  The Fibonacci Hamiltonian, that is a Schr\"{o}dinger operator associated to a
quasiperiodical sturmian potential with respect to the golden mean has been
investigated intensively in recent years. Damanik and Tcheremchantsev developed
a method and find 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 counter example, we exhibit a pathological
irrational number which do 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.
