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Throughout this note 5 denotes a semi-locally-connected continuum and T(S)=S an onto-homeomorphism. If E is a set of 5 such that T(E)=E for some positive integer k, then E has a, finite period, otherwise the period for E is infinite. The set of all images of E under T and its inverse T~ is said to be the orbit of E. If E is of period 4 = 1, then E is invariant under T. We investigate the least invariant A -set which contains the orbit of a cyclic element E, when the period and orbit for E are infinite. This work occupies an intermediate position between a previous paper, wherein certain general results are directed toward the study of finite orbits, and the problem of the action of T(S)=S in general. We follow here the spirit of the work of Ayres begun in his paper, On transformations having periodic properties, Fund. Math. vol. 33 (1939) pp. 95-103. With Ayres we denote the unique cyclic chain between two cyclic elements E and Dby C(E,D).

Minimal 𝐴-sets, infinite orbits, and fixed elements