Weighted thermodynamic formalism and applications

Abstract

Let $(X,T)$ and $(Y,S)$ be two subshifts so that $Y$ is a factor of $X$. For any asymptotically sub-additive potential $\Phi$ on $X$ and $\ba=(a,b)\in\R^2$ with $a>0$, $b\ge 0$, we introduce the notions of $\ba$-weighted topological pressure and $\ba$-weighted equilibrium state of $\Phi$. We setup the weighted variational principle. In the case that $X,Y$ are full shifts with one-block factor map, we prove the uniqueness and Gibbs property of $\ba$-weighted equilibrium states for almost additive potentials having the bounded distortion properties. Extensions are given to the higher dimensional weighted thermodynamic formalism. As an application, we conduct the multifractal analysis for a new type of level sets associated with Birkhoff averages, as well as for weak Gibbs measures associated with asymptotically additive potentials on self-affine symbolic spaces.

43 pages and 1 figures

References in corpus (4)

• Lyapunov spectrum of asymptotically sub-additive potentials
• On the topological pressure of random bundle transformations in sub-additive case
• Weighted equilibrium states for factor maps between subshifts
• Existence of a measurable saturated compensation function between subshifts and its applications

Cited by in corpus (10)

• Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets
• Weighted equilibrium states for factor maps between subshifts
• Multifractal analysis and localized asymptotic behavior for almost additive potentials
• Measures of the full Hausdorff dimension for a general Sierpiński carpet
• Large deviations for the local fluctuations of random walks and new insights into the "randomness" of Pi
• Localized asymptotic behavior for almost additive potentials
• Multifractal analysis for Birkhoff averages on Lalley-Gatzouras repellers
• New approach to weighted topological entropy and pressure
• Infinite Non-Conformal Iterated Function Systems
• The packing spectrum for Birkhoff averages on a self-affine repeller