Abstract
Multifractal analysis studies level sets of asymptotically defined quantities in dynamical systems. In this paper, we consider the u -dimension spectra on such level sets and establish a conditional variational principle for general asymptotically additive potentials by requiring only existence and uniqueness of equilibrium states for a dense subspace of potential functions.
Highlights
The theory of multifractal analysis is a subfield of the dimension theory in dynamical systems
It studies a global dimensional quantity that assigns to each level set a “size” or “complexity”, such as its topological entropy or Hausdorff dimension
In [6], Barreira, Saussol, and Schmeling extended their work to higher-dimensional multifractal spectra, for which they consider the more general u -dimension in place of the topological entropy
Summary
The theory of multifractal analysis is a subfield of the dimension theory in dynamical systems. In [6], Barreira, Saussol, and Schmeling extended their work to higher-dimensional multifractal spectra, for which they consider the more general u -dimension in place of the topological entropy. Let D ( X ) ⊂ C ( X ) be the family of continuous functions with a unique equilibrium measure, they obtain the following result: Theorem 1. In [7], Barreira and Doutor study the spectrum of the u -dimension for the class of almost additive sequences with a unique equilibrium measure and establish a conditional variational principle for the dimension spectra in the context of the nonadditive thermodynamic formalism. This paper is devoted to the study of higher-dimensional multifractal analysis for the class of asymptotically additive potentials. We consider the multifractal behavior of u -dimension spectrum of level sets and establish the conditional variational principle under the assumption proposed by Climenhaga.
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