Shadowing in Dynamical Systems: Zero-dimensional Extensions and Inverse Limits

Abstract

Shifts of finite type (SFTs) play a central role in the theory of shadowing. Good and Meddaugh showed that SFTs serve as basic building blocks in the structural theory of shadowing systems; in particular, every compact metric dynamical system with shadowing is a factor of the inverse limit of an inverse sequence consisting of SFTs. We first show that, for this factor representation alone, neither shadowing nor metrizability is needed: every compact Hausdorff dynamical system is a factor of the inverse limit of an inverse system consisting of SFTs. Thus, being a factor of an SFT inverse limit is not the structural feature genuinely forced by shadowing. What shadowing provides is stronger stability: in the metric case, every compact shadowing system is a factor of the inverse limit of an inverse sequence of SFTs with surjective bonding maps. Hence the associated zero-dimensional extension still has shadowing. We also prove that every compact Hausdorff shadowing system is conjugate to an inverse limit of metrizable shadowing systems with factor bonding maps.