CHAOTIC PRODUCT FUNCTIONS WITH A NON-CHAOTIC COMPONENT

It is known that a chaotic function in the Devaney sense has three main properties: namely, topological transitivity, density of periodic points, and sensitive dependence on initial conditions. The product method is a main tool for constructing new discrete dynamical systems as well as topological space. In the theory of discrete dynamical systems, it has been investigated whether the properties of two chaotic functions carry over to their product function, and it has been shown that product of two Devaney chaotic functions may not always be chaotic in the Devaney sense. In this work, we consider the reverse problem: if the product system is chaotic, what can be said about the chaotic behavior of its components? So, we construct chaotic functions in the product form φ×γ, where the second component is chaotic while first is not. Additionally, we show that φ×γ is chaotic in the Li-Yorke sense if γ is Li-Yorke chaotic, even though φ is not.