Finite type as fundamental objects even non-single-valued and non-continuous

Abstract: In this paper, we investigate the shadowing property for general upper semicontinuous set-valued maps, inspired by Good and Meddaugh [11], we identify the fundamental objects underpinning the shadowing theory for such maps. Specifically, we explore the relationships among shifts of finite type, the simplest set-valued maps, and the shadowing property. Our main result establishes that if $X$ is a compact, totally disconnected Hausdorff space, and $F$ is an upper semicontinuous closed-valued map on $X$ with the set-valued shadowing property, the following hold: (1) If $F$ is expansive, the orbit system of $(X,F)$ is conjugate to a shift of finite type. (2) Without expansiveness, $(X,F)$ is conjugate to an inverse limit of a net of the simplest set-valued systems satisfying the Mittag-Leffler condition. Additionally, the orbit system of $(X,F)$ is conjugate to an inverse limit of a net of shifts of finite type, also satisfying the Mittag-Leffler condition As a consequence, if an upper semicontinuous set-valued map on a totally disconnected space has the set-valued shadowing property, its orbit system also exhibits the single-valued shadowing property.