The growth function of the volume of geodesic balls in Riemannian manifolds of hyperbolic type

Let (M;g) be a compact Riemannian manifold of hyperbolic type and X be its universal Riemannian covering. We study in this paper, the growth function of the geodesic balls of X. We show that the critical exponent of the group of deck transformations of X is equal to the volume entropy hg of M

VOLUME GROWTH AND CLOSED GEODESICS ON RIEMANNIAN MANIFOLDS OF HYPERBOLIC TYPE

We study the volume growth function of geodesic spheres in the universal Riemannian covering of a compact manifold of hyperbolic type. Furthermore, we investigate the growth rate of closed geodesics in compact manifolds of hyperbolic type

Asymptotic behaviour of mild solution of nonlinear stochastic partial functional equations

This paper presents conditions to assure existence, uniqueness and stability for impulsive neutral stochastic integrodifferential equations with delay driven by Rosenblatt process and Poisson jumps. The Banach fixed point theorem and the theory of resolvent operator developed by Grimmer [R.C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc., 273(1):333–349, 1982] are used. An example illustrates the potential benefits of these results

Existence and exponential stability for neutral stochastic integro-differential equations with impulses driven by a Rosenblatt process

The existence and uniqueness of mild solution of an impulsive stochastic system driven by a Rosenblatt process is analyzed in this work by using the Banach fixed point theorem and the theory of resolvent operator developed by R. Grimmer in R. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273 (1982), 333–349. Furthermore, the exponential stability in mean square for the mild solution to neutral stochastic integro-differential equations with Rosenblatt process is obtained by establishing an integral inequality. Finally, an example is exhibited to illustrate the abstract theory