ng to the injective branch $f_{\textrm{B}}$. This choice is of special interest to us as it is the class that generates the $G$-function on $\mathcal{R}_{+}$ in graphical convergence. This allows us to express $\omega(A)$ as $$\omega(A)=\{ x\in X\!:(\exists n_{k}\in\mathbb{N})(n_{k}\rightarrow\infty)(f_{\textrm{B}}^{-n_{k}}(x)=x_{-k}\textrm{ converges in }(X,\mathcal{U}))\};\label{Eqn: Attractor_R+}$$ note that the $x_{-k}$ of this equation and the $x_{k}$ of Eq. (\[Eqn: Def: omega(A)\]) are, in general, quite different points.

A simple illustrative example of the construction of $\omega(A)$ for the positive injective branch of the homeomorphism $(4x^{2}-1)/3$, $-1\leq x\leq1$, is shown in Fig. \[Fig: omega\], where the arrow-heads denote the converging sequences $f^{n_{i}}(x_{i})\rightarrow x$ and $f^{n_{i}-m}(x_{i})\rightarrow x_{-m}$ which proves invariance of $\omega(A)$ for a homeomorphic $f$; here continuity of the function and its inverse is explicitly required for invariance. Positive invariance of a subset $A$ of $X$ implies that for any $n\in\mathbb{N}$ and $x\in A$, $f^{n}(x)=y_{n}\in A$, while negative invariance assures that for any $y\in A$, $f^{-n}(y)=x_{-n}\in A$. Invariance of $A$ in both the forward and backward directions therefore means that for any $y\in A$ and $n\in\mathbb{N}$, there exists a $x\in A$ such that $f^{n}(x)=y$. In interpreting this figure, it may be useful to recall from Def. 4.1 that an increasing number of injective branches of $f$ is a necessary, but not sufficient, condition for the occurrence of chaos; thus in Figs. \[Fig: log357\](a) and \[Fig: omega\], increasing noninjectivity of $f$ leads to constant valued limit functions over a connected $\mathcal{D}_{+}$ in a manner similar to that associated with the classical Gibb’s phenomenon in the theory of Fourier series.

Graphical convergence of an increasingly nonlinear family of functions implied by its increasing non-injectivity may now be combined with the requirements of an attractor to lead to the concept of a chaot