1.1 Background

Since the early work by Lagrange and Gauss linking regular continued fractions (CFs) to algebra and dynamical systems, an extensive and ongoing effort has focused on expanding the scope of CF theory to new algorithms. While regular CFs represent the fractional part $x-\lfloor x \rfloor $ of a real number $x\in \mathbb {R}$ as a descending iterated fraction:

(1.1) $$ \begin{align} \cfrac{1}{a_1+\cfrac{1}{a_2+\cdots}} \end{align} $$

with positive integer digits, a menagerie of one-dimensional CF variants have been formed by modifying various aspects of this simple construction: whether by changing positive quantities to negative, altering the set of allowable digits, or selecting a different set of numbers to have expansions. (See §1.2 for an introduction to many of these variants.)

After over 200 years of study, the one-dimensional CFs are largely well understood. Most of them inherit the essential properties of regular CFs from the viewpoints of algebra, dynamics, and geometry: Lagrange’s theorem, shift map ergodicity, and Diophantine interpretation, respectively. The study of one-dimensional CFs has been facilitated by a connection to hyperbolic geometry, pioneered by Artin [Reference Artin3] and developed by Series [Reference Series57], Katok and Ugarcovici [Reference Katok and Ugarcovici33–Reference Katok and Ugarcovici35], and others. In particular, Artin observed that the Gauss map for regular CFs can be identified with a section of geodesic flow in a finite cover of the modular surface; leading to extensive developments in both CF theory and the study of geodesics on hyperbolic manifolds.

In trying to extend these properties beyond one-dimensional CFs, one is immediately confronted by the question of how to generalize one-dimensional CFs to more than one dimension. Several algorithms, such as those of Jacobi–Perron, Brun, and Selmer, act by building up the CF expansion to several different real values simultaneously [Reference Schweiger55]. Other algorithms, such as the Hurwitz complex CF algorithm [Reference Hurwitz30] or the Heisenberg CF algorithm studied previously by the authors [Reference Lukyanenko and Vandehey38], treat points in these spaces as single entities with a single continued fraction expansion. (Yet another type of CF-like algorithm deriving more from geometric properties can be seen in [Reference Hayward24, Reference Hersonsky and Paulin27].) This is analogous to how complex points can be understood either via their real and complex part (that is, essentially in $\mathbb {R}^2$ ) or as an element in complex space (in $\mathbb {C}$ ). In this paper, we will generally be interested in the latter form of higher-dimensional CF expansion, as it has a more natural connection to hyperbolic geometry.

The story of these higher-dimensional CFs has been markedly different from their real CF cousins. Despite interest in these topics stretching back to the 1850s [Reference Hamilton22, Reference Hamilton23], only a small number of algorithms are known to be well behaved. Among them is the A. Hurwitz complex CF [Reference Hurwitz30], which represents a complex number z with real and imaginary parts both in $[-1/2,1/2)$ as a descending iterated fraction:

(1.2)

(See §1.2.6 for a full description.) Proofs of, for example, ergodicity for these well-behaved algorithms are extremely delicate [Reference Nakada45]: the space of the algorithm has a serendipitous decomposition, which results in a finite range property among other features, and this allows high-powered results (such as those in [Reference Ito and Yuri32, Reference Schweiger56]) to be applied. Should the algorithm be perturbed, even slightly (see §1.2.6 and Figure 2), the decomposition will break down and the methods will no longer apply.

As a major goal of this paper is to prove properties like ergodicity for a larger variety of higher-dimensional CFs (including perturbed variations of standard algorithms), let us discuss some of the roadblocks to using traditional techniques. First of all, one does not expect the structure of the cylinder sets (the sets of numbers whose expansions all start with the same sequence of digits) to have a simple structure, so methods like those cited above will not apply. Second, the natural extension of even some one-dimensional CF variants (see [Reference Arnoux and Schmidt2, §7]) as well as simple higher-dimensional CFs (see [Reference Ei, Ito, Nakada and Natsui18, Reference Hiary and Vandehey28]) is already fractal in nature, which makes it difficult to prove results about the natural extension, let alone about the simpler algorithm. Third, making a precise connection between CF digits and geodesic coding is ‘fraught with minor discrepancies’ and in its strong form would imply properties such as Serret’s tail equivalence theorem [Reference Panti50] that are known to fail for higher-dimensional CFs. Indeed, the geodesic coding approach has long been considered ‘intrinsically two-dimensional’ [Reference Adler and Flatto1].

