Multibranch multifractality and the phase transitions in time series of mean interevent times

Empirical time series of interevent or waiting times are investigated using a modified Multifractal Detrended Fluctuation Analysis operating on fluctuations of mean detrended dynamics. The core of the extended multifractal analysis is the nonmonotonic behavior of the generalized Hurst exponent h(q)-the fundamental exponent in the study of multifractals. The consequence of this behavior is the nonmonotonic behavior of the coarse Hölder exponent α(q) leading to multibranchedness of the spectrum of dimensions. The Legendre-Fenchel transform is used instead of the routinely used canonical Legendre (single-branched) contact transform. Thermodynamic consequences of the multibranched multifractality are revealed. These are directly expressed in the language of phase transitions between thermally stable, metastable, and unstable phases. These phase transitions are of the first and second orders according to Mandelbrot's modified Ehrenfest classification. The discovery of multibranchedness is tantamount in significance to extending multifractal analysis.