ACTA issues

Non-denseness of $2$-graphs of equicontinuous families

L. Bernal-González

Acta Sci. Math. (Szeged) 70:3-4(2004), 839-857

Abstract. The main result contained in this paper is the following: If $X$ is a uniform space with at least two points and ${\cal A}$ is a family of continuous selfmappings of $X$ such that ${\cal A}$ is equicontinuous on some nonempty open subset of $X$, then its second-order graph $\{(x, \varphi(x), \varphi ^2(x)) \in X^3: x \in X, \varphi\in {\cal A}\} $ is not dense in $X^3$. This improves strongly a recent result by C. Pe?a and applies to get results on infinite-dimensional holomorphy and on universal operators. For instance, we prove that if $G$ is a bounded domain in a complex Banach space then the second-order graph of the family of holomorphic selfmappings on $G$ is not dense in $G^3$. We also show that if $E$ is an infinite-dimensional separable Fréchet space then the $\infty $-graph of the family of all hypercyclic operators on $E$ is dense in $E^{{\msbm N}_0}$.

AMS Subject Classification (1991): 54H20, 30F45, 47A16, 54E15, 54H11

Keyword(s): N, -graph, equicontinuity, uniformizable space, universal operator, holomorphic selfmapping, orbit, Helly space

Received November 26, 2003, and in revised form March 18, 2004. (Registered under 5849/2009.)