Summary: The Fibonacci Hamiltonian, that is a Schrödinger operator associated to a quasiperiodical Sturmian potential with respect to the golden mean has been investigated intensively in recent years. D. Damanik and S. Tcheremchantsev developed a method in [J. Anal. Math. 97, 103–131 (2006; Zbl 1132.81018)] and used it to exhibit a non trivial dynamical upper bound for this model. In this paper, we use this method to generalize to a large family of Sturmian operators dynamical upper bounds and show at sufficently large coupling anomalous transport for operators associated to irrational number with a generic Diophantine condition. As a counterexample, we exhibit a pathological irrational number which does not verify this condition and show its associated dynamic exponent only has ballistic bound. Moreover, we establish a global lower bound for the lower box counting dimension of the spectrum that is used to obtain a dynamical lower bound for bounded density irrational numbers.