Applications of transfer operator methods to the dynamics of low-dimensional piecewise smooth maps
Zhang, Yiwei
Date: 25 July 2012
Thesis or dissertation
Publisher
University of Exeter
Degree Title
PhD in Mathematics
Abstract
This thesis primarily concentrates on stochastic and spectral
properties of the transfer operator generated by piecewise expanding
maps (PWEs) and piecewise isometries (PWIs). We also consider the
applications of the transfer operator in thermodynamic formalism.
The original motivation stems from studies of one-dimensional PWEs.
In ...
This thesis primarily concentrates on stochastic and spectral
properties of the transfer operator generated by piecewise expanding
maps (PWEs) and piecewise isometries (PWIs). We also consider the
applications of the transfer operator in thermodynamic formalism.
The original motivation stems from studies of one-dimensional PWEs.
In particular, any one dimensional mixing PWE admits a unique
absolutely continuous invariant probability measure (ACIP) and this
ACIP has a bounded variation density. The methodology used to prove
the existence of this ACIP is based on a so-called functional
analytic approach and a key step in this approach is to show that
the corresponding transfer operator has a spectral gap. Moreover,
when a PWE has Markov property this ACIP can also be viewed as a
Gibbs measure in thermodynamic formalism.
In this thesis, we extend the studies on one-dimensional PWEs in
several aspects. First, we use the functional analytic approach to
study piecewise area preserving maps (PAPs) in particular to search
for the ACIPs with multidimensional bounded variation densities. We
also explore the relationship between the uniqueness of ACIPs with
bounded variation densities and topological transitivity/ minimality
for PWIs.
Second, we consider the mixing and corresponding mixing rate
properties of a collection of piecewise linear Markov maps generated
by composing x to mx mod 1 with permutations in SN. We show that typical permutations preserve the mixing property under
the composition. Moreover, by applying the Fredholm determinant
approach, we calculate the mixing rate via spectral gaps and obtain
the max/min spectral gaps when m,N are fixed. The spectral gaps
can be made arbitrarily small when the permutations are fully
refined.
Finally, we consider the computations of fractal dimensions for
generalized Moran constructions, where different iteration function
systems are applied on different levels. By using the techniques in
thermodynamic formalism, we approximate the fractal dimensions via
the zeros of the Bowen's equation on the pressure functions
truncated at each level.
Doctoral Theses
Doctoral College
Item views 0
Full item downloads 0