21 research outputs found
Topological regluing of rational functions
Regluing is a topological operation that helps to construct topological
models for rational functions on the boundaries of certain hyperbolic
components. It also has a holomorphic interpretation, with the flavor of
infinite dimensional Thurston--Teichm\"uller theory. We will discuss a
topological theory of regluing, and trace a direction in which a holomorphic
theory can develop.Comment: 38 page
Quantization of multidimensional cat maps
In this work we study cat maps with many degrees of freedom. Classical cat
maps are classified using the Cayley parametrization of symplectic matrices and
the closely associated center and chord generating functions. Particular
attention is dedicated to loxodromic behavior, which is a new feature of
two-dimensional maps. The maps are then quantized using a recently developed
Weyl representation on the torus and the general condition on the Floquet
angles is derived for a particular map to be quantizable. The semiclassical
approximation is exact, regardless of the dimensionality or of the nature of
the fixed points.Comment: 33 pages, latex, 6 figures, Submitted to Nonlinearit
Ruelle-Perron-Frobenius spectrum for Anosov maps
We extend a number of results from one dimensional dynamics based on spectral
properties of the Ruelle-Perron-Frobenius transfer operator to Anosov
diffeomorphisms on compact manifolds. This allows to develop a direct operator
approach to study ergodic properties of these maps. In particular, we show that
it is possible to define Banach spaces on which the transfer operator is
quasicompact. (Information on the existence of an SRB measure, its smoothness
properties and statistical properties readily follow from such a result.) In
dimension we show that the transfer operator associated to smooth random
perturbations of the map is close, in a proper sense, to the unperturbed
transfer operator. This allows to obtain easily very strong spectral stability
results, which in turn imply spectral stability results for smooth
deterministic perturbations as well. Finally, we are able to implement an Ulam
type finite rank approximation scheme thus reducing the study of the spectral
properties of the transfer operator to a finite dimensional problem.Comment: 58 pages, LaTe
Automorphisms of Rational Maps
Let f(z) be a rational map, Aut(f) the finite group of Mo.. bius transformations commuting with f. We study the question: when can two kinds of more flexible automorphisms of the dynamics of f be realized in Aut(g) for some deformation g of f? First let Mod(f) denote the group of isotopy classes of quasiconformal maps commuting with f