In this paper, we develop a softer version of geodesic coding, which we refer to as geodesic marking. In a typical geodesic coding (what we describe here is an arithmetic coding, in the terminology of Katok and Ugarcovici [Reference Katok and Ugarcovici33], where codings formed by cutting sequences are related), we look at a given geodesic from two perspectives: first, we have a bi-infinite sequence formed by the continued fraction digits of both the forward and backward endpoints of the geodesic, and second, we have a bi-infinite sequence of intersections of our geodesic with a particular cross-section. A shift in one sequence should correspond to a shift in the other. In particular, returning to the cross-section after flowing along the geodesic should move the CF expansion forward one digit. Our geodesic marking still has the two bi-infinite sequences, but now returning to the cross-section can move the CF expansion forward several digits at a time. Thus the first-return map to the cross-section now corresponds to a jump transformation for the continued fraction. This jump transformation, in practice, skips over strings of small digits. (What counts as a small digit could be made effective, but we do not do so here.) It should be noted that small digits appear to cause some of the roadblocks mentioned above: cylinder sets associated with small digits tend to be irregular, while those of large digits are far better behaved, for instance. So, in essence, geodesic markings skip over the troublesome parts of CF algorithms.

Geodesic marking provides a robust connection between these higher-dimensional CFs and hyperbolic geometry which is preserved even under perturbation of the algorithm. The following theorem illustrates some significant cases to which our work applies.

In particular, this illustrates how our work applies to several different spaces (complex numbers, quaternionic numbers, octonionic numbers, and the Heisenberg group) and many systems within those spaces (folded and $\alpha $ -type variants are discussed in more detail in §1.2, see also Figure 2). Our results also apply to several one-dimensional CF algorithms, such as folded real CFs and some of Nakada’s $\alpha $ -CFs (see §§1.2.3 and 1.2.4).

While convergence follows standard arguments, the ergodicity statement is a substantial breakthrough for higher-dimensional CFs, where it was only previously known for specific complex CF variants, such as the A. Hurwitz and J. Hurwitz CFs. Our approach furthermore provides a flexible, unifying method for understanding both one-dimensional and higher-dimensional CFs.

Theorem 1.1 follows from a more general result concerning CFs on boundaries of rank-one symmetric spaces of non-compact type, which we refer to as Iwasawa inversion spaces. CFs were first extended to this setting by the authors in [Reference Lukyanenko and Vandehey38], where a CF theory on the non-commutative Heisenberg group was proposed. In [Reference Chousionis, Tyson and Urbański9], Chousionis–Tyson–Urbanski, studying conformal iterated function systems, defined Iwasawa continued fractions on the closely related Iwasawa groups (see §1.4). Here, we extend the definition of Iwasawa CFs to an Iwasawa CF algorithm associating a digit sequence to each point in an Iwasawa inversion space and leverage the connection to hyperbolic geometry to prove the following theorem.

We will postpone the full definitions of these terms until §2, but will provide some insight into them now. Discreteness simply says that the modular group $\mathcal M$ associated with our CF algorithm acts discretely on the corresponding hyperbolic space. It is necessary to ensure that we have a finite-volume hyperbolic manifold (generalizing the modular surface) in which to look at geodesic flow. Properness says that the only points under consideration for our CF algorithm have norm bounded away from 1. This guarantees that CF expansions converge quickly, among other properties. Properness also helps us avoid indifferent fixed points in our dynamical system, which have been noted before to cause infinite invariant measures [Reference Dajani, Hensley, Kraaikamp and Masarotto13]. Completeness says that the set of digits for our CF algorithm is maximal in an appropriate sense, the upshot of which is that the sequence of digits in the CF expansion of a point is functionally the only expansion the hyperbolic geometry can see.

In the case where a system is not complete, we can still obtain a partial result.

A full description of central symmetries will appear in §3.7. For the moment, we can consider centrally symmetric systems as ones where the system is incomplete due to the appearance of hidden symmetries, such as $x\mapsto -x$ , as in §1.2.4.

The use of geodesic marking, as opposed to classical geodesic coding, is critical to Theorem 1.2. As noted above, one typical corollary of geodesic coding is Serret’s tail-equivalence theorem for every point. That is, two points lie in the same orbit of the modular group $\mathcal M$ if and only if the tails of their CF digit sequences agree. However, for the A. Hurwitz complex CFs, Lakein [Reference Lakein36] provides an explicit counterexample to tail-equivalence. (Lakein’s counterexample makes use of an element not belonging to $\mathcal M$ , but this can be remedied by multiplying his choice of A by .) Thus, one would not expect for geodesic coding to be available in this case. Geodesic marking, however, avoids certain points which exhibit pathological behavior, those with all small CF digits. This allows for an ergodicity result and leads to the following almost everywhere (a.e.) tail equivalence result for Iwasawa CFs (proven as Theorem 6.17), which is novel for all higher-dimensional algorithms including folded A. Hurwitz complex CFs:

The question of tail-equivalence is being actively researched even for one-dimensional CFs, see [Reference Boca and Merriman5, Reference Panti50]. The importance of small digits versus large digits to tail-equivalence has been noted before in [Reference Nakada and Natsui48].

