Edelstein’s contractivity and attractors

Abstract:

In this article an example is constructed to show that Theorem 1.1 of L. Janos [Canad. Math. Bull. 18 (1975), no. 5, 675-678] is false. A proper formulation is obtained as follows. Theorem. If $(X,\tau )$ is a metrizable topological space, $f:X \to X$ is continuous, and $a \in X$, then the following statements are equivalent: (1) There exists a metric d compatible with $\tau$ such that f is contractive with respect to d and the sequence $({f^n}(x))_{n = 1}^\infty$ converges to a for every $x \in X$. (2) The singleton {a} is an attractor for compact subsets under f. Furthermore, under this proper formulation, we show that Theorem 3.2 Janos [Proc. Amer. Math. Soc. 61 (1976), 161-175] and Theorem 2.3 Janos and J. L. Solomon [ibid. 71 (1978), 257-262], where the false Theorem 1.1 in [2] has been quoted in the original proofs, remain valid.