16 research outputs found
On conformal measures and harmonic functions for group extensions
We prove a Perron-Frobenius-Ruelle theorem for group extensions of
topological Markov chains based on a construction of -finite conformal
measures and give applications to the construction of harmonic functions.Comment: To appear in Proceedings of "New Trends in Onedimensional Dynamics,
celebrating the 70th birthday of Welington de Melo
The TIGA technique for detecting gravitational waves with a spherical antenna
We report the results of a theoretical and experimental study of a spherical
gravitational wave antenna. We show that it is possible to understand the data
from a spherical antenna with 6 radial resonant transducers attached to the
surface in the truncated icosahedral arrangement. We find that the errors
associated with small deviations from the ideal case are small compared to
other sources of error, such as a finite signal-to-noise ratio. An in situ
measurement technique is developed along with a general algorithm that
describes a procedure for determining the direction of an external force acting
on the antenna, including the force from a gravitational wave, using a
combination of the transducer responses. The practicality of these techniques
was verified on a room-temperature prototype antenna.Comment: 15 pages, 14 figures, submitted to Physical Review
The arithmetic-geometric scaling spectrum for continued fractions
To compare continued fraction digits with the denominators of the
corresponding approximants we introduce the arithmetic-geometric scaling. We
will completely determine its multifractal spectrum by means of a number
theoretical free energy function and show that the Hausdorff dimension of sets
consisting of irrationals with the same scaling exponent coincides with the
Legendre transform of this free energy function. Furthermore, we identify the
asymptotic of the local behaviour of the spectrum at the right boundary point
and discuss a connection to the set of irrationals with continued fraction
digits exceeding a given number which tends to infinity.Comment: 22 pages, 1 figur
Growth gap in hyperbolic groups and amenability
International audienceWe prove a general version of the amenability conjecture in the unified setting of a Gromov hyperbolic group G acting properly cocompactly either on its Cayley graph, or on a CAT(-1)-space. Namely, for any subgroup H of G, we show that H is co-amenable in G if and only if their exponential growth rates (with respect to the prescribed action) coincide. For this, we prove a quantified, representation-theoretical version of Stadlbauer's amenability criterion for group extensions of a topologically transitive subshift of finite type, in terms of the spectral radii of the classical Ruelle transfer operator and its corresponding extension. As a consequence, we are able to show that, in our enlarged context, there is a gap between the exponential growth rate of a group with Kazhdan's property (T) and the ones of its infinite index subgroups. This also generalizes a well-known theorem of Corlette for lattices of the quaternionic hyperbolic space or the Cayley hyperbolic plane