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Self-affine multiplicity fluctuation is investigated by using the two-dimensional factorial moment methodology and the concept of the Hurst exponent (H). Investigation on the experimental data of compound particles and target fragments emitted in 84Kr?AgBr interactions at 1.7 A GeV reveals that the best power law behaviours are exhibited at H = 0.7 and 0.6 respectively, and the data for shower particles produced in 84Kr-emulsion interactions at 1.7 A GeV indicate that the best power law behaviour occurs at H = 0.6, all of which show the self-affine multiplicity fluctuation patterns. The multifractality and the non-thermal phase transition occurring during producing the compound particles, the target fragments, and the shower particles in the 84Kr?AgBr interaction and the 84Kr?emulsion interaction are also discussed. The multifractality is observed during producing compound particles, target fragments, and shower particles. In the target fragment production, an evidence of non-thermal phase transition is observed, but in the shower particle production and the compound particle production, no evidence of non-thermal phase transition is observed.

Very recently, via the covariant form of the adiabatic invariant I = pidqi instead of I = ∫pidqi, an equally spaced spectroscopy of a Schwarzschild black hole was derived. The emphasis was given to the covariant of results. In this paper, we extend that work in a spherically symmetric spacetime to the case of a rotating Banados—Teitelboim—Zanelli (BTZ) black hole. It is noteworthy that the adiabatic covariant action I = pidqi gives the same value for the black hole spectroscopy in different coordinates. The result shows that the area spectrum is ΔA = 8πl2P, which confirms Bekenstein's initial proposal. And the result is consistent with that already obtained by other methods.