49 research outputs found
Quantitative spectral perturbation theory for compact operators on a Hilbert space
We introduce compactness classes of Hilbert space operators by grouping
together all operators for which the associated singular values decay at a
certain speed and establish upper bounds for the norm of the resolvent of
operators belonging to a particular compactness class. As a consequence we
obtain explicitly computable upper bounds for the Hausdorff distance of the
spectra of two operators belonging to the same compactness class in terms of
the distance of the two operators in operator norm.Comment: 26 page
A numerical study of rigidity of hyperbolic splittings in simple two-dimensional maps
Chaotic hyperbolic dynamical systems enjoy a surprising degree of rigidity, a fact which is well known in the mathematics community but perhaps less so in theoretical physics circles. Low-dimensional hyperbolic systems are either conjugate to linear automorphisms, that is, dynamically equivalent to the Arnold cat map and its variants, or their hyperbolic structure is not smooth. We illustrate this dichotomy using a family of analytic maps, for which we show by means of numerical simulations that the corresponding hyperbolic structure is not smooth, thereby providing an example for a global mechanism which produces non-smooth phase space structures in an otherwise smooth dynamical system
Anomalous dynamics in symmetric triangular irrational billiards
We identify a symmetry induced mechanism which dominates the long time behaviour in symmetric triangular billiards. We rigorously prove the existence of invariant sets in symmetric irrational billiards on which the dynamics is governed by an interval exchange transformation. Counterintuitively, this property of symmetric irrational billiards is analogous to the case of general rational billiards, and it highlights the non-trivial impact of symmetries in non-hyperbolic dynamical systems. Our findings provide an explanation for the logarithmic subdiffusive relaxation processes observed in certain triangular billiards. In addition we are able to settle a long standing conjecture about the existence of non-periodic and not everywhere dense trajectories in triangular billiards
The resonance spectrum of the cusp map in the space of analytic functions
We prove that the Frobenius--Perron operator of the cusp map
, (which is an approximation of the
Poincar\'e section of the Lorenz attractor) has no analytic eigenfunctions
corresponding to eigenvalues different from 0 and 1. We also prove that for any
the spectrum of in the Hardy space in the disk
\{z\in\C:|z-q|<1+q\} is the union of the segment and some finite or
countably infinite set of isolated eigenvalues of finite multiplicity.Comment: Submitted to JMP; The description of the spectrum in some Hardy
spaces is adde
Complete spectral data for analytic Anosov maps of the torus
Using analytic properties of Blaschke factors we construct a family of
analytic hyperbolic diffeomorphisms of the torus for which the spectral
properties of the associated transfer operator acting on a suitable Hilbert
space can be computed explicitly. As a result, we obtain explicit expressions
for the decay of correlations of analytic observables without resorting to any
kind of perturbation argument.Comment: 19 pages, 4 figure