Set Of Periodic Points Research Articles

Special α-limit points and unilateral γ-limit points for graph maps

Let G be a graph and f: G → G be a continuous map. Denote by P(f), R(f), SA(f) and UΓ(f) the sets of periodic points, recurrent points, special α-limit points and unilateral γ-limit points of f, respectively. In this paper, we show that R(f) ⊂ SA(f) = UΓ(f) ∪ P(f) ⊂ \(\overline {R(f)} \).

Recurrent points and non-wandering points of graph maps

Let G be a graph and f : G → G be a continuous map. Denote by P ( f ) , R ( f ) and Ω ( f ) the sets of periodic points, recurrent points and non-wandering points of f, respectively. In this paper we show that: (1) If L = ( x , y ) is an open arc contained in an edge of G such that { f m ( x ) , f k ( y ) } ⊂ ( x , y ) for some m , k ∈ N , then R ( f ) ∩ ( x , y ) ≠ ∅ ; (2) Any isolated point of P ( f ) is also an isolated point of Ω ( f ) ; (3) If x ∈ Ω ( f ) − Ω ( f n ) for some n ∈ N , then x is an eventually periodic point. These generalize the corresponding results in W. Huang and X. Ye (2001) [9] and J. Xiong (1983, 1986) [17,19] on interval maps or tree maps.

Periodic points of equivariant maps

We assume that $X$ is a compact connected polyhedron, $G$ is a finite group acting freely on $X$, and $f:X\to X$ a $G$-equivariant map. We find formulae for the least number of $n$-periodic points in the equivariant homotopy class of $f$, i.e., $\inf_h |(\mathrm{Fix}(h^n)|$ (where $h$ is $G$-homotopic to $f$). As an application we prove that the set of periodic points of an equivariant map is infinite provided the action on the rational homology of $X$ is trivial and the Lefschetz number $L(f^n)$ does not vanish for infinitely many indices $n$ commeasurable with the order of $G$. Moreover, at least linear growth, in $n$, of the number of points of period $n$ is shown.

Relatively pointwise recurrent graph map

Let f be a self-continuous map of a graph G. Let P(f) and R(f) denote the sets of periodic points and recurrent points respectively. We say that the map f is relatively recurrent if R(f) = G. In this paper, it is shown that f is relatively recurrent if and only if one of the following statements holds: (a) G is a circle and f is a homeomorphism topologically conjugate to an irrational rotation of the unit circle S 1 ; (b) P(f) = G. Part (b) extends a result of Blokh.

The [formula omitted] closing lemma for generic [formula omitted] endomorphisms

Given a compact m-dimensional manifold M and 1⩽r⩽∞, consider the space Cr(M) of self mappings of M. We prove here that for every map f in a residual subset of C1(M), the C1 closing lemma holds. In particular, it follows that the set of periodic points is dense in the nonwandering set of a generic C1 map. The proof is based on a geometric result asserting that for generic Cr maps the future orbit of every point in M visits the critical set at most m times.

Diffraction of limit periodic point sets

Limit periodic point sets are aperiodic structures with pure point diffraction supported on a countably, but not finitely generated Fourier module that is based on a lattice and certain integer multiples of it. Examples are cut and project sets with p-adic internal spaces. We illustrate this by explicit results for the diffraction measures of two examples with 2-adic internal spaces. The first and well-known example is the period doubling sequence in one dimension, which is based on the period doubling substitution rule. The second example is a weighted planar point set that is derived from the classic chair tiling in the plane. It can be described as a fixed point of a block substitution rule.

Differential properties and attracting sets of a simplest skew product of interval maps

For a skew product of interval maps with a closed set of periodic points, the dependence of the structure of its -limit sets on its differential properties is investigated. An example of a map in this class is constructed which has the maximal differentiability properties (within a certain subclass) with respect to the variable , is -smooth in the -variable and has one-dimensional -limit sets. Theorems are proved that give necessary conditions for one-dimensional -limit sets to exist. One of them is formulated in terms of the divergence of the series consisting of the values of a function of ; this function is the -norm of the deviation of the restrictions of the fibre maps to some nondegenerate closed interval from the identity on the same interval. Another theorem is formulated in terms of the properties of the partial derivative with respect to of the fibre maps. A complete description is given of the -limit sets of certain class of -smooth skew products satisfying some natural conditions.Bibliography: 33 titles.

Perfect, strongly eutactic lattices are periodic extreme

We introduce a parameter space for periodic point sets, given as unions of m translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a periodic sphere packing with greater density. This applies in particular to the densest known lattice sphere packings in dimension d ⩽ 8 and d = 24 .

