Abstract
We survey the theoretical findings and a set of applications in several different fields concerning two-dimensional maps characterized by a vanishing denominator. This family of maps, and the related dynamic properties, were originally brought to the attention of the researchers through their appearance in an economic application. Nowadays such maps have also been used in other research areas such as ecology or biology confirming a broader application of the theoretical results about plane maps with vanishing denominator to various research areas.
Highlights
In this paper we consider a particular class of maps that are not defined in the whole space
We will survey applications in Biology and Ecology, besides the well-known application to Economics and we show how maps with vanishing denominator may appear in a class of fractional rational maps that may have multiple practical applications
When the focal point is simple there exists a one-to-one correspondence between the slope m of the arc γ in Q and the point (F(Q), y) in which its image T(γ ) crosses the prefocal set δQ
Summary
In this paper we consider a particular class of maps that are not defined in the whole space. We summarize the results concerning two-dimensional maps not defined in the whole plane because of the presence of a denominator that can vanish. These maps are called maps with vanishing denominator and the consequences of their features for the global properties of such maps have already been surveyed (see for instance 1 and more recently 2). What differentiates this survey from the others is the focus on the different fields of application of such maps. The paper proceeds as follows: in Section 2 we introduce the most important definitions and the main results; Section 3 is devoted to the applications while in Section 4 we make some conclusions
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