Variational principles of partial pre-imageentropy and conditional pre-image entropy

Abstract

In this paper, wedefine and study two entropy-like invariants for non-invertible map,which are called partial pre-image entropy and conditional pre-imageentropy, respectively, and prove some variational principles forthem. More precisely, let $(X,~T)$ be a topological dynamicalsystem and $\xi$ be a measurable partition of $X$ with $T^{-1}\xi\preceq~\xi$. Then$h_{\mathrm{top}}(T\mid~[\xi]^-)~\geq~\sup_{\mu~\in~\mathcal{M}(X,~T)}h_{\mu}(T\mid~[\xi]^-).$ Moreover, if $[\xi]^-$ can be generated by a upper semi-continuous partition of $X$, then$h_{\mathrm{top}}(T\mid~[\xi]^-)=\sup_{\mu~\in~\mathcal{M}(X,~T)}h_{\mu}(T\mid~[\xi]^-).$ As an application, we study the relativizationof pre-image entropy and obtain other formulas about Cheng-Newhousepre-image entropy.