5 research outputs found
Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Hyperbolic Sets
It is known that the famous Feigenbaum-Coullet-Tresser period doubling
universality has a counterpart for area-preserving maps of {\fR}^2. A
renormalization approach has been used in \cite{EKW1} and \cite{EKW2} in a
computer-assisted proof of existence of a "universal" area-preserving map
-- a map with orbits of all binary periods 2^k, k \in \fN. In this paper, we
consider maps in some neighbourhood of and study their dynamics.
We first demonstrate that the map admits a "bi-infinite heteroclinic
tangle": a sequence of periodic points , k \in \fZ, |z_k|
\converge{{k \to \infty}} 0, \quad |z_k| \converge{{k \to -\infty}} \infty,
whose stable and unstable manifolds intersect transversally; and, for any N
\in \fN, a compact invariant set on which is homeomorphic to a
topological Markov chain on the space of all two-sided sequences composed of
symbols. A corollary of these results is the existence of {\it unbounded}
and {\it oscillating} orbits.
We also show that the third iterate for all maps close to admits a
horseshoe. We use distortion tools to provide rigorous bounds on the Hausdorff
dimension of the associated locally maximal invariant hyperbolic set: 0.7673
\ge {\rm dim}_H(\cC_F) \ge \varepsilon \approx 0.00044 e^{-1797}.$
Regularity of critical invariant circles of the standard nontwist map
We study critical invariant circles of several noble rotation numbers at the edge of break-up for an area-preserving map of the cylinder, which violates the twist condition.These circles admit essentially unique parametrizations by rotational coordinates. We present a high accuracy computation of about 107 Fourier coefficients. This allows us to compute the regularity of the conjugating maps and to show that, to the extent of numerical precision, it only depends on the tail of the continued fraction expansion.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/49075/2/non5_3_013.pd