Arnoux-Rauzy Subshifts: Linear Recurrence, Powers, and Palindromes

We consider Arnoux-Rauzy subshifts $X$ and study various combinatorial questions: When is $X$ linearly recurrent? What is the maximal power occurring in $X$? What is the number of palindromes of a given length occurring in $X$? We present applications of our combinatorial results to the spectral theory of discrete one-dimensional Schr"odinger operators with potentials given by Arnoux-Rauzy sequences.