Graph maps whose periodic points form a closed set

A continuous map f from a graph G to itself is called a graph map. Denote by P ( f ) , R ( f ) , ω ( f ) , Ω ( f ) and CR ( f ) the sets of periodic points, recurrent points, ω-limit points, non-wandering points and chain recurrent points of f respectively. It is well known that P ( f ) ⊂ R ( f ) ⊂ ω ( f ) ⊂ Ω ( f ) ⊂ CR ( f ) . Block and Franke (1983) [5] proved that if f : I → I is an interval map and P ( f ) is a closed set, then CR ( f ) = P ( f ) . In this paper we show that if f : G → G is a graph map and P ( f ) is a closed set, then ω ( f ) = R ( f ) . We also give an example to show that, for general graph maps f with P ( f ) being a closed set, the conclusion ω ( f ) = R ( f ) cannot be strengthened to Ω ( f ) = R ( f ) or ω ( f ) = P ( f ) .

The limit of a pedal sequence of triangles

Several authors have studied the dynamics of the pedal sequence in which a triangle ABC is replaced by its pedal triangle (whose vertices are the feet of the perpendiculars from each vertex to the opposite side) and this procedure is repeated indefinitely. They found cases where the shape of the triangle repeats periodically and showed that the map is ergodic and modelled by the full shift on four symbols. We show here that the parameter space of shapes of triangles is a tetrahedron and that this has as a double cover (branched over four points) the two-torus where the dynamics is multiplication by 2. The pedal sequence thus inherits its dense set of periodic points and also the mixing property from this toral endomorphism. We find an expression for the limit point of a pedal sequence and show that it depends continuously but not differentiably on the shape of the original triangle.

Dynamics groups of asynchronous cellular automata

We say that a finite asynchronous cellular automaton (or more generally, any sequential dynamical system) is ?-independent if its set of periodic points are independent of the order that the local functions are applied. In this case, the local functions permute the periodic points, and these permutations generate the dynamics group. We have previously shown that exactly 104 of the possible $2^{2^{3}}=256$ cellular automaton rules are ?-independent. In the article, we classify the periodic states of these systems and describe their dynamics groups, which are quotients of Coxeter groups. The dynamics groups provide information about permissible dynamics as a function of update sequence and, as such, connect discrete dynamical systems, group theory, and algebraic combinatorics in a new and interesting way. We conclude with a discussion of numerous open problems and directions for future research.

A topological lens for a measure-preserving system

AbstractWe introduce a functor which associates to every measure-preserving system (X,ℬ,μ,T) a topological system $(C_2(\mu ),\tilde {T})$ defined on the space of twofold couplings of μ, called the topological lens of T. We show that often the topological lens ‘magnifies’ the basic measure dynamical properties of T in terms of the corresponding topological properties of $\tilde {T}$. Some of our main results are as follows: (i) T is weakly mixing if and only if $\tilde {T}$ is topologically transitive (if and only if it is topologically weakly mixing); (ii) T has zero entropy if and only if $\tilde {T}$ has zero topological entropy, and T has positive entropy if and only if $\tilde {T}$ has infinite topological entropy; (iii) for T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney).

Preservation and destruction of periodic orbits by symplectic integrators

We investigate what happens to periodic orbits and lower- dimensional tori of Hamiltonian systems under discretisation by a symplec- tic one-step method where the system may have more than one degree of freedom. We use an embedding of a symplectic map in a quasi-periodic non- autonomous flow and a KAM result of Jorba and Villaneuva (J Nonlinear Sci 7:427-473, 1997) to show that periodic orbits persist in the new flow, but with slightly perturbed period and an additional degree of freedom when the map is non-resonant with the periodic orbit. The same result holds for lower- dimensional tori with more degrees of freedom. Numerical experiments with the two degree of freedom Henon-Heiles system are used to show that in the case where the method is resonant with the periodic orbit, the orbit is destroyed and replaced by two invariant sets of periodic points—analogous to what is understood for one degree of freedom systems.

Nonwandering set of points of skew-product maps with base having closed set of periodic points

The aim of the present paper is to study the structure of the nonwandering set of points Ω ( ⋅ ) for the skew-product maps C Δ ∗ ( I 2 ) of the unit square I 2 = [ 0 , 1 ] × [ 0 , 1 ] , ( x , y ) → ( f ( x ) , g ( x , y ) ) , with base f having closed set of periodic points. For every F ∈ C Δ ∗ ( I 2 ) and every point ( x , y ) with x periodic of period p x by f and y not chain recurrent of F p x | I x , where I x = { x } × I , we prove that ( x , y ) ∉ Ω ( F ) . On the other hand we construct a map F 0 ∈ C Δ ∗ ( I 2 ) with an isolated fixed point x 0 of f and y 0 ∉ Ω ( F | I x 0 ) such that ( x 0 , y 0 ) ∈ Ω ( F 0 ) .