Figure 1

Different Iwasawa CF algorithms can be thought of as variations on the regular CF algorithm: the equations for the shift map, digit extraction algorithm, and recombination algorithm remain the same, while the underlying data are adjusted.

Figure 2

Four variants of the Hurwitz complex CF algorithm. The fundamental domain K in each case is displayed inside the unit circle (fixed by the inversion $\iota _c$ ), and is decomposed into rank- $1$ cylinder sets. The lattice $\mathcal Z=\mathbb {Z}^2$ is extended by the rotation $(x,y)\mapsto (-x,-y)$ in the folded variant.

1.2 Key examples of CF algorithms

We now describe a number of well-known variants of continued fractions, primarily in the one-dimensional case, which are of interest to us. We will discuss the algorithms in an increasing order of complexity (see Figure 1 for a diagram), pointing out the variations that motivate the definition of Iwasawa CFs: namely, the choice of underlying space, inversion, digit sequence, and fundamental domain for the corresponding lattice; as well as the definitions of properness, completeness, and discreteness. A more thorough discussion of the class of Iwasawa CFs is provided in §3, along with a more complete list of known Iwasawa CF algorithms in Table 1.

1.2.1 Regular CFs

The regular continued fraction representation of a number $x\in [0,1)$ represents it as a limit

$$ \begin{align*} x=\cfrac{1}{a_1+\cfrac{1}{a_2+\cdots}}, \end{align*} $$

where $a_i\in \mathbb {N}$ . (In the introduction, we ignore the behavior of points with finite CF expansion for simplicity.) The digits $a_i$ are extracted from x by repeated applications of the Gauss map $T(x)=1/x-\lfloor 1/x\rfloor $ :

$$ \begin{align*} a_i = \bigg\lfloor \frac{1}{T^{i-1}x} \bigg\rfloor. \end{align*} $$

The Gauss map is famously ergodic with an invariant measure given by the density $({1}/{\log 2})({1}/({1+x}))$ . (See [Reference Dajani and Kraaikamp14] for a fuller treatment.)

In the framework of Iwasawa CFs, regular CFs are described using the following data:

1. (1) the underlying space X is $\mathbb {R}$ ;

2. (2) the inversion used is $\iota (x)=1/x$ ;

3. (3) the allowed digits are elements of the lattice $\mathcal Z=\mathbb {Z}$ ;

4. (4) the set of ‘fractional points’ is $K=[0,1)$ , which tiles $\mathbb {R}$ under integer translations.

As we show below, many standard and novel algorithms can be described by adjusting the above data and leaving the formulas above essentially unchanged.

1.2.2 Backwards CFs

The backwards CF (sometimes called Rényi CF) reverses the domain of the Gauss map to produce the Rényi map

$$ \begin{align*} T_R(x)=T_G(1-x)=\frac{1}{1-x}-\bigg\lfloor \frac{1}{1-x}\bigg\rfloor. \end{align*} $$

The CF digits of $x\in [0,1)$ are then extracted in an analogous manner to the one used for standard CFs, via

(1.3) $$ \begin{align} a_i = \bigg\lfloor \frac{1}{1-T^{i-1}x} \bigg\rfloor, \end{align} $$

and recombined as

$$ \begin{align*} x=1-\cfrac{1}{a_1+1-\cfrac{1}{a_2+\cdots}}. \end{align*} $$

The shift map $T_R$ is ergodic, but due to the presence of indifferent fixed points, the corresponding invariant measure is infinite [Reference Adler and Flatto1].

With a small adjustment, backwards CFs fit into the framework of Iwasawa CFs, as follows.

The mapping $x\mapsto (1-x)$ conjugates backwards CFs to an equivalent system known as the D-backwards CF with $D=[0,1)$ , see Masarotto [Reference Masarotto41]. The resulting shift map is then given by $T_D(x)={-1}/{x}-\lfloor {-1}/{x}\rfloor $ . Adjusting Masorotto’s notation by using negative integer digits $a_i<-1$ , we take

(1.4) $$ \begin{align} a_i = \bigg\lfloor \frac{-1}{T^{i-1}x} \bigg\rfloor \end{align} $$

and recombine the digits as

$$ \begin{align*} x=\cfrac{-1}{a_1+\cfrac{-1}{a_2+\cdots}}. \end{align*} $$

All three CF algorithms discussed so far are real algorithms looking at points in $[0,1]$ which use integers for their digits. The only difference between them is the choice of inversion: $x\mapsto {1}/{x}$ for regular CFs, $x\mapsto {1}/({1-x})$ for backwards CFs, and $x\mapsto {-1}/{x}$ for D-backwards CFs.

In the Iwasawa CF formalism, we will assume that inversions send $0$ to $\infty $ and preserve the unit circle. While the backwards CF algorithm a priori does not fit this requirement, the conjugate D-backwards system is an Iwasawa CF.

