On the continuity of the pressure for monotonic mod one transformations

If f : [0, 1] → R is strictly increasing and continuous define Tfx = f(x) (mod 1). A transformation T̃ : [0, 1] → [0, 1] is called ε-close to Tf , if T̃ x = f̃(x) (mod 1) for a strictly increasing and continuous function f̃ : [0, 1] → R with ‖f̃ − f‖∞ < ε. It is proved that the topological pressure p(Tf , g) is lower semi-continuous, and an upper bound for the jumps up is given. Furthermore the continuity of the maximal measure is shown, if a certain condition is satisfied. Then it is proved that the topological pressure is upper semi-continuous for every continuous function g : [0, 1] → R, if and only if 0 is not periodic or 1 is not periodic. Finally it is shown that the topological entropy is continuous, if htop(Tf ) > 0.