The new face of multifractality: Multi-branchness and the phase transitions in time series of inter-event times Jaros law Klamut ∗ and Ryszard Kutner Faculty of Physics, University of Warsaw, Pasteur Str. 5, PL-02093 Warsaw, Poland Tomasz Gubiec Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215 USA and Faculty of Physics, University of Warsaw, Pasteur Str. 5, PL-02093 Warsaw, Poland Zbigniew R. Struzik University of Tokyo, Bunkyo-ku, Tokyo 113-8655, Japan and Advanced Center for Computing and Communication, RIKEN, 2-1 Hirosawa, Wako 351-0198, Saitama, Japan We develop an extended multifractal analysis based on the Legendre-Fenchel transform (some- times referred to as Legendre multi-branched one) rather than the routinely used canonical Legendre transform. In our variant of coarse-graining pre-processing, the local detrending of time series has been replaced by an appropriate averaging over days combined with properly-suited detrending on a daily time scale. This new approach is devoid of troublesome artifacts in the form of innumer- able faults of these local trends that can deform the hierarchy of fluctuations and hence the final multifractality. Notably, our analysis is sensitive to the change of time scale as it should be. This analysis has developed, e.g., for empirical time series of inter-event or waiting times, which are an essential element of the popular continuous-time random walk formalism. The core of this extended multifractal analysis is the non-monotonic behavior of the generalized Hurst exponent – the funda- mental exponent of the study – and hence a multi-branched spectrum of dimensions, which for our case is additionally of the left-sided one. We examine the main thermodynamic consequences of the existence of this type of multifractality. They can be expressed directly in the language of thermally stable, metastable, and unstable phases, and phase transitions between them as well. These phase transitions are of the first and second orders according to the modified Ehrenfest classification, sometimes called the Mandelbrot one. PACS numbers: 89.65 Gh, 05.40.-a, 89.75.Da I. INTRODUCTION A. General remarks The concept of extended scale invariance referred to as multifractality, has become a routinely ap- plied but still intensively developed methodology for studying both complex systems [1–5, 7] and nonlin- ear (e.g., chaotic with a low degree of freedom) dy- namical ones [8]. It is a rapidly evolving and inspir- ing approach to nonlinear science in many different fields stretching far beyond traditional physics [9]. The direct inspiration of the present work is our earlier results presented in papers [10, 11]. In these publications, we found the left-sided multifractal- ity on financial markets as a direct result of a non- analytic behavior of the R´ enyi exponent. We indi- cated that a broad distribution of inter-event times is * Jaroslaw.Klamut@fuw.edu.pl responsible for the existence of left-sided multifrac- tality. In the present work, we suggest that primarily nonlinear long-term autocorrelations bear responsi- bility for the multifractality observed. Attention was first drawn to the existence of left- sided multifractality by Mandelbrot and coauthors [12, 13]. This multifractality was generated by the binomial cascade, which produces singularity in the R´ enyi exponent or stretched exponential decay of the smallest coarse-grained probability. Blumenfeld and Aharony [14] discovered an exciting breakdown of multifractality in diffusion-limited aggregation. They found strongly asymmetric spectra of singular- ity depending on the size of the growing aggregate in DLA, showing an apparent tilt to the left as a sig- nature of the phase transition to non-multifractality. Earlier, the multifractals with the right part of the spectrum of singularities not well defined (caused by a phase transition), were mimicked by a random version of the paradigmatic two-scale Cantor set and also in the domain of DLA [15–18] (and refs therein). arXiv:1809.02674v3 [q-fin.ST] 13 Jul 2019