We will make use of both of the allowed inversions $\iota _+(x)=1/x$ and $\iota _-(x)=-1/x$ throughout the paper.

Interestingly, backwards continued fractions are the more natural system within the framework of Iwasawa CFs. The inversion $\iota _-$ is an orientation-preserving linear-fractional mapping, and is an element of the modular group $PSL(2,\mathbb {Z})$ , while $\iota _+$ is orientation-reversing, which forces us to consider the larger group $PGL(2,\mathbb {Z})$ . This leads us to the question of completeness of the digit set, see §1.2.4.

1.2.4 Folded CFs

We now discuss in more detail the nearest-integer system based on the shift map $T(x)=\vert 1/x \vert -[\vert 1/x \vert ]$ . Because the mapping $x\mapsto \vert 1/x \vert $ is 2-to-1, the standard approach is to keep track both of the integer digit and the choice made when taking the absolute value

$$ \begin{align*} b_i = [\vert 1/x \vert], \quad c_i=\text{sign}(x), \end{align*} $$

so that one reconstructs

$$ \begin{align*} x = \cfrac{c_1}{b_1+\cfrac{c_2}{b_2+\cdots}}. \end{align*} $$

To maintain similarity to previous algorithms, we combine the integer digit $b_i$ and the sign $c_i$ into a single datum, namely the linear mapping $a_i(x)=c_i(x+b_i)$ . This allows us to rewrite the fraction in the format

$$ \begin{align*} x = \cfrac{1}{a_1\bigg(\cfrac{1}{a_2(\cdots)}\bigg)} = \lim_{n\rightarrow \infty}\iota_+a_1\iota_+a_2\ldots\iota_+a_n(0), \end{align*} $$

where each $a_i$ is now a function and we take the convention of suppressing parentheses and composition signs.

We thus transition from thinking of digits as elements of $\mathbb {Z}$ to thinking of them as automorphisms of $\mathbb {R}$ . In the Iwasawa CF framework, we will assume that these automorphisms are isometries of the underlying space, which is indeed the case here.

Therefore, for the algorithm under discussion, we are now interested in digits in the expanded lattice $\mathcal Z$ generated by integer translations and negation, that is, $\mathcal Z=\langle x\mapsto x+1, x\mapsto -x\rangle $ .

Inconveniently, moving the set of ‘fractional’ points $K=[-1/2, 1/2)$ around by the group $\mathcal Z$ causes overlaps, and we therefore exclude this CF variant from the class of Iwasawa CFs.

Adjusting to the interval under consideration to $K=[0,1/2]$ provides a non-overlapping tiling of $\mathbb {R}$ (that is, K is a fundamental domain for the action of $\mathcal Z$ ), giving the folded CF (see Marmi–Moussa–Yoccoz [Reference Marmi, Moussa and Yoccoz40]) that now does fit in the Iwasawa CF framework.

The folded CF algorithm is defined by the following data:

1. (1) the underlying space X is $\mathbb {R}$ ;

2. (2) the inversion used is $\iota _+(x)=1/x$ ;

3. (3) the group $\mathcal Z$ of allowed digits is generated by $x\mapsto x+1$ and $x\mapsto -x$ ;

4. (4) the set of ‘fractional points’ is $K=[0,1/2)$ , which tiles $\mathbb {R}$ under the action of $\mathcal Z$ .

Given these data, we obtain a rounding function $x\mapsto [x]\in \mathcal Z$ that now provides the unique linear mapping $[x]\in \mathcal Z$ combining an integer translation and possibly a negation such that $[x]^{-1}(x) \in [0,1/2)$ . For example, we have that $[5.1](x)=5+x$ and $[5.1]^{-1}(x) = x-5$ , while $[5.7](x)=-(x-6)$ and $[5.7]^{-1}(x) = -(x-6)$ .

For a point $x\in [0,1/2)$ , we can then write the forward shift map as $T(x)=[1/x]^{-1}(1/x)$ and extract the digits as $a_i = [1/T^{i-1}(x)]$ . The point x is reconstructed from the digits by writing

$$\begin{align*}x = \lim_{n\rightarrow \infty} \cfrac{1}{a_1\bigg(\cfrac{1}{a_2(\ldots a_n(0))}\bigg)},\end{align*}$$

or more compactly as $x=\lim _{n\rightarrow \infty } a_1 \iota a_2 \iota \ldots a_n(0)$ .

The absolute value mapping from $(-1/2,1/2)$ to $(0,1/2)$ then provides a semiconjugacy between the $\vert \cdot \vert $ -based nearest-integer fractions and the folded CFs. Ergodicity passes down (but not up!) through semiconjugacy, so folded CFs are ergodic. See Marmi–Moussa–Yoccoz [Reference Marmi, Moussa and Yoccoz40] for the corresponding invariant measure.

