Tag Archives: contact structure

Short 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_con 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.

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