Expansiveness of homeomorphisms and dimension (Q1915535)

\textit{R. Mañé} [Trans. Am. Math. Soc. 252, 313-319 (1979; Zbl 0362.54036)]\ proved that if a compact metric space \(X\) admits an expansive homeomorphism, then \(X\) is finite dimensional. In this paper, we define the notion of ``barriers'' of a homeomorphism \(f: X\to X\) and an index \(B(f)\). We are interested in the relation between the index \(B(f)\) and the dimension of \(X\). The following theorem is proved: If \(f: X\to X\) is a continuum-wise expansive homeomorphism of a compact metric space \(X\), then \(\dim X\leq B(f)\leq 2\cdot \dim X< \infty\). As a corollary, if \(f: X\to X\) is a continuum-wise fully expansive homeomorphism, then \(\dim X\leq B(f)\leq 2\).