The entropies and multifractal spectrum of some compact systems

Abstract

In the present paper, the following two compact systems and their extensions are studied. (i) A compact system ( X , f ) and its inverse limit ( X ¯ , f ¯ ) . (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 ( X ¯ , f ¯ ) is obtained, i.e., h ( f | Z ) = h ( f ¯ | π 0 - 1 Z ) , where Z is any subset of X and π 0 the projection of X ¯ to X such that π 0 ( x 0 , x 1 , … ) = x 0 . For case (ii), we obtain a similar result. Using these results, we show that ( X , f ) and ( X ¯ , f ¯ ) (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 ( X ¯ , f ¯ ) (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.