200 research outputs found
H\"older-differentiability of Gibbs distribution functions
In this paper we give non-trivial applications of the thermodynamic formalism
to the theory of distribution functions of Gibbs measures (devil's staircases)
supported on limit sets of finitely generated conformal iterated function
systems in . For a large class of these Gibbs states we determine the
Hausdorff dimension of the set of points at which the distribution function of
these measures is not -H\"older-differentiable. The obtained results
give significant extensions of recent work by Darst, Dekking, Falconer, Li,
Morris, and Xiao. In particular, our results clearly show that the results of
these authors have their natural home within thermodynamic formalism.Comment: 13 pages, 2 figure
On the asymptotics of the -Farey transfer operator
We study the asymptotics of iterates of the transfer operator for
non-uniformly hyperbolic -Farey maps. We provide a family of
observables which are Riemann integrable, locally constant and of bounded
variation, and for which the iterates of the transfer operator, when applied to
one of these observables, is not asymptotic to a constant times the wandering
rate on the first element of the partition . Subsequently, sufficient
conditions on observables are given under which this expected asymptotic holds.
In particular, we obtain an extension theorem which establishes that, if the
asymptotic behaviour of iterates of the transfer operator is known on the first
element of the partition , then the same asymptotic holds on any
compact set bounded away from the indifferent fixed point
A Fr\'{e}chet law and an Erd\"os-Philipp law for maximal cuspidal windings
In this paper we establish a Fr\'{e}chet law for maximal cuspidal windings of
the geodesic flow on a Riemannian surface associated with an arbitrary finitely
generated, essentially free Fuchsian group with parabolic elements. This result
extends previous work by Galambos and Dolgopyat and is obtained by applying
Extreme Value Theory. Subsequently, we show that this law gives rise to an
Erd\"os-Philipp law and to various generalised Khintchine-type results for
maximal cuspidal windings. These results strengthen previous results by
Sullivan, Stratmann and Velani for Kleinian groups, and extend earlier work by
Philipp on continued fractions, which was inspired by a conjecture of Erd\"os
Strong renewal theorems and Lyapunov spectra for -Farey and -L\"uroth systems
In this paper we introduce and study the -Farey map and its
associated jump transformation, the -L\"uroth map, for an arbitrary
countable partition of the unit interval with atoms which accumulate
only at the origin. These maps represent linearised generalisations of the
Farey map and the Gauss map from elementary number theory. First, a thorough
analysis of some of their topological and ergodic-theoretic properties is
given, including establishing exactness for both types of these maps. The first
main result then is to establish weak and strong renewal laws for what we have
called -sum-level sets for the -L\"uroth map. Similar results
have previously been obtained for the Farey map and the Gauss map, by using
infinite ergodic theory. In this respect, a side product of the paper is to
allow for greater transparency of some of the core ideas of infinite ergodic
theory. The second remaining result is to obtain a complete description of the
Lyapunov spectra of the -Farey map and the -L\"uroth map in
terms of the thermodynamical formalism. We show how to derive these spectra,
and then give various examples which demonstrate the diversity of their
behaviours in dependence on the chosen partition .Comment: 29 pages, 16 figure
Fractal analysis for sets of non-differentiability of Minkowski's question mark function
In this paper we study various fractal geometric aspects of the Minkowski
question mark function We show that the unit interval can be written as
the union of the three sets ,
, and does
not exist and The main result is that the Hausdorff
dimensions of these sets are related in the following way.
Here, refers to the level set of the
Stern-Brocot multifractal decomposition at the topological entropy
of the Farey map and
denotes the Hausdorff dimension of the measure of maximal entropy of the
dynamical system associated with The proofs rely partially on the
multifractal formalism for Stern-Brocot intervals and give non-trivial
applications of this formalism.Comment: 22 pages, 2 figure
Radon--Nikodym representations of Cuntz--Krieger algebras and Lyapunov spectra for KMS states
We study relations between --KMS states on Cuntz--Krieger algebras
and the dual of the Perron--Frobenius operator .
Generalising the well--studied purely hyperbolic situation, we obtain under
mild conditions that for an expansive dynamical system there is a one--one
correspondence between --KMS states and eigenmeasures of
for the eigenvalue 1. We then consider
representations of Cuntz--Krieger algebras which are induced by Markov fibred
systems, and show that if the associated incidence matrix is irreducible then
these are --isomorphic to the given Cuntz--Krieger algebra. Finally, we
apply these general results to study multifractal decompositions of limit sets
of essentially free Kleinian groups which may have parabolic elements. We
show that for the Cuntz--Krieger algebra arising from there exists an
analytic family of KMS states induced by the Lyapunov spectrum of the analogue
of the Bowen--Series map associated with . Furthermore, we obtain a formula
for the Hausdorff dimensions of the restrictions of these KMS states to the set
of continuous functions on the limit set of . If has no parabolic
elements, then this formula can be interpreted as the singularity spectrum of
the measure of maximal entropy associated with .Comment: 30 pages, minor changes in the proofs of Theorem 3.9 and Fact
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