Automatic continued fractions are transcendental or quadratic

We establish new combinatorial transcendence criteria for continued fraction expansions. Let $\alpha =[0;a_1,a_2,\mathrm{.}\mathrm{.}\mathrm{.}]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients $(a_{\mathrm{\ell }}{)}_{\mathrm{\ell }\ge 1}$ of $\alpha$ cannot be generated by a finite automaton, and that the complexity function of $(a_{\mathrm{\ell }}{)}_{\mathrm{\ell }\ge 1}$ cannot increase too slowly.