Suppose $\eta$ is a continuous function with a continuous inverse on the circle $S^1$ such that it behaves as a conjugacy between two maps $T_1:S^1 \to S^1$ and $T_2: S^1 \to S^1$. That is, $\eta \circ T_1 \circ \eta^{-1} = T_2$.
A conjecture is if for a set $A$, $\lambda(\eta(A)) = 1$, then, $(\eta)_*\lambda = \lambda$, where $\lambda$ is the Lebesgue measure. I can see that this is not true for a generic continuous function $\eta$ because the Lebesgue measure of any subset is not the same as the Lebesgue measure of the range of the subset under $\eta$, in general. I am wondering if there is something special about conjugacies which might make this conjecture true.
Thank you very much for your time!
EDIT: Here is some context for this question. I am interested in the Lebesgue measure of a certain nice set. The nice set $N$ is the set of all points in $S^1$ for which Birkhoff averages under $T_1$ converge to expected values wrt to a certain measure (that happens to be Lebesgue) that is invariant under both $T_1$ and $T_2$ and is the SRB measure of $T_2$. What I really want is to find necessary and sufficient conditions for the Lebesgue measure of $N$ to be 1. Now if the conjugacy did not preserve the Lebesgue measure, then, I know that my nice set automatically has measure 0 (since it can only have measure 0 or 1). On the other hand, if $\lambda(N)= 1$, does it imply that $\eta$ preserves $\lambda$ ?(this was the conjecture in the question).