This problem isn't legible.
Hardly legible. Seems related to a result of M. Herman (in ?) about ergodicity in families $f_b = f +b$, for $f$ homeomorphism of the circle.
Herman or Henon?
Seems that he is asking if $f_b$ is ergodic with respect to Lebesgue measure, in the sense that $f_b(A) = A$ implies that $A$ has measure 0 or 1.
So is the question: "Let $f$ be a homeomorphism of the circle (maybe a specific two-slope example). For each $b$, define $f_b(x)=f(x)+b$. Say that $f_b$ is ergodic with respect to Lebesgue if $f_b(A)=A$ implies $A$ has measure 0 or 1 (no assumption of invariance of Lebesgue measure here). For a fixed $f$, what can be said about the set of $b$ such that $f_b$ is ergodic?"