Comparison theorems and orbit counting in hyperbolic geometry
نویسندگان
چکیده
منابع مشابه
Comparison Theorems and Orbit Counting in Hyperbolic Geometry
In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a mixture of ideas from both “thermodynamic” ergodic theory and the automaton associated to strongly Markov gr...
متن کاملOrbit-counting in Non-hyperbolic Dynamical Systems
There are well-known analogs of the prime number theorem and Mertens’ theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest non-hyperbolic algebraic systems. The asymptotic behaviour of the orbit-counting function is governed by a rotation on an associated compact group, and in simple examples we exhibit uncountably many different asymptoti...
متن کاملRelative volume comparison theorems in Finsler geometry and their applications
We establish some relative volume comparison theorems for extremal volume forms of Finsler manifolds under suitable curvature bounds. As their applications, we obtain some results on curvature and topology of Finsler manifolds. Our results remove the usual assumption on S-curvature that is needed in the literature.
متن کاملComparison Theorems in Riemannian Geometry
The subject of these lecture notes is comparison theory in Riemannian geometry: What can be said about a complete Riemannian manifold when (mainly lower) bounds for the sectional or Ricci curvature are given? Starting from the comparison theory for the Riccati ODE which describes the evolution of the principal curvatures of equidistant hypersurfaces, we discuss the global estimates for volume a...
متن کاملrelative volume comparison theorems in finsler geometry and their applications
we establish some relative volume comparison theorems for extremal volume forms of finsler manifolds under suitable curvature bounds. as their applications, we obtain some results on curvature and topology of finsler manifolds. our results remove the usual assumption on s-curvature that is needed in the literature.
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1998
ISSN: 0002-9947,1088-6850
DOI: 10.1090/s0002-9947-98-01756-5