Sensitivity and strong sensitivity on induced dynamical systems

Document Type : Research Paper


1 Academia de Matematicas, Universidad Autonoma de la Ciudad de Mexico, Calz. Ermita Iztapalapa S/N, Col. Lomas de Zaragoza 09620, Mexico City, Mexico.

2 Departamento de Matematicas, Universidad Autonoma Metropolitana, Av. San Rafael Atlixco 186, Col. Vicentina, Del. Iztapalapa, C.P. 09340, Mexico City, Mexico


Given a metric space X, we consider the family of all normal upper semicontinuous fuzzy sets on X, denoted by $\mathcal{F}(X)$, and a discrete dynamical system $(X,f)$. In this paper, we study when $(\mathcal{F}(X), \widehat{f})$ is (strongly) sensitive, where $\widehat{f}$ is the Zadeh's extension of f and $\mathcal{F}(X)$ is equipped with different metrics: The uniform metric, the Skorokhod metric, the sendograph metric and the endograph metric. We prove that the sensitivity in the induced dynamical system $(\mathcal{K}(X),\overline{f})$ is equivalent to the sensitivity in $ \widehat{f} :\mathcal{F}(X)\to \mathcal{F}(X) $ with respect to the uniform metric, the Skorokhod metric and the sendograph metric. We also show that the following conditions are equivalent:
\item {\rm a)} $(X,f)$ is strongly sensitive;
\item {\rm b)} $(\mathcal{F}(X), \widehat{f})$ is strongly sensitive, where $\mathcal{F}(X)$ is equipped with the uniform metric;
\item {\rm c)} $(\mathcal{F}(X), \widehat{f})$ is strongly sensitive, where $\mathcal{F}(X)$ is equipped with the Skorokhod metric;
\item {\rm d)} $(\mathcal{F}(X), \widehat{f})$ is strongly sensitive, where $\mathcal{F}(X)$ is equipped with the sendograph metric.