Two-Point Preimage Entropy For Random Group Endomorphisms

In this paper, we study the preimage entropy for random group endormorphisms,where dynamical system for random group endormorphism refers to iterate of different group endor-morphism and preimage entropy considers the exponential increase of "backward orbits" to describe the uncertainty.The most important result in this paper is variational principle for random group endomor-phisms: Letβis the Borelσ-algebra on X andφis a topological RGE on X overθ. If we denote the set ofφ-invariant probability measures by M(φ), then sup₍μ∈M(φ) h_pre,μ(φ) = h_pre(φ). There are four sections in this paper as following:In the first section (introduce), we give some background for dynamical system and preimage entropy.In the second section (§1), we introduce measure-theoretic two-point preimage entropy for random group endormorphisms. And then we give some important properties for measure-theoretic two-point preimage entropy.for example Shannon-McMillan-Breiman theorem: For any finite partitionαof X, we have (?) and in L¹(Ω×X,μ). Particularly, ifμis ergodic,(?) With Shannon-McMillar.-Breiman theorem we get the formula for entropy between random group endormorphismsφand the skew product transformationinduced byφ:(?)In the third section (§2), we define topological two-point preimage entropy for topological random group endormorphisms . In this section, we prove the mensurability for the function sup_x∈X s(ω,n,ε,φ^-1(k,ω})[x])ω, and then we get topological two-point preimage entropy for topological random group endormorphisms using separated set, after that give an equivalent definition using open covers.In the forth section (§3), a variational principle is given. It relates measure-theoretic two-point preimage entropy for random group endormorphisms in§1 and topological two-point preimage entropy for topological random group endormorphisms in§2, i.e.sup_{μ∈M(φ)} h_pre,μ(φ) = h_pre(φ).