Simultaneous multifractal decompositions for the spectra of local entropies and ergodic averages

We consider different multifractal decompositions of the form Kαi={x:gi(x)=αi},i=1,2,…,d,Kαi={x:gi(x)=αi},i=1,2,…,d, and we study the dimension spectrum corresponding to the multiparameter decomposition Kα=⋂i=1dKαi,α=(α1,…,αd). Then for an homeomorphism f : X → X and potentials φ, ψ : X → R we analyze the decompositions Kα+={x:limn→∞1n(Sn+(φ))(x)=α},Kβ-={x:limn→∞1n(Sn-(ψ))(x)=β}, where 1n(Sn+(φ)),1n(Sn-(ψ)) are ergodic averages using forward and backward orbits of f respectively. We must emphasize that the analysis, in any case, is done without requiring conditions of hyperbolicity for the dynamical system or Hölder continuity on the potentials. We illustrate with an application to galactic dynamics: a set of stars (which do not interact among them) moving in a galactic field.