2,603 research outputs found
Integrable mappings and polynomial growth
We describe birational representations of discrete groups generated by
involutions, having their origin in the theory of exactly solvable
vertex-models in lattice statistical mechanics. These involutions correspond
respectively to two kinds of transformations on matrices: the
inversion of the matrix and an (involutive) permutation of the
entries of the matrix. We concentrate on the case where these permutations are
elementary transpositions of two entries. In this case the birational
transformations fall into six different classes. For each class we analyze the
factorization properties of the iteration of these transformations. These
factorization properties enable to define some canonical homogeneous
polynomials associated with these factorization properties. Some mappings yield
a polynomial growth of the complexity of the iterations. For three classes the
successive iterates, for , actually lie on elliptic curves. This analysis
also provides examples of integrable mappings in arbitrary dimension, even
infinite. Moreover, for two classes, the homogeneous polynomials are shown to
satisfy non trivial non-linear recurrences. The relations between
factorizations of the iterations, the existence of recurrences on one or
several variables, as well as the integrability of the mappings are analyzed.Comment: 45 page
Stickiness in Hamiltonian systems: from sharply divided to hierarchical phase space
We investigate the dynamics of chaotic trajectories in simple yet physically
important Hamiltonian systems with non-hierarchical borders between regular and
chaotic regions with positive measures. We show that the stickiness to the
border of the regular regions in systems with such a sharply divided phase
space occurs through one-parameter families of marginally unstable periodic
orbits and is characterized by an exponent \gamma= 2 for the asymptotic
power-law decay of the distribution of recurrence times. Generic perturbations
lead to systems with hierarchical phase space, where the stickiness is
apparently enhanced due to the presence of infinitely many regular islands and
Cantori. In this case, we show that the distribution of recurrence times can be
composed of a sum of exponentials or a sum of power-laws, depending on the
relative contribution of the primary and secondary structures of the hierarchy.
Numerical verification of our main results are provided for area-preserving
maps, mushroom billiards, and the newly defined magnetic mushroom billiards.Comment: To appear in Phys. Rev. E. A PDF version with higher resolution
figures is available at http://www.pks.mpg.de/~edugal
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