While Marmi–Moussa–Yoccoz do describe folded variants of all $\alpha $ -CFs, it is only the nearest-integer variant $\alpha =1/2$ that fits within the Iwasawa CF framework, since the other systems continue to operate with fractional sets K that are not fundamental domains for any relevant lattice.

As it turns out, the folded CFs also arise naturally from the regular CF construction, where we have $\mathcal Z=\mathbb {Z}$ and $\iota =\iota _+$ . Since the shift map combines both elements of $\mathcal Z$ and the mapping $\iota $ , analysis of the shift map revolves around understanding the group $\mathcal M = \langle \mathbb {Z}, \iota _+\rangle $ . The group $\mathcal M$ includes the negation mapping $x\mapsto -x$ since

(1.5) $$ \begin{align} \dfrac{1}{1+\dfrac{1}{-1+\dfrac{1}{1+x}}}=-x, \end{align} $$

so that the subgroup $\mathcal Z'\subset \mathcal M$ of linear transformations (that is, the stabilizer of $\infty $ ) is the group ${\mathcal Z}'=\langle x\mapsto x+1, x\mapsto -x\rangle $ . We thus have that the group of allowed digits $\mathcal Z=\mathbb {Z}$ is smaller than the natural group ${\mathcal Z}'$ of linear transformations, giving what we call an incomplete system. Expanding the set of digits to $\langle \mathbb {Z}, x\mapsto -x\rangle $ while also contracting the fundamental domain to $[0,1/2)$ provides a completion of the system, again giving us the folded fractions.

1.2.6 Complex CFs

We now briefly touch on higher-dimensional CFs, in the planar case. For more higher-dimensional CFs, including quaternionic and Heisenberg CFs, see the discussion in §3.

Our primary example is the A. Hurwitz complex CF, first defined in [Reference Hurwitz31]. It is described by the following Iwasawa CF data:

1. (1) the underlying space X is $\mathbb {C}$ ;

2. (2) the inversion used is $\iota (z)=1/z$ ;

3. (3) the group $\mathcal Z$ of allowed digits is the group of Gaussian integers, ;

4. (4) the specified fundamental domain K of $\mathcal Z$ is the unit square centered at the origin.

Thus, the shift map is given by $T(z)=1/z - [1/z]$ , where $[\cdot ]$ finds the nearest Gaussian integer; the digits are extracted via $a_i = [1/T^{i-1}z]$ , and reconstructed as

$$ \begin{align*} z=\cfrac{1}{a_1+\cfrac{1}{a_2+\cdots}} \end{align*} $$

It is common to write the system in real coordinates, with corresponding data:

1. (1) the underlying space X is $\mathbb {R}^2$ ;

2. (2) the inversion used is $\iota (x,y)={(x,-y)}/({x^2+y^2}),$ (we will denote such conjugate- reflections by $\iota _c$ );

3. (3) the group $\mathcal Z$ of allowed digits is the group $\mathbb {Z}^2$ ;

4. (4) the specified fundamental domain K of $\mathcal Z$ is the unit cube centered at the origin, i.e. $[-1/2,1/2)\times [-1/2,1/2)$ .

Both the real and complex descriptions of the Hurwitz CF are quite natural: the mappings $\iota $ and $\mathbb {Z}^2$ both lift to isometries of real hyperbolic 3-space, while the corresponding modular group $\mathcal M=\langle \mathcal Z, \iota \rangle $ is shown to be discrete by embedding into , see Proposition 3.15.

Ergodicity of the Hurtwitz CF was shown by Nakada in [Reference Nakada45] (cf, [Reference Hensley26]).

As in the case of $\iota _+$ real CFs, the system is not complete, since the stabilizer of $\infty $ in $\mathcal M$ contains the unexpected mapping $z\mapsto -z$ :

$$ \begin{align*}\dfrac{1}{1+\dfrac{1}{-1+\dfrac{1}{1+z}}}=-z.\end{align*} $$

As in the case of real folded fractions, we can create a folded variant by extending $\mathcal Z$ to include negation and reducing K correspondingly. For example, one could take $K=[-1/2,1/2)\times [0,1/2)$ . In general, we will call a CF algorithm a folded variant if $\mathcal Z$ is expanded to the stabilizer of $\infty $ in $\mathcal M$ , and K is similarly reduced.

One can likewise create $\alpha $ -type variants by shifting the location of the fundamental domain, that is, replacing K with $K+\alpha $ ; or create more exotic variants by choosing a different fundamental domain entirely, e.g. by choosing a tetromino to create the Tetris CFs. However, it is not the case that we can arbitrarily shift folded variants. For example, the set $[-1/2,1/2)\times [-1/4,1/4)$ is not a fundamental domain for the group .

See Figure 2 for illustrations of these algorithms and some of their cylinder sets. The finite range condition appears to fail in these cases. We recover ergodicity for folded variants. For centrally symmetric systems like the Tetris variant, we are able to bound the number of ergodic components by 2. The $\alpha $ -variant with $\alpha =0.3$ shown in the figure is not complete and not centrally symmetric with respect to $z\mapsto -z$ , and thus none of the results of this paper apply to it.

