Let $X$ be a compact metric space. We define $\phi_t: X \rightarrow X$ such that $\phi_o = id, \phi_t \circ \phi_s = \phi_{s+t}$. Also, $\phi_t$ is a bijection with inverse $\phi_{-t}$. Also, each $\phi_t$ is a Holder continuous function.
Let $f: X \rightarrow \mathbb R_{>0}$ be a Holder continuous function. We define the function $\kappa_f: X \times \mathbb R \rightarrow \mathbb R$ given by$$\kappa_r(x,t):= \int_{0}^tf(\phi_s(x)) ds.$$Prove that the function $\kappa_f(x, \cdot): \mathbb R \rightarrow \mathbb R$ is an increasing homeomorphism for all $x \in X.$
My attempt: Since $f$ is Holder continuous, and since $X$ is compact, $f$ has a positive minimum and for every $x \in X$. Now I cannot proceed further. Can anyone please help me?