Quasi-energy function for diffeomorphisms with wild separatrices

We consider the class G 4 of Morse—Smale diffeomorphisms on $$ \mathbb{S} $$ 3 with nonwandering set consisting of four fixed points (namely, one saddle, two sinks, and one source). According to Pixton, this class contains a diffeomorphism that does not have an energy function, i.e., a Lyapunov function whose set of critical points coincides with the set of periodic points of the diffeomorphism itself. We define a quasi-energy function for any Morse—Smale diffeomorphism as a Lyapunov function with the least number of critical points. Next, we single out the class G 4,1 ⊂ G 4 of diffeomorphisms inducing a special Heegaard splitting of genus 1 of the sphere $$ \mathbb{S} $$ 3. For each diffeomorphism in G 4,1, we present a quasi-energy function with six critical points.

Density of periodic points, invariant measures and almost equicontinuous points of cellular automata

Revisiting the notion of μ-almost equicontinuous cellular automata introduced by R. Gilman, we show that the sequence of image measures of a shift ergodic measure μ by iterations of such automata converges in Cesàro mean to an invariant measure μ c . Moreover the dynamical system (cellular automaton F, invariant measure μ c ) has still the μ c -almost equicontinuity property and the set of periodic points is dense in the topological support of the measure μ c . We also show that the density of periodic points in the topological support of a measure μ occurs for each μ-almost equicontinuous cellular automaton when μ is an invariant and shift ergodic measure. Finally using most of these results we give a non-trivial example of a couple ( μ-equicontinuous cellular automaton F, shift and F-invariant measure μ) such that the restriction of F to the topological support of μ has no equicontinuous points.

On Ω-blow-ups in the simplest C 1-smooth skew products of interval mappings

The authors prove a criterion (necessary and sufficient condition) for the emergence of the C0-Ω-blow-up for C1-smooth skew products of interval mappings with closed set of periodic points. An example of the mapping with given properties that admits the C0-Ω-blow-up is presented. It is proved that the C1-Ω-blow-up is impossible for mappings of such a type (in the space of C1-smooth skew products of interval mappings). It is proved that there is no one-parameter family of C1-smooth skew products of interval mappings with closed set of periodic points C1-smoothly depending on the parameter in which from one fixed point, periodic orbits with periods 2 and 4 simultaneously arise.

Stability of dynamical structures under perturbation of the generating function

We consider the set valued functions C, NW and ℒ taking f in 𝒞(I, I) to its centre C( f ), its set of nonwandering points NW( f ) and its collection of ω-limit sets ℒ( f ) = {ω (x, f ) : x ∈ I}, and consider how these sets are affected by pertubations of f. Our main results characterize those functions g in 𝒞(I, I) at which C, NW and ℒ are continuous. In particular, we show that either of the maps C and NW is continuous at g if and only if one of the following conditions is satisfied: (i) The map ω which takes a function f to its set ω ( f ) of ω-limit points is continuous at g; (ii) the periodic orbits of g which are p-stable, i.e. stable with respect to small perturbations of g, are dense in the set CR(g) of chain recurrent points of g; (iii) CR(g) = ω (g) and the p-stable periodic orbits of g are dense in the set of periodic points of g.

Scaling dynamics of a cubic interval-exchange transformation

We study the dynamics of renormalization of a specific interval-exchange transformation which features exact scaling (the cubic Arnoux–Yoccoz model). Using a symbolic space that describes both dynamics and scaling, we characterize the periodic points of the scaling map in terms of generalized decimal expansions, where the base is the reciprocal of a Pisot number and the digits are algebraic integers. The set of periodic points has a rich arithmetic and geometric structure: we establish rigorously some facts, and use extensive numerical experimentation to formulate a conjecture.

About a low complexity class of cellular automata

Extending to all probability measures the notion of μ-equicontinuous cellular automata introduced for Bernoulli measures by Gilman, we show that the entropy is null if μ is an invariant measure and that the sequence of image measures of a shift ergodic measure by iterations of such automata converges in Cesàro mean to an invariant measure μ c . Moreover, this cellular automaton is still μ c -equicontinuous and the set of periodic points is dense in the topological support of the measure μ c . The last property is also true when μ is invariant and shift ergodic. To cite this article: P. Tisseur, C. R. Acad. Sci. Paris, Ser. I 346 (2008).