Multidimensional Spectra and Number Theory

Part of the Progress in Mathematics book series (PM, volume 272)


In the theory of dynamical systems we are sometimes interested in more than one local quantity at the same time. Examples include Lyapunov exponents, local entropy, and pointwise dimension. However, the theory of multifractal analysis described in the former chapters only considers separately each of these local quantities. This led Barreira, Saussol and Schmeling to develop in 20 a multidimensional version of the theory of multifractal analysis. For example, we can consider intersections of level sets of Birkhoff averages of different functions, and describe their multifractal properties, including their “size” in terms of topological entropy and of Hausdorff dimension. It turns out that the corresponding multidimensional multifractal spectra exhibit several nontrivial phenomena that are absent in the one-dimensional case. A unifying element continues to be the use of the thermodynamic formalism.


Number Theory Lyapunov Exponent Cohomology Class Multifractal Analysis Multifractal Spectrum 
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© Birkhäuser Verlag AG 2008

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