A NEW PROOF OF THE SINGULAR CONTINUITY OF THE MINKOWSKI $?$-FUNCTION

Abstract We give a new proof of the singular continuity of Minkowski’s $?$ -function. Our proof follows by showing that the maximal Lyapunov exponent of a specific pair of $3\times 3$ nonnegative integer matrices related to Stern’s diatomic sequence is strictly greater than $\log 2$ .