Expansive homeomorphisms and plane separating continua (Q924298)

A homeomorphism \(h: X\to X\) of a metric space \((X,d)\) onto itself is said to be expansive if there exists a constant \(c> 0\) and for any distinct points \(x,y\in X\) there exists an integer \(n\), such that \(d(h^n(x),h^n(y))> c\). It is shown that if \(X\) is a one-dimensional continuum that separates the plane into two pieces, then \(h\) cannot be expansive. The corresponding statement is still unknown for 3-separating plane continua. Another question which remains to be answered is: Does there exist a 2-dimensional non-separating plane continuum that admits an expansive homeomorphism?