contact structure | Pengfei's Notes

8. (Alejandro) Let $f:X\to X$ be an arbitrary transitive homeomorphism and $u:X\to(0,1/4)$ be an arbitrary non-constant continuous function. Then, let’s define $c(x):=u(x)-u(fx)+1$ , $x\in X$ , and consider the suspension flow $f_t:X_c\to X_c$ on $X_c$ . Note that for each $x\in X$ and $t\in(0,1/4)$ : $f_1(x,t+u(x))=(x,t+u(x)+1)=(x,t+u(fx)+c(x))=(fx,t+u(fx))$ . So the compact set $\lbrace(x,t+u(x)):x\in X\rbrace$ is $f_1$ -invariant for every $t\in(0,1/4)$ , and $f_1$ is not transitive. Notice the function c is not constant because $f$ is transitive and $u$ is not constant itself.