Quasi-shadowing for Z^(d)-actions

A diffeomorphism is non-uniformly partially hyperbolic if it preserves an ergodic measure with at least one zero Lyapunov exponent.We prove that a C^(1)-smooth Z^(d)-action has the quasishadowing property if one of the generators is C^(1+α)(α>0)non-uniformly partially hyperbolic.

Definition of Measure-theoretic Pressure Using Spanning Sets

We introduce a new definition of measure-theoretic pressure for ergodic measures of con- tinuous maps on a compact metric space.This definition is similar to those of topological pressure involving spanning sets.As an application,for C^(1+α)(α>0)diffeomorphisms of a compact manifold, we study the relationship between the measure-theoretic pressure and the periodic points.

Unique Ergodicity for Zero-entropy Dynamical Systems with the Approximate Product Property

We show that for every topological dynamical system with the approximate product property, zero topological entropy is equivalent to unique ergodicity. Equivalence of minimality is also proved under a slightly stronger condition. Moreover, we show that unique ergodicity implies the approximate product property if the system has periodic points.

Some Ergodic Theorems for a Parabolic Anderson Model

In this paper, we study some ergodic theorems of a class of linear systems of interacting diffusions, which is a parabolic Anderson model. First, under the assumption that the transition kernel a = （a（i,j））i,j∈s is doubly stochastic, we obtain the long-time convergence to an invariant probability measure Vh starting from a bounded a-harmonic function h based on self-duality property, and then we show the convergence to the invariant probability measure Uh holds for a broad class of initial distributions. Second, if （a（i, j））i,j∈s is transient and symmetric, and the diffusion parameter c remains below a threshold, we are able to determine the set of extremal invariant probability measures with finite second moment. Finally, in the case that the transition kernel （a（i,j））i,j∈s is doubly stochastic and satisfies Case I （see Case I in [Shiga, T.： An interacting system in population genetics. J. Math. Kyoto Univ., 20, 213-242 （1980）]）, we show that this parabolic Anderson model locally dies out independent of the diffusion parameter c.

The Continuity of Barrier Function with Respect to the Parameter

The authors study the continuity of barrier function Be（x） with respect to the parameter. A sufficient condition which makes Be（x） be continuous with respect to c is obtained, and an example of discontinuity when the condition is not satisfied is also constructed.