2023/06/09 by Lucas Backes, Fagner B. Rodrigues
Mathematics · Biochemistry, Genetics and Molecular Biology · #Mathematical Dynamics and Fractals #Advanced Topology and Set Theory #Caveolin-1 and cellular processes
paper · doi:10.1109/tit.2023.3284613
We prove a variational principle for the upper and lower metric mean dimension of level sets <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX"> \x∈ X: lim n→ ∞ \frac 1n∑ j=0n-1φ (fj(x))=α \ </tex-math></inline-formula> associated to continuous potentials <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">φ:X→ \mathbb R </tex-math></inline-formula> and continuous dynamics <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">f:X→ X </tex-math></inline-formula> defined on compact metric spaces and exhibiting the specification property. This result relates the upper and lower metric mean dimension of the above mentioned sets with growth rates of measure-theoretic entropy of partitions decreasing in diameter associated to some special measures. Moreover, we present several examples to which our result may be applied to. Similar results were previously known for the topological entropy and for the topological pressure.