The entropies and multifractal spectrum of some compact systems

In the present paper, the following two compact systems and their extensions are studied. (i) A compact system (X,f) and its inverse limit . (ii) A compact system (X,f) and its corresponding symbolic system (Σ,σ), where f is an expansive homeomorphism. For case (i), a relationship of topological entropy of (X,f) and is obtained, i.e., , where Z is any subset of X and π0 the projection of to X such that π0(x0,x1,…)=x0. For case (ii), we obtain a similar result. Using these results, we show that (X,f) and (resp. (X,f) and (Σ,σ)) have the same multifractal spectrum relative to the entropy spectrum. Moreover, as some applications of these results, we obtain that (a) The main result in Takens and Verbitski (1999) [Takens F, Verbitski E. Multifractal analysis of local entropies for expansive homeomorphism with specification. Commun Math Phys 1999;203:593–612] holds under weaker conditions. (b) (X,f) and (resp. (X,f) and (Σ,σ)) have the same multifractal analysis of local entropies. (c) For two positive expansive compact systems (X,f) and (Y,g), if they are almost topologically conjugate, then they have the same multifractal spectrum for local entropies.From a physical point of view, the numerical study of dynamical systems and multifractal spectra is also a very useful tool.