1.3 Theorem 1.2 in a special case

We now outline our proof of ergodicity in the case of nearest-integer CFs with inversion $\iota _-$ , where some simplifications are possible (cf. Remark 1.5). Ergodicity is certainly not new in the nearest-integer case, and connections to geodesic flow have been used since at least the work of Adler–Flatto [Reference Adler and Flatto1]. (From a historical perspective, using ergodicity of geodesic flow to prove the ergodicity of a CF alogrithm is backwards. The ergodicity of regular CFs was shown first, and the ergodicity of geodesic flow in the modular surface was proven using this [Reference Artin3, Reference Hedlund25].) For a more thorough treatment of these techniques in the regular CF case, we recommend [Reference Einsiedler and Ward19, §9.6].

We start by viewing $\mathbb {R}$ as the real axis in $\mathbb {C}$ , and interpret the upper half-plane as the hyperbolic plane $\mathbb {H}^2_{\mathbb {R}}$ . Both the integer shifts $\mathcal Z$ and the inversion $\iota $ on $\mathbb {R}$ extend to the half-plane, where they now act by isometries. The modular group $\mathcal M$ generated by $\mathcal Z$ and $\iota $ acts on $\mathbb {H}^2_{\mathbb {R}}$ discretely, and gives rise to a tiling of the space by translates of the tile $\mathcal T$ bounded by the vertical lines $x=\pm 1/2$ and the unit circle $\mathbb S$ . Notably, each of the lines $x=\pm 1/2$ are equal to $M\mathbb S$ for an appropriate $M\in \mathcal M$ . We will study hyperbolic geodesics $\gamma $ , which takes the form of either a vertical line or a semi-circle that intersects $\mathbb {R}$ at right angles.

We will derive ergodicity for the CF shift map from the ergodicity of the geodesic flow on the modular surface $\mathcal M \backslash \mathbb {H}^2_{\mathbb {R}}$ , which we can think of as the tile $\mathcal T$ with ‘opposite sides’ identified. That is, the sides $x=\pm 1/2$ are identified by the translation $z\mapsto z+1$ , and the two halves of the circular arc at the bottom are identified via $z\mapsto -1/z$ . By Mautner’s Theorem 6.3, geodesic flow in $\mathbb {H}^2_{\mathbb {R}}$ is ergodic. In particular, a generic geodesic $\gamma $ is dense in $\mathcal M \backslash \mathbb {H}^2_{\mathbb {R}}$ , see Figure 3.

Figure 3

Wall-crossings of the vertical geodesic $x=0.1795$ can be used to renormalize it to always stay within the fundamental domain for $\mathcal M$ .

It appears to be intuitively clear that, for a geodesic $\gamma \subset \mathbb {H}$ , the continued fraction expansion of the forward endpoint $\gamma _+\in \mathbb {R}$ can be immediately read off from the sequence of tiles that $\gamma $ traverses in $\mathbb {H}^2$ , or, equivalently, from the sequence of elements of $\mathcal M$ that are used to normalize it back to the starting tile. Indeed, it appears that the inversion corresponds to $\gamma $ crossing $\mathbb S$ and the digits count the number of vertical lines crossed before returning to $\mathbb S$ after an inversion. Our goal will be to formalize this relationship in sufficient detail to prove the ergodicity of the shift map T from the ergodicity of the geodesic flow on $\mathcal M\backslash \mathbb {H}^2_{\mathbb {R}}$ , doing so without relying on two-dimensional geometry, which has been central to previous approaches.

Let $\gamma $ be a vertical geodesic as in Figure 3. Let $a_1=[-1/\gamma _+]$ be the first nearest-integer CF digit of $\gamma _+$ and $M_1^{-1}(z)=-1/z-a_1$ the corresponding element of $PSL(2,\mathbb {Z})$ enacting the nearest-integer CF shift $T(\gamma _+)$ . Applying $M_1^{-1}$ to all of $\gamma $ , we obtain Figure 4a. We denote the natural elements of $PSL(2,\mathbb {Z})$ enacting $T^i$ by $M_i^{-1}$ .

Figure 4

Constructing the section $\mathcal {C}_{\mathbb {W}}$ of geodesic flow.

