How to define metric d' in the following example?

Question

In Peter Walters'- An Introduction to Ergodic Theory, I'm fine with rest of all, but unable to define metric d'. How this metric should be defined ? Any idea or approach?

Answer 1

What the author seems to be getting at, is essentially that you can define a piecewise linear homeomorphism $h: (0, \infty) \to \mathbb{R}$ such that $h(2x) = h(x) + 1$ and then take $d'(x, y) = |h(x) - h(y)|$. That looks possible, but overcomplicated.

Instead, I would suggest defining $d''(x,y) = \left|\log x - \log y\right|$. This metric is topologically equivalent to $d$ because $\log: (0,\infty) \to \mathbb{R}$ is a homeomorphism, but it is not uniformly equivalent because $\log$ is not uniformly continuous. It is easy to see that for any $a > 0$ the mapping $T_a: x \mapsto ax$ is an isometry with respect to $d''$, since $$ d''(ax, ay) = \left|(\log a + \log x) - (\log a + \log y)\right| = \left|\log x - \log y\right| = d''(x, y). $$