Shift with orbit basis and realization of one dimensional maps
The author defines a special class of one-sided shift transformations called shifts with orbit basis. A description of this class as the images of the Kakutani towers over two-sided shifts is given. Next, it is shown that the topological entropy of shifts with free orbit basis B is equal to - log t(B), where \(t(B)=\sup\{t\geq 0:\) D(t) converges and is non- negativ\(e\}\), and \(D(t)=1-a_ 0t-a_ 1t^ 2-..\). is the reciprocal of the Artin-Mazur zeta function. The coefficient \(a_ n\) in that formula is the number of words of the length n in the orbit basis B. Finally, it is shown that any piecewise continuous map of the unit interval can be represented as the union of shifts with orbit basis. In particular, the structure of \(\beta\) -transformations and of unimodal continuous maps of the unit interval is discussed in this context. Shifts with orbit basis were earlier studied by the author in [Sci. Pap. Coll. Gen. Educ. Univ. Tokyo 80, 11-22 (1980; Zbl 0465.58023)], where they were defined as the images of the Kakutani towers.
scientific article; zbMATH DE number 3872340