On conjugacy of homeomorphisms of the circle possessing periodic points

We give a necessary and sufficient condition for topological conjugacy of homeomorphisms of the circle having periodic points. As an application we get the following theorem on the representation of homeomorphisms. The homeomorphism has a periodic point of period n iff there exist a positive integer q<n relatively prime to n and a homeomorphism such that the lift of Φ⁻¹○F○Φ restricted to [0,1] has the form

     

Homotopical theory of periodic points of periodic homeomorphisms on closed surfaces

In this work we show that the Wecken theorem for periodic points holds for periodic homeomorphisms on closed surfaces, which therefore completes the periodic point theory in such a special case. Using it we derive the set of homotopy minimal periods for such homeomorphisms. Moreover we show that the results hold for homotopically periodic self-maps of closed surfaces. This let us to re-formulate our results as a statement on properties of elements of finite order in the group of outer automorphisms of the fundamental group of a surface with non-positive Euler characteristic.     

Existence of periodic points of endomorphisms of a circle

A theorem of Block and Franke is improved on the existence of periodic points for a map of a circle to itself and a proof which seems more understandable is given. Project supported by the National Natural Science Foundation of China (Grant No. 19531070).     

On the growth of the number of periodic points for smooth self-maps of a compact manifold

Let be a continuous self-map of a smooth compact connected and simply-connected manifold of dimension . We show that in the homotopy class of there is a map with less then periodic points, up to any given fixed period .

     

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).     

Least number of periodic points of self-maps of Lie groups

There are two algebraic lower bounds of the number of n-periodic points of a self-map f :M → M of a compact smooth manifold of dimension at least 3:N Fn(f) = min{#Fix(gn); g ～f; g is continuous} and N J Dn(f) = min{#Fix(gn); g ～ f; g is smooth}.In general,N J Dn(f) may be much greater than N Fn(f).If M is a torus,then the invariants are equal.We show that for a self-map of a nonabelian compact Lie group,with free fundamental group,the equality holds  all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.     

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 {fm(x),fk(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(fn) 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] and [19] on interval maps or tree maps.     

Remark on Krasnosielski’s guiding functions and Srzednicki’s periodic segments

We show that the existence of the complete set of guiding functions guarantees the existence of periodic isolating segment that carries the same information concerning periodic solutions of the non-autonomous periodic equations x^′=f(x,t)$x^{\prime }=f(x,t)$.     

Periodic segment implies infinitely many periodic solutions

In this note we show that the existence of a periodic segment for a non-autonomous ODE with periodic coefficients implies the existence of infinitely many periodic solutions inside this segment provided that a sequence of Lefschetz numbers of iterations of an associated map is not constant. In the case when this sequence is bounded we have to impose a geometric condition on the segment to get solutions by use of symbolic dynamics.

     

On some properties of orientation-preserving surjections on the circle

Some properties of orientation-preserving surjections with nonempty set of periodic points are studied. In particular, orientation-preserving homeomorphisms of the whole circle S ¹ are considered.      

On the triple points of singular maps

The number of triple points (mod 2) of a self-transverse immersion of a closed 2n-manifold M into 3n-space are known to equal one of the Stiefel-Whitney numbers of M. This result is generalized to the case of generic (i.e. stable) maps with singularities. Besides triple points and Stiefel-Whitney numbers, a certain linking number of the manifold of singular values with the rest of the image is involved in the generalized equation which corrects an erroneous formula in [9].? If n is even and the closed manifold is oriented then the equations mentioned above make sense over the integers. Together, the integer- and mod 2 generalized equations imply that a certain Stiefel-Whitney number of closed oriented 4k-manifolds vanishes. This Stiefel-Whitney number is in fact the first in a family which vanish on such manifolds. Received: October 12, 2001     

On the existence of periodic solutions for differential inclusions

In this paper we study the existence of periodic solutions for differential inclusions. We prove existence theorems under various sets of hypotheses for both the nonconvex and convex problems. Also we show the existence of extreme solutions. Some feedback control systems are also considered.     

一类二阶迭代泛函微分方程的周期解

本文研究一类二阶迭代泛函微分方程周期解的存在性问题.利用Schauder和Banach不动点定理,获得此类方程周期解的存在唯一性及稳定性的结果,推广了已有结论.     

General theorem for the existence of iterative roots of homeomorphisms with periodic points

We apply Zdun’s factorization theorem (see Zdun (2008) [3]) to give the conditions for the existence and the form of continuous and orientation-preserving iterative roots of homeomorphisms of the circle with a rational rotation number. Our theorem generalizes the previous results given by Jarczyk (2003) in [2], Zdun (2008) in [3] and Solarz (2003, 2009) in [4] and [5].     

Estimating Mahler measures using periodic points for the doubling map

The Mahler measure m(P)$m\left(P\right)$ of a polynomial P$P$ is a numerical value which is useful in number theory, dynamical systems and geometry. In this article we show how this can be written in terms of periodic points for the doubling map on the unit interval. This leads to an interesting algorithm for approximating m(P)$m\left(P\right)$ which we illustrate with several examples.     

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.     

Conjugations between circle maps with a single break point

Consider two circle homeomorphisms fi∈C2+α(S?{bi}), α>0, i=1,2 with a single break point bi i.e. a discontinuity in the derivative Dfi, and identical irrational rotation number ρ. Suppose the jump ratios and do not coincide. Then the map ψ conjugating f₁ and f₂ is a singular function i.e. it is continuous on S¹ and Dψ(x)=0 a.e. with respect to Lebesgue measure.