Monotone periodic orbits for torus homeomorphisms

Let be a homeomorphism of the torus isotopic to the identity and suppose that there exists a periodic orbit with a non-zero rotation vector . Then has a topologically monotone periodic orbit with the same rotation vector.

     

Rotation sets and monotone periodic orbits for annulus homeomorphisms

Iff is a homeomorphism of the annulus andp/q is a rational in lowest terms that is contained in the rotation set off thenf has a (p, q)-topologically monotone periodic orbit. In addition, iff has ap/q-period orbit that is not topologically monotone then the Farey interval ofp/q is contained in the rotation set off.     

Characterizations of almost periodic homeomorphisms

In this paper we study recurrent and almost periodic homeomorphisms on the Euclidean space Rm; we give conditions under which recurrent implies periodic. On the other hand we give properties of elements of compact groups of diffeomorphisms on manifolds.     

The numbers of periodic orbits hidden at fixed points of 2-dimensional holomorphic mappings

Let △n be the ball |x| 1 in the complex vector space C n , let f :△n→ C n be a holomorphic mapping and let M be a positive integer. Assume that the origin 0 = (0, . . . , 0) is an isolated fixed point of both f and the M-th iteration f M of f. Then the (local) Dold index P M (f, 0) at the origin is well defined, which can be interpreted to be the number of virtual periodic points of period M of f hidden at the origin: any holomorphic mapping f 1 :△n→ C n sufficiently close to f has exactly P M (f, 0) distinct periodic points of period M near the origin, provided that all the fixed points of f M 1 near the origin are simple. Therefore, the number O M (f, 0) = P M (f, 0)/M can be understood to be the number of virtual periodic orbits of period M hidden at the fixed point. According to the works of Shub-Sullivan and Chow-Mallet-Paret-Yorke, a necessary condition so that there exists at least one virtual periodic orbit of period M hidden at the fixed point, i.e., O M (f, 0)≥1, is that the linear part of f at the origin has a periodic point of period M. It is proved by the author recently that the converse holds true. In this paper, we will study the condition for the linear part of f at 0 so that O M (f, 0)≥2. For a 2 × 2 matrix A that is arbitrarily given, the goal of this paper is to give a necessary and sufficient condition for A, such that O M (f, 0)≥2 for all holomorphic mappings f :△2 → C 2 such that f(0) = 0, Df(0) = A and that the origin 0 is an isolated fixed point of f M .     

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.     

Weak hyperbolicity on periodic orbits for polynomials

We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like n^5+ε with the period, for some ε>0, then the Julia set of the polynomial is locally connected when it is connected. As a consequence for a polynomial the presence of a Cremer cycle implies the presence of a sequence of repelling periodic orbits with “small” multipliers. Somewhat surprisingly the proof is based on measure theorical considerations. To cite this article: J. Rivera-Letelier, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1113–1118.     

Volume preserving flows with cyclic winding numbers groups and without periodic orbits on 3-manifolds

We construct examples of volume preserving non-singular C ¹ vector fields on closed orientable 3-manifolds, which have cyclic winding numbers groups with respect to the preserved volume element, but have no periodic orbits. Received: 17 January 1998 / Revised version: 31 March 1998     

Existence of periodic orbits for high-dimensional autonomous systems

We give a result on existence of periodic orbits for autonomous differential systems with arbitrary finite dimension. It is based on a Poincaré-Bendixson property enjoyed by a new class of monotone systems introduced in [L.A. Sanchez, Cones of rank 2 and the Poincaré-Bendixson property for a new class of monotone systems, J. Differential Equations 216 (2009) 1170-1190]. A concrete application is done to a scalar differential equation of order 4.     

A bound for the topological entropy of homeomorphisms of a punctured two-dimensional disk

We consider homeomorphisms ƒ of a punctured 2-disk D ² \ P, where P is a finite set of interior points of D ², which leave the boundary points fixed. Any such homeomorphism induces an automorphism ƒ _* of the fundamental group of D ² \ P. Moreover, to each homeomorphism ƒ, a matrix B _ƒ (t) from GL(n, ℤ[t, t ⁻¹]) can be assigned by using the well-known Burau representation. The purpose of this paper is to find a nontrivial lower bound for the topological entropy of ƒ. First, we consider the lower bound for the entropy found by R. Bowen by using the growth rate of the induced automorphism ƒ _*. Then we analyze the argument of B. Kolev, who obtained a lower bound for the topological entropy by using the spectral radius of the matrix B _ƒ (t), where t ∈ ℂ, and slightly improve his result. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 47–55, 2005.     

Geodesics onS 2 and periodic points of annulus homeomorphisms

Summary We show that an area preserving homeomorphism of the open or closed annulus which has at least one periodic point must in fact have infinitely many interior periodic points. A consequence is the theorem that every smooth Riemannian metric onS ² with positive Gaussian curvature has infinitely many distinct closed geodesics.In this paper we investigate area preserving homeomorphisms of the annulus and their periodic points. The main result is that an area preserving homeomorphism of the annulus which has at least one periodic point (perhaps on the boundary) must in fact have infinitely many interior periodic points.The motivation and main application of this result is the furthering of a program begun by Birkhoff [B] in his book Dynamical Systems. There he shows that for many Riemannian metrics onS ², including those with positive curvature, the problem of finding closed geodesics reduces to finding periodic points of a certain area preserving homeomorphism of the annulus. The annulus map in question can be shown to have a periodic point so our main result above can be applied to show the existence of infinitely many distinct closed geodesics whenever this annulus map exists. This is done in Sect. 4 Other quite different approaches to the problem of finding infinitely many geodesics have been successful in handling the cases which do not reduce to the investigation of an annulus homeomorphism (see [Ba]).Oblatum 20-III-1991 & 6-XI-1991     

Stabilizability and a separation principle for periodic orbits

In this paper, we derive a result for stabilizability and a separation principle for periodic orbits. First, using degree theory, we derive a necessary condition for local asymptotic stabilizability of periodic orbits. This condition is similar to the famous Brockett's necessary condition (1983) for local asymptotic stabilizability for equilibria. Next, we derive a separation principle for periodic orbits. Our separation principle states that if a state feedback system defined in the neighborhood of a periodic orbit is asymptotically stabilizable and if an exponentially good state estimator is known, then the composite state feedback-state estimator scheme is locally orbitally asymptotically stable.     

Lifting of homeomorphisms to 4-sheeted branched coverings of a disk

Let p:X→D be a simple, possibly not connected, 4-sheeted branched covering of a closed 2-dimensional disk D with n branch values A₁,…,An. The isotopy classes of homeomorphisms of D which are fixed on the boundary of D and permute the branch values form a braid group Bn. Some of these homeomorphisms can be lifted to homeomorphisms of X. They form a subgroup L(p) of finite index in Bn. For each equivalence class of coverings we find a set of generators for L(p) which contains between n and n+4 elements, depending on the equivalence class of the covering, and the generators are powers of half-twists.