I read books A First Course in Discrete Dynamical Systems by Richard A. Holmgren and An Introduction to Chaotic Dynamical Systems by Robert L. Devaney.
I want to understand which concepts of "chaos" lead mathematicians to place these three conditions for a function to be chaotic. If you can please explain your Idea on an example.
Definition 8.5. Let $V$ be a set. $F: V \to V$ is said to be chaotic on $V$ if
1. $F$ has sensitive dependence on initial conditions.
2. $F$ is topologically transitive.
3. periodic points are dense in $V$.
Definition 8.2. $F:J\to J$ has sensitive dependence on initial conditions if there exists $δ > 0$ such that, for any $x \in J$ and any neighborhood $N$ of $x$, there exists $y\in N$ and $n > 0$ such that $|F^n(x) - F^n(y)| > δ$.
Definition 8.1. $F:J\to J$ is said to be topologically transitive if for any pair of open sets $U, V \subset J$ there exists $k > 0$ such that $F^k(U) \cap V \neq 0$.