Hof A., Knill O., Simon.B. Singular Continuous Spectrum for Palindromic Schr\"odinger Operators (37K, LaTeX) ABSTRACT. We give new examples of discrete Schr\"odinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hull $X$ of the potential is strictly ergodic, then the existence of just one potential $x$ in $X$ for which the operator has no eigenvalues implies that there is a generic set in $X$ for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such an $x$ is that there is a $z\in X$ that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset in $X$. The class includes well-known substitutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely continuous spectrum for all $x\in X$ if $X$ derives from a primitive substitution. For potentials defined by circle maps, $x_n = 1_J (\theta_0+ n\alpha)$, we show that the operator has purely singular continuous spectrum for a generic subset in $X$ for all irrational $\alpha$ and every half-open interval $J$.