Shadowing properties and chaotic properties of non-autonomous product systems

This paper examines how properties such as shadowing properties, transitivity, and accessibility in non-autonomous discrete dynamical systems carry over to their product systems. The paper establishes a proof that the product system exhibits the pseudo-orbit shadowing property (PSP) if, and only if, both factor systems possess PSP. This relationship, which is both sufficient and necessary, also holds for the average shadowing property (ASP) and accessibility. Consequently, in practical problem scenarios, certain chaotic properties of two-dimensional systems can be simplified to those observed in one-dimensional systems. However, it should be noted that while the point-transitivity, transitivity, or mixing of the product system can be deduced from the factor systems, the reverse is not true. In particular, this paper constructs counterexamples to demonstrate that some of the theorems presented herein do not hold when considering their inverses.