The periodic orbit structure of orientation preserving diffeomorphisms on D2 with topological entropy zero (original) (raw)

Asymptotic linking of periodic orbits for diffeomorphisms of the two-disk

Let f be a C'lf orientation preserving diffeomorphism of the two-disk with positive topological entropy. We define for f an interval of topological invariants. Each point in this interval describes the way the elements of an infinite sequence of periodic orbits with arbitrarily large periods, are asymptotically linked one around the other, AMS classification scheme numbers: 3405.54EO. 58F15

Diffeotopically trivial periodic diffeomorphisms

Inventiones Mathematicae, 1970

In this paper, we settle negatively an old question as to whether all free periodic diffeomorphisms that are diffeotopic to the identity can be found by restricting circle group actions to finite cyclic subgroups. More precisely, we construct examples of periodic diffeomorphisms of D 2" for n>3 which on S z"-~ are free and cannot be obtained from any piecewise linear (PL) circle group action on S 2"-1 (cf. Gluck [5]).

Dynamical Ordering of Non-Birkhoff Orbits and Topological Entropy in the Standard Mapping

Progress of Theoretical Physics, 2002

The standard mapping is an analytical, reversible monotone twist mapping. The appearance ordering (i.e. the so-called dynamical ordering), of symmetric non-Birkhoff periodic orbits (SNBO) in the standard mapping is derived. Essential use is made of the reversibility. After the establishment of various properties of the symmetry axes under the mapping, two theorems connecting the dynamical ordering are proved. Then, the braids for SNBOs are constructed with the aid of techniques developed in braid group theory. A lower bound of the topological entropy of a system possessing an SNBO is obtained using the eigenvalue of the reduced Burau matrix representation of the braid constructed from the SNBO. The behavior of the topological entropy in the integrable limit is discussed.

Symmetric Periodic Orbits and Topological Entropy in the Two-Dimensional Cubic Map

Progress of Theoretical Physics, 2005

We show that a threefold horseshoe appears in the doubly reversible two-dimensional cubic map including one parameter, a. It is found that the lower bound of the topological entropy h top is ln 3 after the completion of this threefold horseshoe. For a = 0, this map is integrable, and the relation h top = 0 holds. We demonstrate that for the case a > 0, h top ≥ ln 2 holds. The dynamical orderings for the symmetric periodic orbits are also derived.

Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps

Nonlinearity, 2013

We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the initial heteroclinic tangency and prove that there are infinitely sequences (cascades) of bifurcations of birth of asymptotically stable and unstable as well as elliptic periodic orbits.

Periodic orbits for dissipative twist maps

Ergodic Theory and Dynamical Systems, 1987

We develop simple topological criteria for the existence of periodic orbits in maps of the annulus. These are applied to one-parameter families of dissipative twist maps of the annulus and their attractors. It follows that many of the motions found by variational methods in area preserving twist maps also occur in the dissipative case.

Non-periodic bifurcations for surface diffeomorphisms

Transactions of the American Mathematical Society, 2015

We prove that a “positive probability” subset of the boundary of the set of hyperbolic (Axiom A) surface diffeomorphisms with no cycles H \mathcal {H} is constituted by Kupka-Smale diffeomorphisms: all periodic points are hyperbolic and their invariant manifolds intersect transversally. Lack of hyperbolicity arises from the presence of a tangency between a stable manifold and an unstable manifold, one of which is not associated to a periodic point. All these diffeomorphisms that we construct lie on the boundary of the same connected component of H \mathcal {H} .

Non-Symmetric Non-Birkhoff Period-2 Orbits in the Standard Mapping

Progress of Theoretical Physics, 2001

Period-2 badly ordered orbits (non-Birkhoff orbits) are studied in the standard mapping. Points of symmetric non-Birkhoff orbits appear on symmetry axes due to the saddle-node bifurcation, and non-symmetric non-Birkhoff orbits appear due to the equi-period bifurcation of symmetric non-Birkhoff orbits. The braids of symmetric non-Birkhoff orbits are constructed, and the topological entropy is estimated.

Minimal Periodic Orbit Structure of 2DimensionalHomeomorphisms

Journal of Nonlinear Science, 2005

We present a method for estimating the minimal periodic orbit structure, the topological entropy, and a fat representative of the homeomorphism associated with the existence of a finite collection of periodic orbits of an orientation-preserving homeomorphism of the disk D2. The method focuses on the concept of fold and recurrent bogus transition and is more direct than existing techniques. In particular, we introduce the notion of complexity to monitor the modification process used to obtain the desired goals. An algorithm implementing the procedure is described and some examples are presented at the end.