The tent map ’ : 1⁄20; 1 ! 1⁄20; 1 ; x 7! minf2x; 2ð1 xÞg, the baker map ’ : 1⁄20; 1 ! 1⁄20; 1 ; x 7! 2x ð0 x 1=2Þ; 2x 1 ð1=2 x 1Þ, and the quadratic dynamical system ’ : 1⁄20; 1 ! 1⁄20; 1 ; x 7! 4xð1 xÞ are fundamental maps in the chaos theory, and have an intriguing property ‘‘For any infinite sequence !0; !1; !2; . . . each term !i of which is either 1⁄20; 1=2 ð1⁄4 AÞ or 1⁄21=2; 1 ð1⁄4 BÞ, there exists an initial point x0 2 !0 such that ’ðx0Þ 2 !1; ’ð’ðx0ÞÞ 2 !2; ’ð’ð’ðx0ÞÞÞ 2 !3; . . .’’. It is easy to recognize that this property guarantees the variety of their orbits. Then, if we assume that A and B mean the head and the tail of a coin, respectively, we can see that for any indeterministic time series obtained by coin tossing, all members of the series are traced deterministically. Accordingly, one may raise the general problem about the relation between determinism and indeterminism. Also, from the fact that any phenomenon expressed by the time series is totally described by such a simple law, one may raise the fundamental problem ‘‘What are the laws describing phenomena?’’. Also, assuming that A and B are alternative selections chosen successively, one may see such a situation that any history of actions, choosing selections, is recognized (like a destiny) by such a simple law, and may raise the problem ‘‘What is the choices of selections?’’. However, this property is too restricted in that the number of the selections A and B is only two. Therefore, the following generalized property (P) is naturally proposed as a primitive chaotic behavior. (P) For any infinite sequence !0; !1; !2; . . . ; there exists an initial point x0 2 !0 such that f!0 ðx0Þ 2 !1; f!1 ð f!0 ðx0ÞÞ 2 !2; . . . : Here, each !i is an element of a family fX ; 2 g of nonempty subsets of a set X, and each fX is the map from X to X. In this note, let us explore sufficient conditions for guaranteeing the existence of this chaotic behavior for the purpose of revealing the essence of its existence. As a result, we can see the emergence of dendrite known in the fields of the materials science and the fractal theory. In the beginning, we prepare the following lemma, whose proof is the same as that in the previous article. Lemma 1. If X is a countably compact space, fX ; 2 g is a family of nonempty closed subsets of X, and each fX is a continuous map from the subspace X of X onto X, the property (P) holds. Accordingly, the chaotic behavior is guaranteed by such a space X, a family fX ; 2 g, and maps fX ; 2 . However, the existence of the continuous onto maps is by no means a weak condition but an artificial one. Therefore, it is natural that we explore the sufficient conditions for this existence, and we can recall the following lemma obtained from the Hahn–Mazurkiewicz Theorem and the Tietze Extension Theorem.