Multifractal Analysis Of The Asympyotically Additive Potentials

The nonadditive thermodynamic formalism developed by Barriera in1996is a gen-eralization of the classical thermodynamic formalism in which the topological pressureP (ψ) of a single function is replaced by the topological pressure P (Φ) of a sequenceof functions Φ=(ψ_n)_n. This is an important improvement in multifractal analysis.From then on, people are interested in the multifractal analysis of additive, sub-additive, super-additive, almost additive sequences, assuming the equilibrium state isunique. Climenhaga also got some multifractal analysis results of the additive functionsequences under the weaker condition that there is a dense subset D C(X), in whichfor every continuous function, it has an unique equilibrium state.For general asymptotically additive potential on general topological dynamicalsystems, we establish some variational principle for the u-dimension spectra on theconditions that the entropy function is upper-continuous and that there is a densesubset in which for every continuous function, it has an unique equilibrium state.