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We construct first a class of Moran fractals in R^d with countably manygenerators and non-stationary contraction rates; at each step n, thecontractions depend on n-truncated sequences, and are related to asymptoticletter frequencies. In some cases the sets of contractions may be infinite ateach step. We show that the Hausdorff dimension of such a fractal is equal tothe zero h of a pressure function. We prove that the dimensions of these setsdepend real analytically on the frequencies. Next, we apply the aboveconstruction to obtain non-stationary fractals E(x; f) \subset R^d, associatedto f-expansions of real numbers x, and study the dependence of these fractalson x. We consider for instance beta-expansions, the continued fractionexpansion and other f-expansions. By employing the Ergodic Theorem forinvariant absolutely continuous measures and equilibrium measures, and usingsome probabilities for which the digits become independent random variables, westudy the function x \to dim_H(E(x; f)) on the respective set of quasinormalnumbers x \in [0; 1). We investigate also another class of fractals \tilde E_f(x) \subset R^d, for which both the non-stationary contraction vectors and theasymptotic frequencies depend on the f-representation of x. We obtain then someproperties of the digits of x, related to \tilde E_f (x) and to equilibriummeasures.

A class of measures and non-stationary fractals, associated to f-expansions