Inverse Limits with Generalized Markov Interval Functions

Inverse limits with countably Markov interval functions

“…In the following definition we generalize the notion of Markov interval functions from the case where A is finite to the case where A is countable (including the case when A is countably infinite). One can easily see that every (generalized) Markov interval function (as defined in [3] and [1]) is also a countably Markov interval function. The set A is finite, therefore also countable and the set A ′ in this case is empty.…”

“…Holte proved under which conditions two inverse limits with Markov interval bonding functions are homeomorphic [3]. A generalization of Markov interval maps was introduced in [1], where authors defined so-called generalized Markov interval functions. They are a non-trivial generalization of single-valued mappings from I = [x, y] to I to set-valued functions from I to 2 I .…”

“…An upper semicontinuous function f from I = [x, y] to 2 I is a generalized Markov interval function with respect to A = {a 0 , a 1 I. Banič and T. Lunder also introduced the conditions when such functions follow the same pattern and, with this notion, proved that two generalized inverse limits with generalized Markov interval bonding functions are homeomorphic, if the bonding functions follow the same pattern. More precisely, let {f n } ∞ n=1 be a sequence of upper semicontinuous functions from I = [a 0 , a m ] to 2 I with surjective graphs, which are all generalized Markov interval functions with respect to A = {a 0 , a 1 , .…”

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

“…In 2002 S. E. Holte ([9]) introduced the notation of the same pattern between Markov interval maps, and showed that any two inverse limits of inverse sequences whose bonding maps are Markov interval maps with the same pattern are homeomorphic. There are many generalizations of Markov interval maps ( [1,2,5,7,17]). In 2013 I. Banič and T. Lunder ( [5]) extended the notation of Markov interval maps to upper-semi continuous functions having interval-valued images on finitely many points of their domain, named generalized Markov interval functions.…”

“…There are many generalizations of Markov interval maps ( [1,2,5,7,17]). In 2013 I. Banič and T. Lunder ( [5]) extended the notation of Markov interval maps to upper-semi continuous functions having interval-valued images on finitely many points of their domain, named generalized Markov interval functions. They also defined the notation of the same pattern between generalized Markov interval functions and showed the corresponding theorem for generalized inverse limits to Holte's theorem.…”