203 research outputs found
An analogue of Bauerâs theorem for closed orbits of skew products
In this article we prove an analogue of Bauerâs theorem from algebraic number theory in the context of hyperbolic systems
Logarithm laws for equilibrium states in negative curvature
Let be a pinched negatively curved Riemannian manifold, whose unit
tangent bundle is endowed with a Gibbs measure associated to a potential
. We compute the Hausdorff dimension of the conditional measures of .
We study the -almost sure asymptotic penetration behaviour of locally
geodesic lines of into small neighbourhoods of closed geodesics, and of
other compact (locally) convex subsets of . We prove Khintchine-type and
logarithm law-type results for the spiraling of geodesic lines around these
objects. As an arithmetic consequence, we give almost sure Diophantine
approximation results of real numbers by quadratic irrationals with respect to
general H\"older quasi-invariant measures
Properties of measures supported on fat Sierpinski carpets
In this paper we study certain conformal iterated function schemes in two dimensions that are natural generalizations of the Sierpinski carpet construction. In particular, we consider scaling factors for which the open set condition fails. For such âfat Sierpinski carpetsâ we study the range of parameters for which the dimension of the set is exactly known, or for which the set has positive measure
A note on the shrinking sector problem for surfaces of variable negative curvature
Given the universal cover for a compact surface of variable negative curvature we consider, for each k â„ 0, the limiting probability for the directions in which a narrowing sector of unit area contains exactly k in the orbit of the covering group. This can be done relative to any Gibbs measure and follows directly from the strong mixing propert
Addendum: an analogue of Artin reciprocity for closed orbits of skew products
One of the unfulfilled aims of the authors of the preceding paper [W. Parry and M. Pollicott. An analogue of Bauerâs theorem for closed orbits of skew products. Ergod. Th. & Dynam. Sys. 28 (2008), 535â546] was to find a dynamical analogue of Artin reciprocity. In this addendum, we present one such version, suggested by work of Sunada
Unique Bernoulli g-measures
We improve and subsume the conditions of Johansson and \"Oberg [18] and
Berbee [2] for uniqueness of a g-measure, i.e., a stationary distribution for
chains with complete connections. In addition, we prove that these unique
g-measures have Bernoulli natural extensions. In particular, we obtain a unique
g-measure that has the Bernoulli property for the full shift on finitely many
states under any one of the following additional assumptions. (1)
\sum_{n=1}^\infty (\var_n \log g)^20$,
\sum_{n=1}^\infty e^{-(\{1}{2}+\epsilon) (\var_1 \log g+...+\var_n \log
g)}=\infty,\var_n \log g=\ordo{\{1}{\sqrt{n}}}, \quad n\to \infty.
That the measure is Bernoulli in the case of (1) is new. In (2) we have an
improved version of Berbee's condition (concerning uniqueness and
Bernoullicity) [2], allowing the variations of log g to be essentially twice as
large. Finally, (3) is an example that our main result is new both for
uniqueness and for the Bernoulli property. We also conclude that we have
convergence in the Wasserstein metric of the iterates of the adjoint transfer
operator to the g-measure
Appolonius circle counting
Circle packings in the plane date back over 2000 years and have attracted the attention of princesses and Nobel Prize winners alike. We give a brief history and describe some recent developments
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