A homeomorphism f of a metric space (M,d) is said to be positively (negatively) expansive, if there is a real number \(e>0\) such that for any x,y\(\in M\) (x\(\neq y)\) one has \(d(f^ mx,f^ my)>e\) for some \(m>0\) \((m<0)\). If f is both positively and negatively expansive, f is called doubly expansive. An old and interesting problem is the question of the existence of expansive homeomorphisms on some manifolds. T. O’Brein and W. Reddy [Pac. J. Math. 35, 737-741 (1970; Zbl 0187.449)] have shown that each compact orientable surface of positive genus admits an expansive homeomorphism.
The second author [J. Peking Univ. Nat. Sci., No.3, 22-40 (1983)] has proved, that the n-dimensional (n\(\geq 2)\) open ball admits a positively expansive \(C^ r\)-diffeomorphism. The authors construct a doubly- expansive analytic diffeomorphism that can be imbedded in an analytic flow on the oben ball.