My question concerns the Bernoulli map (a.k.a. the shift map and as dyadic transformation).
I understand that most numbers $x \in [0,1)$ do not have a periodic orbit under this transformation, and thus, they densely cover the lines $y=2x$ (for $0 \le x<1/2$) and $y=2x-1$ (for $1/2 \le x<1$). I also understand that a uniform measure on $x$ is invariant under this transformation. Yet, my interest is the density of points on the lines. I would like to know whether the density of points on these lines, if defined at all, is uniform.
I also read the answer to the third part of this question, but couldn't deduce any statement about uniformity of the points on these lines. I'm trained as a physicist, and consequently, lack strong mathematical knowledge. So I would be extra grateful if you consider these facts while writing your response. Thank you.