Typical continuous functions are not chaotic in the sense of Devaney (Q1852348)

Let \(C(M,M)\) be the set of continuous functions \(f: M\to M\) on a compact metric space \(M\) with a metric \(d\). The iterates of \(f\) are defined inductively by \(f^{n+1}(x)= f(f^n(x))\). A function \(f: M\to M\) is transitive if for every pair of nonempty open sets \(U,V\subset M\) there is a positive integer \(k\) such that \(f^k(U)\cap V\neq\emptyset\). For \(f\in C(M,M)\) a point \(x\in M\) is called chain recurrent if for all \(\varepsilon> 0\) there exists a finite sequence of points \((x_0,x_1,\dots, x_n)\), such that \(x_0= x_n= x\) and \(d(f(x_{i-1}, x_i)< \varepsilon\), \(i= 0,1,\dots, n\). The set of all chain recurrent points is denoted by \(CR(f)\). Denote a residual subset of \(C(M,M)\) by \(S\). The main results of the paper: 1. There exists \(S\subset C(M,M)\) such that every function in \(S\) is not transitive. Hence, there exists \(S\subset C(M,M)\) such that all functions in \(S\) are not chaotic in the sense of Devaney, and if \(M\) is a compact interval there exists \(S\subset C(M,M)\) such that every function in \(S\) is Li-Yorke chaotic but is not chaotic in the sense of Devaney. 2. There exists \(S\subset C(M,M)\) such that all functions in \(S\) have an asymptotically stable set (attractor). Hence, if \(M\) is connected, there exists \(S\subset C(M,M)\) such that \(CR(f)\neq M\) for all \(f\in S\). 3. If \(M\) is locally connected and connected, \(M\) has fixed point property and absolute retract property, then the set of functions in \(C(M,M)\) which have every point chain recurrent, is nowhere dense in \(C(M,M)\). [See also Errata to this paper: the author, ibid. 27, No. 1, 397-399 (2002; Zbl 1012.37010), reviewed below].