Dynamical properties of shift maps on inverse limits with a set valued function

Abstract

Set-valued functions from an interval into the closed subsets of an interval arise in various areas of science and mathematical modeling. Research has shown that the dynamics of a single-valued function on a compact space are closely linked to the dynamics of the shift map on the inverse limit with the function as the sole bonding map. For example, it has been shown that with Devaney’s definition of chaos the bonding function is chaotic if and only if the shift map is chaotic. One reason for caring about this connection is that the shift map is a homeomorphism on the inverse limit, and therefore the topological structure of the inverse-limit space must reflect in its richness the dynamics of the shift map. In the set-valued case there may not be a natural definition for chaos since there is not a single well-defined orbit for each point. However, the shift map is a continuous single-valued function so it together with the inverse-limit space form a dynamical system which can be chaotic in any of the usual senses. For the set-valued case we demonstrate with theorems and examples rich topological structure in the inverse limit when the shift map is chaotic (on certain invariant sets). We then connect that chaos to a property of the set-valued function that is a natural generalization of an important chaos producing property of continuous functions.

Topologically Transitive and Mixing Properties of Set-Valued Dynamical Systems

“…However, knowing how the points of the system move is not sufficient as there are problems and applications that require one to know how the subsets of the system move. In recent years, several works and research on the topological dynamics of set-valued dynamical systems can be found (see [13][14][15][16][17][18][19][20]). However, there are many properties for the dynamics of set-valued dynamical systems yet to be discovered.…”

Topological entropy of Markov set-valued functions

“…This is especially true when studying inverse limits of continuous functions on the unit interval, such as unimodal maps. Although most work in the area of set-valued inverse limits has focused on the topological aspects (particularly, investigating the various continua that can arise as inverse limits), some recent work has focused on the dynamical aspects and, in particular, the topological entropy, [5,8,9].…”

Entropy conjugacy for Markov multi-maps of the interval

“…In the past decade multi-maps have been studied extensively, with a particular focus on the topological structure of the associated space of trajectories or a related inverse limit space; see [15]. This development has also led to a renewed interest in the dynamics of multi-maps [17,18,13,11]. Additionally, multi-maps are the topological analogues of random maps of the interval, which have received substantial attention, e.g., [10,14,23,4].…”