arXiv subjects

Xinxing Wu

Publications and source records attributed to Xinxing Wu.

39 records · Page 3Linked to original sources

arXiv2016

On the large deviations theorem of weaker types

In this paper, we introduce the concepts of the large deviations theorem of weaker types, i.e., type I, type I', type II, type II', type III, and type III', and present a systematic study of the ergodic and chaotic properties of dynamical systems satisfying the large eviations theorem of various types. Some characteristics of the ergodic measure are obtained and then applied to prove that every dynamical system satisfying the large deviations theorem of type I' is ergodic, which is equivalent to the large deviations theorem of type II' in this regard, and that every uniquely ergodic dynamical system restricted on its support satisfies the large deviations theorem. Moreover, we prove that every dynamical system satisfying the large deviations theorem of type III is an $E$-system. Finally, we show that a dynamical system satisfying the central limit theorem, introduced in [Y. Niu, Y. Wang, Statist. Probab. Lett., {\bf 80} (2010), 1180--1184], does not exist.

math.DS↗

arXiv2016

Topological dynamics of Zadeh's extension on the space of upper semi-continuous fuzzy sets

In this paper, some characterizations about transitivity, mildly mixing property, $\mathbf{a}$-transitivity, equicontinuity, uniform rigidity and proximality of Zadeh's extensions restricted on some invariant closed subsets of the space of all upper semi-continuous fuzzy sets with the level-wise metric are obtained. In particular, it is proved that a dynamical system is weakly mixing (resp., mildly mixing, weakly mixing and $\mathbf{a}$-transitive, equicontinuous, uniformly rigid) if and only if the Zadeh's extension is transitive (resp., mildly mixing, $\mathbf{a}$-transitive, equicontinuous, uniformly rigid).

math.DS↗

arXiv2014

On various definitions of shadowing with average error in tracing

In this paper we present a systematic study of shadowing properties with average error in tracing such as (asymptotic) average shadowing, $\underline{d}$-shadowing, $\overline{d}$-shadowing and almost specification. As the main tools we provide a few equivalent characterizations of the average shadowing property, which also partly apply to other notions of shadowing. We prove that almost specification on the whole space induces this property on the measure center. Next, we show that always (e.g. without assumption that the map is onto) almost specification implies asymptotic average shadowing, which in turn implies the average shadowing property and consequently also $\underline{d}$-shadowing and $\overline{d}$-shadowing. Finally, we study connections among sensitivity, transitivity, equicontinuity and (average) shadowing.

math.DS↗