Consider now the subsegment $\gamma '$ of $\gamma $ strictly between the intersection with $\mathbb S$ and before the intersection with $M_1\mathbb S$ . The intersection of $\gamma $ with $M_1\mathbb S$ can be used to recover the first CF digit of $\gamma _+$ , since we have that $M_1^{-1}(z)=-1/z+a_1$ . However, intersections of $\gamma '$ with other $\mathcal M$ -translates of $\mathbb S$ do not correspond to digits of $\gamma _+$ . We wish to find a subset of $\mathbb S$ for $\gamma $ to intersect with that does not detect these ‘spurious’ intersections of $\gamma '$ seen in Figure 4a, but does continue to detect (most of) the crossings of $\gamma $ with $M_i\mathbb S$ . In this way, intersections of $\gamma $ with $\mathcal M\mathbb S$ correspond strongly to iterations of the shift map $T^i$ on $\gamma _+$ , and so the behavior of geodesic flow will strongly correlate to the behavior of the shift map T. We will do this in the unit tangent bundle of $\mathbb {H}^2_{\mathbb {R}}$ , by restricting the allowed unit vectors over $\mathbb S$ . The process is summarized in Figure 4.

We first quickly prove that $M_1^{-1}\gamma $ in fact crosses $\mathbb S$ . Indeed, we have that $M_1^{-1}\gamma _+$ is inside $\mathbb S$ , while $\vert a_i \vert \geq 2$ , so that $M_1^{-1}(\gamma _-)=M_1^{-1}(\infty )=-a_i$ is outside of $\mathbb S$ . While this bound appears to deteriorate to $\vert M_i^{-1}\gamma _- \vert \geq 1$ with additional iterations, by looking at the permissible digits, one shows that $\vert M_i^{-1}\gamma _- \vert $ is bounded below by the golden ratio $\phi $ . See Remark 1.5 for the more general approach to this step.

In Figure 4(a), we see that $\gamma '$ has (at least) two intersections that we do not want to code: the intersection with the sphere centered at the point $(1,0)$ and the intersection with the vertical line $x=1.5$ . The first of these is avoided simply by restricting to the vectors in $T^1\mathbb S$ that point towards $K=[-1/2,1/2)$ (that is, whose corresponding geodesics terminate in K). By completeness, any (non-identity) $\mathcal Z$ -translate of these vectors must land outside of K. In particular, the corresponding vectors on the sphere centered at $(1,0)$ point to the interval $[1/2,3/2)$ , whereas we know that $M_1^{-1}\gamma _+\in K$ . Thus this intersection is avoided.

The second spurious intersection in our example requires more work, and we rule it out by making two observations about $M_1^{-1}\gamma '$ that are predicated on the use of horoheight and horoballs (shown in green in Figure 4, cf. Ford circles). We may measure horoheight either from $\infty $ , in which case horoheight is simply the y coordinate and a horoball is a set of the form $\{(x,y):y>y_0\}$ , or from a (rational) point on the x-axis, in which case horoheight can be thought of as the depth into the corresponding cusp, and horoballs appear as Euclidean disks tangent to the x-axis. For our first observation, the fact that $\vert M^{-1}_1\gamma _+ \vert \le 1/2$ and $\vert M^{-1}_1 \gamma _- \vert \ge \phi $ implies that the intersection of $M^{-1}_1\gamma '$ with $\mathbb S$ occurs away from the x-axis, in the smaller ‘wall’ region (see Figure 4b):

$$ \begin{align*}\mathbb{W}=\big\{z\in \mathbb S \;:\; \text{Im}(z)>\sqrt{\tfrac{3}{2}(5\sqrt{5}-11)}\approx 0.52\big\},\end{align*} $$

and that the intersection with $M^{-1}_1\mathbb S$ is likewise bounded away from the x-axis. That is, $M_1^{-1}\gamma '$ is contained in a horoball $\mathcal B=\{y>\epsilon \}$ for some $\epsilon>0$ . For our second observation, consider a mapping $M\in \mathcal M$ that sends the line $x=1.5$ to $\mathbb S$ . Normalizing $M_1^{-1}\gamma '$ further by M, we see (Figure 4c) that $MM_1^{-1}\gamma ' \subset M \mathcal B$ is now contained in one of the horoballs based at a finite rational point. In particular, this provides (see Corollary 5.6) an upper bound $c=(\tfrac 32(5\sqrt {5}-11))^{-1/2}\approx 1.923$ on how far $MM_1^{-1}\gamma '$ travels away from the x-axis. We can reject the intersection of $MM_1^{-1}\gamma '$ with $\mathbb S$ (and thus the intersection of $M_1^{-1}\gamma '$ with the line $x=1.5$ ) by restricting to the vectors in $T^1\mathbb {W}$ that are returning from a cusp excursion towards $\infty $ of depth at least c. We will denote by $\mathcal {C}_{\mathbb {W}}$ the set of such vectors that also point towards K. This is our desired refinement of $\mathbb S$ (Figure 4d).

