Shadowing Property of Continuous Maps with Zero Topological Entropy

Abstract

Abstract. The study of the shadowing property has a long history but for interval maps it is rather new. Recent research in this direction is mainly focused on the positive entropy maps and work for zero entropy is still seldom to be found in the literature. In this paper we give a characterization of zero topological entropy maps which have the shadowing property. Moreover, our condition is necessary for any continuous function to have the shadowing property.

Properties of invariant measures in dynamical systems with the shadowing property

“…Again by (1), it is not hard to see that F D is a shrink function. Therefore F D has the shadowing property, by the Main Theorem from [17], which we present below. By Theorem (4.3) we can approximate every invariant measure µ with supp(µ) ⊂ Q by an ergodic measure supported on an almost 1-1 extension of an odometer.…”

“…Before we can explain why F D has the shadowing property, we will need three definitions form [17]. If a, b ∈ [0, 1] and a = b then we write a, b to denote interval spanned by a and b.…”

Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval

“…In [2] it is proved that if (I, f 1,∞ ) is LYC then f is LYC provided it has the shadowing property. But this condition eliminates maps f with h(f ) = 0, see [8]. On the other hand, by Theorem B, if h(f ) > 0, then f 1,∞ must be even DC1.…”

“…Since a point in P cannot be periodic it has a preimage in P so that P is countably infinite. Since the intervals in J are periodic, there are j n ∈ N such that (8) z j ∈ O n if j ≥ j n , and…”