Circularly chainable continua that annularly inscribe squares

Let A denote the annulus { x ∈ R 2 : 1 ≤ | | x | | ≤ 1 + 2 } . In this paper we prove that if X is a circularly chainable continuum essentially embedded in A , then X contains the four vertices of an Euclidean square of side length at least 2 . Furthermore, let us say that a plane continuum X satisfies the property of annular inscription in A if whenever X is essentially embedded in A , the image of such embedding admits an inscribed square of side length at least 2 . We show that every circularly chainable, not chainable plane continuum satisfies the property of annular inscription in A and, moreover, that the property of annular inscription in A is generic among continua that separate the plane.