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The Three-Dimensional Gauss Algorithm Is Strongly Convergent Almost Everywhere

A proof that the three-dimensional Gauss algorithm is strongly convergent almost everywhere is given. This algorithm is equivalent to Brun's algorithm and to the modified lacobi-Perron algorithm considered by Podsypanin and Schweiger. The proof involves the rigorouscomputer assisted estimation of the largest Lyapunov exponent of a cocycle associated to the algorithm. To the best of my knowledge, this is the first proof of almost everywhere strong convergence of a Jacobi-Perron type algorithm in dimension greater than two.