Asymptotic (statistical) periodicity in two-dimensional maps

<p style='text-indent:20px;'>In this paper we give a new sufficient condition for the existence of asymptotic periodicity of Frobenius–Perron operators corresponding to two–dimensional maps. Asymptotic periodicity for strictly expanding systems, that is, all eigenvalues of the system are greater than one, in a high-dimensional dynamical system was already known. Our new result enables one to deal with systems having an eigenvalue smaller than one. The key idea for the proof is to use a function of bounded variation defined by line integration. Finally, we introduce a new two-dimensional dynamical system numerically exhibiting asymptotic periodicity with different periods depending on parameter values, and discuss the application of our theorem to the example.</p>

Rate of Convergence and Periodicity of the Expected Population Structure of Markov Systems that Live in a General State Space

In this article we study the asymptotic behaviour of the expected population structure of a Markov system that lives in a general state space (MSGS) and its rate of convergence. We continue with the study of the asymptotic periodicity of the expected population structure. We conclude with the study of total variability from the invariant measure in the periodic case for the expected population structure of an MSGS.

On Weak Asymptotic Periodicity

We introduce the asymptotic property associated with recurrence-like behavior of orbits in dynamical systems in general metric spaces. We define a notion of weak asymptotic periodicity and determine its elementary properties and relations including the invariance by topological conjugacy. We use the equicontinuity and the topology of the space to describe necessary and sufficient conditions for the existence of such a behavior.

On asymptotically periodic-like motions in flows

We study three properties associated to the recurrence of orbits in flows: asymptotic periodicity, positive asymptotic periodicity and G-asymptotic periodicity. We determine which implications between these notions hold and which do not. We also show how these notions are related to Lyapunov stability.