Empirical time series of inter-event 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 non-monotonic behavior of the generalized Hurst exponent $h(q)$ -- the fundamental exponent in the study of multifractals. The consequence of this behavior is the non-monotonic behavior of the coarse Hölder exponent $\alpha (q)$ leading to multi-branchedness 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 multi-branched multifractality are revealed. The results  are presented for the high-frequency data from Polish stock market (Warsaw Stock Exchange) for intertrade times for KGHM - one of the most liquid stocks there.
 J. Klamut, R. Kutner, T. Gubiec, and Z. R. Struzik, 'Multibranch multifractality and the phase transitions in time series of mean interevent times', Phys. Rev. E 101, 063303 (2020)