Markov-like set-valued functions on intervals and their inverse limits

Abstract

We introduce Markov-like functions on intervals as a generalization of generalized Markov interval functions and define the notation of the same pattern between Markov-like functions. Then we show that two generalized inverse limits with Markov-like bonding functions having the same pattern are homeomorphic. This result gives a generalization of the results of S. Holte ([9]) and I. Banič and T. Lunder ([5]). ←− {X n , f n } of the inverse sequence {X n , f n } n∈N is defined by lim ←− {X n , f n } := {(x 1 , x 2 ,. . .) ∈ ∞ n=1 X n | x n ∈ f n (x n+1) for any n ∈ N}.

Markov-like set-valued functions on finite graphs and their inverse limits

“…From Theorem 1.3.3, we can assume both I and J are the unit interval [0,1]. We now construct a map H from lim…”

“…Let graphs of f : T → 2 T and g : S → 2 S be defined by The notations of generalized Markov interval maps and having the same pattern were introduced by Banič and Lunder [BL] and the author [1] gave further generalization. In both [BL] and [1] they considered only interval maps and defined the notation of having the same pattern by defining correspondence from a Markov partition to the other Markov partition in accordance with the order of real numbers. However, finite graphs do not have an appropriate order.…”

“…Hence we can give a unified definition including one by [BL] and [1]. In particular, in both [BL] and [1], they considered the same pattern with respect to a pair (h, h) of a piecewise linear homeomorphism h as the above example. In Section 2.6, we will give examples of Markov-like functions having the same pattern in the new sense.…”