Measure-theoretic Pressure And Multi-fractal Analysis For Non-additive Potentials

Let(X,d,T)be a topological dynamical system,?={?n?1 a non-additive poten-tial on X,and let M(X,T)denote the set of all T-invariant Borel probability measures.For each x ? X and n ?N,consider measure and let V(x)be the limit-point set of ?n(x).This thesis first uses(n,?)-separated set and(?,n,?)-separated set to define the measure-theoretic pressure for non-additive potentials,and proves that these two defini-tions are equivalent provided that the measure is ergodic and ? is a subadditive potential on X.In addition,when the dynamical system(X,d,T)satisfies the uniform separation condition and the ergodic measures are entropy dense,these two definitions are equivalent for all invariant measures and asymptotically additive potentials.Furthermore,if the dy-namical system(X,d,T)has the uniform separation and the g-almost product properties,we have PGK(T,?)? inf{h(T,?)+ ?*(?),??K}.where K(?)M(X,T)is a compact connected subset,GK:= {x ? X:V(x)= K} and PGK(T,?)is the topological pressure on the set GK.Finally,we give an application of above the results to multi-fractal analysis for asymptotically additive potentials.