Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors

P Grassberger, R Badii, A Politi

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414 Citations (Web of Science)


The analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems. Prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective Liapunov exponents. For hyperbolic attractors, the latter determines the metric entropies and provides one constraint on the partial dimensions. For nonhyperbolic attractors, there are important modifications. We discuss them for the examples of the logistic and Hénon map. We show, in particular, that the generalized dimensions have singularities with noncontinuous derivative, similar to first-order phase transitions in statistical mechanics.
Original languageEnglish
Pages (from-to)135-178
Number of pages44
JournalJournal of Statistical Physics
Issue number1-2
Publication statusPublished - 1 Apr 1988


  • dynamical systems
  • generalized dimensions and entropies
  • Liapunov exponents
  • scaling functions
  • hyperbolicity
  • phase transitions


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