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

MESON, ALEJANDRO M.; VERICAT, FERNANDO

CHAOS, SOLITONS AND FRACTALS

Elsevier

Lugar: Amsterdam; Año: 2007

0960-0779

We consider different multifractal decompositions of the form K_{¥á_{i}}={x:g_{i}(x)=¥á_{i}}, i=1,2,,...,d, and we study the dimension spectrum corresponding to the multiparameter decomposition K_{¥á}=⋂_{i=1}^{d}K_{¥á_{i}}, ¥á=(¥á©û,...,¥á_{d}). Then for an homeomorfism f:X¡æX and potentials ϕ,¥÷:X¡æ R we analyze the decompositions K_{¥á}⁺={x:lim_{n¡æ¡Ä}(1/n)(S_{n}⁺(ϕ))(x)=¥á}, K_{¥â}⁻={x:lim_{n¡æ¡Ä}(1/n)(S_{n}⁻(¥÷))(x)=¥â}, where (1/n)(S_{n}⁺(ϕ)), (1/n)(S_{n}⁻(¥÷)) 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 Holder 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.