Consider the doubling map $x\mapsto 2x$ on $\mathbb R/\mathbb Z$. I'm interested in the collection of all right inverses of this map whose image is an interval. In other words, I want to know about "branches" of the halving multi-map $x\mapsto\tfrac x2$. More generally, for an integer $n\geq2$, I'm interested in branches of $x\mapsto\tfrac xn$.
To be precise, for each $\theta\in\mathbb R/\mathbb Z$, we can define $f_{n,\theta}:\mathbb R/\mathbb Z\to \mathbb R/\mathbb Z$ by:
- $f_{n,\theta}(x)$ is the unique $y\in\left[\theta,\theta+\tfrac1n\right)$ such that $ny=x$.
(Here, intervals like $\left[\theta,\theta+\tfrac1n\right)$ are understood in the circular order on $\mathbb R/\mathbb Z$. You know what I mean.)
I want to think of $f_{n,\theta}$ as a discrete dynamical system. Depending on where $\theta$ is, $f_{n,\theta}$ may have an attracting fixed point, or an attracting cycle. Does it ever have stranger behavior than that?
For example, if $n=2$, the following attracting cycles are possible for different ranges of $\theta$ values.
- $0$
- $\frac{1}{31}, \frac{16}{31}, \frac{8}{31}, \frac{4}{31}, \frac{2}{31}$
- $\frac{1}{15}, \frac{8}{15}, \frac{4}{15}, \frac{2}{15}$
- $\frac17,\frac47,\frac27$
- $\frac{5}{31}, \frac{18}{31}, \frac{9}{31}, \frac{20}{31}, \frac{10}{31}$
- $\frac13,\frac23$
...as well as the additive inverses of the above. This is not even close to an exhaustive list; there are infinitely many other possibilities.
However, not every (repelling) cycle of the doubling map can be reversed as an (attracting) cycle of a branch of the halving map! For example, there is no branch with cycle $\frac{3}{15},\frac{9}{15},\frac{12}{15},\frac{6}{15}$, because those numbers are too spread out to be contained in a single interval of length $\frac12$.
I don't want to reinvent the wheel, so my main question is: What is this family of functions called, and where can I read more about it?
To elaborate, I'd like to have answers to questions like: Given a fixed $n$ and $x_0$, as we vary $\theta$, what are all the possible behaviors of the orbit of $x_0$ under the different $f_{n,\theta}$ maps? Given a fixed $n$ and $x_0$, what is the order type of the set of equivalence classes of $\theta$, where each equivalence class is a cyclic interval of $\theta$ values whose $f_{n,\theta}$ maps all have symbolically equivalent orbits starting with $x_0$? Is this order type the same for all irrational $x_0$, and if so, what is that order type?
Here's my motivation: The answers to these questions will dictate the order type of the set of vertices of the convex hull of each meridional slice through a $p$-adic solenoid, in its "regular" embedding in three dimensions. I'm not trying to solve that larger problem on Stackexchange! But I do suspect that the behavior of these little circle maps might be a known topic, which I'd like to study.
I think I've found my way to the literature I wanted! The key phrases are ordered orbit and rotational set.
To be precise, it turns out that rotational sets, as defined in the below references, only coincide with branch attractors in the $n=2$ case. For $n\geq3$, they are defined in such a way that all branch attractors are rotational sets, but not all rotational sets are branch attractors. For example, as noted in Blokh et al., the set $\{\frac14,\frac34\}$ is a rotational set for $n=3$, i.e. the tripling map $x\mapsto3x$, but it is not contained in an interval of length $\frac13$.
So, although the literature on classifying rotational sets doesn't address all my questions head-on, it does contain enough results to answer some of the most important questions. In particular, regarding:
The answer is very much yes. For $n=2$ and $n=3$, there are known to be continuum-many minimal invariant sets of the doubling or tripling map that are Cantor sets contained within a closed interval of length $\frac12$ or $\frac13$, respectively. Given that, I believe that a branch of the halving or "thirding" multi-map whose range is that that interval (minus one endpoint) will have that same Cantor set (minus that endpoint) as a universal strange attractor. In fact, it's suggested that the same may apply for all $n$, although there is no particular theorem in the cited literature that guarantees that.
This means that my further questions on order types are partially answered, too, enough so that I see that the answer isn't what I had hoped for, so I'll need to steer my wider topic of research in a different direction. And for now, that's good enough!
Now here's the literature:
First, via the magic of Googling for some representative sequences, especially the cycle $\frac{5}{31}, \frac{18}{31}, \frac{9}{31}, \frac{20}{31}, \frac{10}{31}$, I found a decent reference for this question in the $n=2$ case. We start with Jenkinson, who studies a related question concerning vertices of the convex set of barycentres of measures on the circle $S^1\subset\mathbb C$ that are invariant under doubling:
The relevant material is also reprised in Oliver Jenkinson. "Frequency locking on the boundary of the barycentre set." Experiment. Math. 9 (2) 309 - 317, 2000. https://projecteuclid.org/journals/experimental-mathematics/volume-9/issue-2/Frequency-locking-on-the-boundary-of-the-barycentre-set/em/1045952354.full
For these results, Jenkinson cites a cluster of sources, principally Bullett & Sentenac:
Searching Google Scholar for works that cite Bullett & Sentenac, we finally get more recent sources that generalize $n\geq2$, notably:
I have only just barely skimmed Blokh et al, but that article looks like it might just answer all my questions!
From there, one can find other references that look promising. For example: