Relation between Expanding Circle Map $x\mapsto Nx \ \operatorname{mod} \ 1$ and the Metric $d(x,y) = \inf_{k\in\mathbb{Z}}|x-y+k|$ on $[0,1[$.

Question

This question is related to a question I asked yesterday, but formulated more directly.

If we endow $[0,1[$ with the metric $$d(x,y) := \inf_{k\in\mathbb{Z}} |x-y+k|,$$ then $[0,1[$ is homeomorphic to $S^{1}$. For a fixed $N\in\mathbb{N}$ we consider the expanding map $$f\colon [0,1[\rightarrow [0,1[, \qquad f(x):= Nx - \lfloor Nx\rfloor = Nx \mod 1.$$

Let $n$ a positive integer and assume that $d(x,y)\leq 1/2N^{n}$. I have the following three related questions:

1. Can we conclude that $$d(f^{n}(x),f^{n}(y)) = N^{n}d(x,y)?$$
2. Does the distance between the first $n$ iterations increase, i.e. $$d(f^{n}(x),f^{n}(y)) \geq d(f^{n-1}(x),f^{n-1}(y)) \geq \ldots \geq d(f(x),f(y)) \geq d(x,y)?$$ If 1. is true for all $0\leqslant j\leqslant n$, then this follows easily.
3. Can we also conclude that $$d(x,y) = |x-y|?$$

I know how to visualize these properties on the circle, but I don't know how to prove this rigorously. Any help would be appreciated!

Answer 1

Yes, but $k$ is relative number so the difference of floor is relative so taking the min on just $N^n(x-y) $ is enough.