Physics Letters A 318 (2003) 452–456 www.elsevier.com/locate/pla Specific heat log-periodicity from multifractal energy spectra Danyel J.B. Soares a , Marcelo L. Lyra b,∗ , Luciano R. da Silva a a Departamento de Física, Universidade Federal do Rio Grande do Norte, 59072-970 Natal, RN, Brazil b Departamento de Física, Universidade Federal de Alagoas, 57072-970 Maceió, AL, Brazil Received 7 July 2003; accepted 29 August 2003 Communicated by J. Flouquet Abstract In this work, we investigate the emergence of log-periodic oscillations in the low-temperature behavior of the specific heat of systems whose energy spectra present a self-similar character. The critical attractor of z-generalized logistic maps are used to generate multifractal energy spectra with tunable singularity spectra. We study the relationship between the average value and amplitude of the log-periodic oscillations on the map nonlinearity strength as well as on the scaling exponents characterizing the energy spectrum. Our numerical results show a monotonic decrease of the oscillations amplitude with increasing nonlinearity. Further, we obtain that the average low-temperature specific heat is directly related to the minimum singularity strength governing the scaling behavior of the most concentrated energy range.  2003 Elsevier B.V. All rights reserved. PACS: 05.20.-y; 61.43.Hv; 65.40.+g Keywords: Multifractal energy spectral; Nonlinear maps; Log-periodic specific heat 1. Introduction Nonlinear one-dimensional dissipative maps have been widely used to describe some relevant aspects re- lated to the emergence of complex behavior in nature. Usually the dynamical attractor presents a transition between periodic and chaotic behavior at which the map displays a strong exponential sensitivity to initial conditions [1]. At the onset of chaos, the attractor ex- hibits a multifractal structure with long-range tempo- ral and spatial correlations [2]. Due to their simplicity, most of the scaling behavior characterizing the transi- * Corresponding author. E-mail address: marcelo@fis.ufal.br (M.L. Lyra). tion to chaos can be obtained with high accuracy and provide important insights on the behavior of more complex systems. The self-similar character of the critical attrac- tor of low-dimensional dissipative maps has been explored to study some thermodynamic features of quasi-crystals. In general, quasi-crystals have proper- ties that are intermediate between periodic and random structures [3,4]. In particular, the energy spectrum has a quite complex structure. Simplified fractals based in the Cantor set and Fibonacci sequence [5–8], as well as the critical attractor of the logistic and circle maps [9,10], have been used to model the energy spectrum of quasi-periodic systems. The thermodynamic behav- ior derived from such self-similar spectra display some anomalous features with the most prominent one being 0375-9601/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.08.056