Sitzungsber. Abt. II (1998) 207: 229^238 The Exponent of Convergence for Brun's Algorithm in two Dimensions By B. R. Schratzberger (Vorgelegt in der Sitzung der math.-nat. Klasse am 17. Dezember 1998 durch das k. M. Fritz Schweiger) Abstract We show that for the two-dimensional multiplicative Brun's algorithm, the exponent of convergence is 1  d , i.e. there is a d > 0 such that for almost all x x 1 ; x 2 ; x i  p t i q t            1 q t  1d i  1; 2.Thus the sec- ond Lyapunov exponent is negative.  In 1993, J. C. Lagarias has shown how to use multiplicative ergodic theorems to determine the approximation exponent 1  d for multi- dimensional continued fractions. Ito, Keane & Ohtsuki 1993 proved that for the two-dimensional modi¢ed Jacobi-Perron algorithm d > 0 (see also Fujita, Ito, Keane & Ohtsuki 1996). In Schweiger 1996, a classical result of Paley & Ursell 1930 was used to determine the expo- nent of convergence of theJacobi-Perron algorithm in two dimensions. Meester 1997 gave another proof for the result on Podsypanin's modi¢ca- tion. In this paper, we will show that a similar method can be applied to Brun's algorithm in two dimensions. Clearly, this is no surprise since Brun's multiplicative algorithm is a factor of the modi¢ed algorithm.