I cite this book, mostly from $37$ onwards until the exercises - this is exercise $12$, page $40$. It is left as an exercise to show that the leading digits of the powers of two contain all digits infinitely often, and I fully intend to finish this exercise - I just am at a loss as to how to get started. The issue is that the problem must be phrased in terms of topological dynamics.
The contents of the book so far, chapters 1-3, covers topological dynamical systems - compact Hausdorff spaces $X$ endowed with a continuous map $\phi:X\to X$, and whether or not they contain recurrent points. By "recurrent", it is meant that $x$ is recurrent when the set $\phi(x),\phi^2(x),\cdots,\phi^n(x),\cdots$ is dense in $X$, and uniformly recurrent whenever the set $\{m\in\Bbb N_1:\phi^m(x)\in U\}$ is syndetic (of bounded gaps) for all open neighbourhoods $U$ of $x$.
The authors, in the example questions, consider some simple systems such as $[0,1)$ with addition modulo $1$, or rotational groups, and a common theme is to extend a system:
The extension of $(X;\phi)$ by a compact space $Y$, along a continuous function $\lambda:X\times Y\to Y$, is the product topological space $X\times Y$ endowed with the dynamic $\psi(x,y)=(\phi(x),\lambda(x,y))$
The group extension of $(X;\phi)$ by a compact group $(G,\circ)$, along a continuous function $\lambda:X\to G$, is the product topological space $X\times G$ endowed with the dynamic $\psi(x,g)=(\phi(x),\lambda(x)\circ g)$.
Arguments in previous examples, such as Diophantine approximation, have broadly followed the lines of:
Consider the dynamical system $K$, show that a certain point $x$ is recurrent. Then extend (possibly multiple times) by another space or group along a convenient function $\lambda$ such that $\psi^n$ inductively follows a nice pattern, (such as $2^n$) and then use theorem from book such as: points recurrent in a system remain recurrent in the extension of that system, and every point of a rotational compact group is uniformly recurrent, to obtain that $x$ is recurrent in the extended system, and therefore: "there exists $n$ such that $\psi^n$ - our nice inductive expression, such as $2^n$, is very close to the desired point for any $\epsilon$", etc.
In this case, I seek to show that the point $m,\,m\in\{1,2,\cdots,9\}$, is uniformly recurrent in some space with a dynamic that involves the leading digits of the powers of two.
How I can actually make the link to a dynamical system, possibly after extension, is beyond me. I suppose $G=\{2^n:n\in\Bbb Z\}$, $a\circ b=ab$ forms a group, but how do I extract the leading digit? Moreover, how I make this topological? - The space must be compact and Hausdorff...
If anyone can point me to a suitable system, or even better, make a meaningful hint so that I can understand how to come up with these things myself, I'd appreciate it. I admit I am punching above my weight with this text, not very fluent in topology, group theory as I am, but I want to keep going as far as possible as it's quite interesting.
