Continuous Mapping Local Entropy And Return To The Entropy Of The Mixed Multifractal Entropy Spectrum

We introduce mixed multifractal spectra associated with Poincare recurrences and invariant measure. By the method of using Caratheodory construction to define local entropies for recurrence time , we establish an exact formula on multifractal spectrum of local entropies for recurrence time and local entropies of invariant measure.