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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Variational principles and mixed multifractal spectra
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by L. Barreira and B. Saussol PDF
Trans. Amer. Math. Soc. 353 (2001), 3919-3944 Request permission


We establish a “conditional” variational principle, which unifies and extends many results in the multifractal analysis of dynamical systems. Namely, instead of considering several quantities of local nature and studying separately their multifractal spectra we develop a unified approach which allows us to obtain all spectra from a new multifractal spectrum. Using the variational principle we are able to study the regularity of the spectra and the full dimensionality of their irregular sets for several classes of dynamical systems, including the class of maps with upper semi-continuous metric entropy. Another application of the variational principle is the following. The multifractal analysis of dynamical systems studies multifractal spectra such as the dimension spectrum for pointwise dimensions and the entropy spectrum for local entropies. It has been a standing open problem to effect a similar study for the “mixed” multifractal spectra, such as the dimension spectrum for local entropies and the entropy spectrum for pointwise dimensions. We show that they are analytic for several classes of hyperbolic maps. We also show that these spectra are not necessarily convex, in strong contrast with the “non-mixed” multifractal spectra.
  • Luis M. Barreira, A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergodic Theory Dynam. Systems 16 (1996), no. 5, 871–927. MR 1417767, DOI 10.1017/S0143385700010117
  • Luis Barreira, Yakov Pesin, and Jörg Schmeling, Dimension and product structure of hyperbolic measures, Ann. of Math. (2) 149 (1999), no. 3, 755–783. MR 1709302, DOI 10.2307/121072
  • L. Barreira and J. Schmeling, Sets of “non-typical” points have full topological entropy and full Hausdorff dimension, Israel J. Math. 116 (2000), 29–70.
  • Rufus Bowen, Some systems with unique equilibrium states, Math. Systems Theory 8 (1974/75), no. 3, 193–202. MR 399413, DOI 10.1007/BF01762666
  • Gerhard Keller, Equilibrium states in ergodic theory, London Mathematical Society Student Texts, vol. 42, Cambridge University Press, Cambridge, 1998. MR 1618769, DOI 10.1017/CBO9781107359987
  • Eric Olivier, Analyse multifractale de fonctions continues, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 10, 1171–1174 (French, with English and French summaries). MR 1650242, DOI 10.1016/S0764-4442(98)80221-8
  • Yakov B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. Contemporary views and applications. MR 1489237, DOI 10.7208/chicago/9780226662237.001.0001
  • David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655
  • Roland Bacher, Dense lattices in dimensions $27$–$29$, Invent. Math. 130 (1997), no. 1, 153–158. MR 1471888, DOI 10.1007/s002220050180
  • J. Schmeling, Entropy preservation under Markov coding, preprint.
  • F. Takens and E. Verbitski, On the variational principle for the topological entropy of certain non-compact sets, preprint.
  • Peter Walters, Equilibrium states for $\beta$-transformations and related transformations, Math. Z. 159 (1978), no. 1, 65–88. MR 466492, DOI 10.1007/BF01174569
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Additional Information
  • L. Barreira
  • Affiliation: Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
  • MR Author ID: 601208
  • Email:
  • B. Saussol
  • Affiliation: LAMFA / CNRS FRE 2270, Université de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens, France
  • Email:
  • Received by editor(s): April 17, 2000
  • Received by editor(s) in revised form: August 30, 2000
  • Published electronically: June 6, 2001
  • Additional Notes: L. Barreira was partially supported by FCT’s Funding Program and NATO grant CRG970161. B. Saussol was partially supported by FCT’s Funding Program and by the Center for Mathematical Analysis, Geometry, and Dynamical Systems.
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 3919-3944
  • MSC (2000): Primary 37D35, 37C45
  • DOI:
  • MathSciNet review: 1837214