamical system, on a compact, T3 space, we study relations among the set of recurrent points of the map, the set of non-wandering points of the map and the intersection of the range sets of all iterations of the map. We define topological version of uniform rigidity and show that on a compact and T3 space any dynamical system is topologically uniformly rigid if it is first countable, almost topologically equicontinuous and transitive or it is second countable, topologically equicontinuous and has a dense set of periodic points. We show that a topologically uniformly rigid dynamical system, on a compact, Hausdorff space, has zero topological entropy. Moreover, we provide necessary examples and counterexamples.

Dynamics of Sofic Shifts

In this paper, we provide a characterization for the subshifts of finite type (SFT) in terms of Cellular automata (CA). In addition, we prove that 1. The following are equivalent for a non-singleton subshift of finite type XF. a) XF is transitive and Per(XF), the set of periodic points of XF, is cofinite b) XF is weak mixing c) XF is mixing. 2. For non-singleton sofic shifts, only the statements (a) and (b) are equivalent.

A relation between entropy and transitivity of Anosov diffeomorphisms

It is known that transitive Anosov diffeomorphisms have a unique measure of maximal entropy (MME). Here we discuss the converse question. Transitivity of Anosov diffeomorphisms can be reached under suitable hypotheses on Lyapunov exponents on the set of periodic points and the structure of the MME. In another way, assuming together the uniqueness of MME and that every point is regular, in Oseledec’s Theorem sense, also we can get the transitivity of Anosov diffeomorphisms in this setting.

Dynamics of monotone maps on regular curves

Let X be a regular curve and let f:X→X be a monotone map. In this paper, the nonwandering set of f and the structure of ω-limit (resp. α-limit) sets for f are investigated. Denote by Per(f) (resp. Ω(f)) the set of periodic points (resp. nonwandering points) of f. We show that whenever the set Per(f) is empty, then f admits a unique minimal set which is totally minimal and either a circle or a Cantor set. If Per(f) is nonempty, we show that Ω(f)=Per(f)‾. In addition, every infinite minimal set is a Cantor set which is an adding machine. On the other hand, we show that f has no Li-Yorke pairs, in particular it has zero entropy. Further results related to the equicontinuity are also obtained.

Periodic points and shadowing for generic Lebesgue measure-preserving interval maps

In this article we study dynamical behaviour of generic Lebesgue measure-preserving interval maps. We show that for each k ⩾ 1 the set of periodic points of period at least k is a Cantor set of Hausdorff dimension zero and of upper box dimension one. Moreover, we obtain analogous results also in the context of generic Lebesgue measure-preserving circle maps. Furthermore, building on the former results, we show that there is a dense collection of transitive Lebesgue measure-preserving interval maps whose periodic points have full Lebesgue measure and whose periodic points of period k have positive measure for each k ⩾ 1. Finally, we show that the generic continuous maps of the interval which preserve the Lebesgue measure satisfy the shadowing and periodic shadowing property.

Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle

This paper introduces class G containing Cartesian products of orientation-changing rough transformations of the circle and studies their dynamics. As it is known from the paper of A. G. Maier non-wandering set of orientation-changing diffeomorphism of the circle consists of 2q periodic points, where q is some natural number. So Cartesian products of two such diffeomorphisms has 4q1q2 periodic points where q1 corresponds to the first transformation and q2 corresponds to the second one. The authors describe all possible types of the set of periodic points, which contains 2q1q2 saddle points, q1q2 sinks, and q1q2 sources; 4 points from mentioned 4q1q2 periodic ones are fixed, and the remaining 4q1q2−4 points have period 2. In the theory of smooth dynamical systems, a very useful result is that, given a diffeomorphism f of a manifold, one can construct a flow on a manifold with dimension one greater; this flow is called the suspension over f. The authors introduce the concept of suspension over diffeomorphisms of class G, describe all possible types of suspension orbits and the number of these orbits. Besides that, the authors prove a theorem on the topology of the manifold on which the suspension is given. Namely, the carrier manifold of the flows under consideration is homeomorphic to the closed 3-manifold T2×[0,1]/φ, where φ:T2→T2. The main result of the paper says that suspensions over diffeomorphisms of the class G are topologically equivalent if and only if corresponding diffeomorphisms are topologically conjugate. The idea of the proof is to show that the topological equivalence of the suspensions ϕt and ϕ′t implies the topological conjugacy of ϕ and ϕ′.