Our choice of $\mathcal {C}_{\mathbb {W}}$ by construction avoids all spurious intersections, but may also inadvertently ignore some intersections corresponding to small digits or even entire geodesics. We say that a geodesic $\gamma $ is markable if it intersects $\mathcal M$ -translates of $\mathcal {C}_{\mathbb {W}}$ infinitely often in both the past and future, and it is easy to see that almost every geodesic is markable, see Corollary 6.6. The markable geodesic Theorem 5.1 records the desired link between the CF digits of a markable geodesic $\gamma $ and its intersections with $\mathcal M$ -translates of $\mathcal {C}_{\mathbb {W}}$ : intersections occur only with walls of the form $M_i\mathcal {C}_{\mathbb {W}}$ , the intersections occur in the desired order, and no other intersections occur.

With the section $\mathcal {C}_{\mathbb {W}}$ in hand, we return to the question of ergodicity. We begin by working through a number of closely related functions acting on different spaces, pulling ergodicity from one function to the next. The ergodicity of geodesic flow on the modular surface $\mathcal M\backslash \mathbb {H}^2_{\mathbb {R}}$ implies the ergodicity of the first return map to the projection of $\mathcal {C}_{\mathbb {W}}$ onto the modular surface. We can then lift this first return map back to $\mathbb {H}^2_{\mathbb {R}}$ to obtain an isomorphic and thus equally ergodic map $\psi :\mathcal {C}_{\mathbb {W}}\to \mathcal {C}_{\mathbb {W}}$ (Proposition 6.9). We then conjugate this system with the projection $\pi $ from the unit tangent bundle of $\mathbb {H}^2_{\mathbb {R}}$ to $\hat {\mathbb {R}}\times \hat {\mathbb {R}}$ that takes any geodesic $\gamma $ to its forward and backward endpoints $(\gamma _+,\gamma _-)$ , obtaining an ergodic mapping $\Psi =\pi \circ \psi \circ \pi ^{-1}$ on $\pi (\mathcal {C}_{\mathbb {W}})$ . This map acts by taking a point $(\gamma _+, \gamma _-)$ to $(M_i^{-1} \gamma _+,M_i^{-1} \gamma _-)$ for some i. Thus, in the first coordinate, $\Psi $ acts by $T^i$ , where i may depend on the value of $\gamma _+$ , that is, this is a jump transformation associated with T.

Although we could conclude that this jump transformation is ergodic, the ergodicity of a jump transformation does not imply the ergodicity of the original map in general. So to recover the ergodicity of T, we step back to $\hat {\mathbb {R}}\times \hat {\mathbb {R}}$ . Namely, we consider a natural-extension-like function $\hat {T}$ on $K\times \hat {\mathbb {R}}$ such that the action of $\hat {T}$ on the first coordinate is simply T. We show that the action of $\hat {T}$ on $\overline {K}=\bigcup _{i=0}^{\infty } \hat {T}^i \pi (\mathcal {C}_{\mathbb {W}})$ is well behaved (Lemma 6.10) and that, in fact, $\Psi $ is simply the map induced by restricting $\hat {T}$ to $\pi (\mathcal {C}_{\mathbb {W}})$ (Lemma 6.11). Induced maps have far greater structure than jump transformations and so we are able to conclude the ergodicity of $\hat {T}$ on $\overline {K}$ from the ergodicity of $\Psi $ (Lemma 6.12) and from there conclude the ergodicity of T by restricting to the first coordinate.

1.4 Further remarks

Iwasawa CFs are the most general setting for our methods, which rely heavily on the fact that Iwasawa inversion spaces are boundaries of rank-one symmetric spaces of non-compact type. Indeed, Iwasawa inversion spaces are precisely the spaces with this property, with the exclusion, due to the break down of vector-space-based techniques, of the exceptional $\mathbb X^1_{\mathbb O}$ that can be defined over the non-associative octonions. Our notion of Iwasawa inversion space differs slightly from the notion of Iwasawa groups of [Reference Chousionis, Tyson and Urbański9], which excludes $\mathbb X^n_{\mathbb {R}}$ and allows $\mathbb X^1_{\mathbb O}$ .

We remark further that boundaries of rank-one symmetric spaces of non-compact type are arguably the most general setting for geometric CFs and Diophantine theory: they are characterized [Reference Cowling, Dooley, Korányi and Ricci12, Reference Le Donne37] as homogeneous geodesic locally compact spaces admitting both a dilation (a notion of fraction) and a well-behaved inversion. (The Cygan metric we work with is not itself geodesic, but gives rise to a geodesic path metric.)

The present work suggests the following further directions of study.

1.5 Outline of the paper

Following this introduction, in §2 we provide the general theory and definitions for Iwasawa inversion spaces. In §3, we define Iwasawa CFs, give further examples (including Table 1), and study conditions that guarantee discreteness, properness, and completeness. In §4, we quickly prove the convergence of Iwasawa CFs. In §5, we will build up the theory surrounding markable geodesics, culminating in the markable geodesic theorem. In §6, we use the markable geodesic theorem to prove the ergodicity of the CF shift map for an Iwasawa CF expansion and, in applications of this result, prove Theorems 1.3 and 1.4.