Simplicity of spectra for certain multidimensional continued fraction algorithms

Abstract

We introduce a new strategy to prove simplicity of the spectrum of Lyapunov exponents that can be applied to a wide class of Markovian multidimensional continued fraction algorithms. As an application we use it for Selmer algorithm in dimension 2 and for the Triangle sequence algorithm and show that these algorithms are not optimal.

Appendix A. Proof of Lemma 1

1.1 A.1 Strategy

In this appendix we give a proof of the ergodicity of Cassaigne algorithm with classical techniques.

First, we recall the definition of ergodicity for multidimensional fraction algorithm given by Schweiger [27] (Definition 21 at p 21).

Definition 7

T is called ergodic if $T^{-1}(E)$ = $E \text { mod } 0$ implies $\lambda (E) = 0$ or $\lambda (B/E) = 0$ for the Lebesgue measure $\lambda$.

Remark 4

One can easily see that this definition is equivalent to the standard definition of ergodicity when the invariant measure is equivalent to the Lebesgue measure.

We follow the strategy suggested by Schweiger in [27] for Selmer algorithm (Theorem 23, p 60):

• (1) describe the associated Markov partition and symbolic coding for the cylinders;

• (2) define the absorbing and so called bad sets (in a sense of distortion properties) and check that the bad set is of measure zero;

• (3) define Schweiger jump transformation as a natural acceleration of the algorithm that takes each point of the simplex except those in the bad set to the absorbing set;

• (4) prove that Schweiger jump transformation has weak bounded distortion property and therefore is ergodic.

• (5) conclude that the original algorithm was ergodic as well.

1.2 A.2 Markov partition

We define a Markov partition for the Cassaigne algorithm in the following way. The alphabet for the coding contains two letters (lets denote them by 1 and 2). After one step of the algorithm we have the following partition of the parameter space $\Delta = (x_1, x_2, x_3: \sum _{i}x_i = 1)$,

$$\begin{aligned} B_1=\{(x_1,x_2,x_3)\in \Delta \mid x_1>x_3\} \end{aligned}$$

and

$$\begin{aligned} B_2=\{(x_1,x_2,x_3)\in \Delta \mid x_1<x_3\}. \end{aligned}$$

A direct calculation reveals that $f = \frac{F_C}{||F_C||}$ maps $B_i$ onto the whole parameter space and that vertices of each simplex are mapped into the vertices. For example, for $(x_1, x_3) \in B_1$, $f(x_1,x_3) = (\frac{x_1-x_3}{1-x_3}, \frac{1-x_1 - x_3}{1-x_3})$.

Iterating the algorithm, we get the Markov partition $\alpha$. Symbolic dynamics is given by a Markov shift defined on the cylinders described by 2 letters.

1.3 A.3 Absorbing set versus bad set

Absorbing set is formed by the cylinders that contain words where both letters are included:

$$\begin{aligned} D = \cup _k (1^{k} 2 \cdots )\cup \cup _k(2^{k} 1\cdots ). \end{aligned}$$

Here $1^{k}$ means a sequence of k times 1.

The bad set is formed by infinite cylinders without change of letters: $H = (1\cdots 1 \cdots )\cup (2\cdots 2 \cdots ).$ Naturally, $H\cup D$ gives the whole parameter space. And the following holds:

Lemma 16

The measure of the set H is 0, the set D is of full measure.

1.4 A.4 A Schweiger jump transformation

For any cylinder B (except those that are in a bad set H) we can define the arrival time to the absorbing set D and denote it by $N_B$. For Cassaigne algorithm, we arrive to the absorbing set once we change letter. For example, for any cylinder $B_n = (1 \cdots 12)$ or $B'_n = (2\cdots 21)$ this time is equal to the number of 1s (respectively, 2s). It allows us to define an accelerated version of the algorithm (an analogue of Zorich acceleration for the Rauzy induction) that contains all the consequent stages of the original algorithm coded by the same letter and the first stage after the change. We denote this accelerated algorithm by $\hat{F}_C: \hat{F}_C = F^{N_B}_C.$

1.5 A.5 Distortion properties

We recall some definitions from [1] (see Chapter 4).

Definition 8

For a given Markov map T acting on a Polish space X with Markov partition $\alpha$ and measure m the distortion at $(x,y)\in Dv_a\times DV_a$ is the ratio

$$\begin{aligned} {\mathcal {G}}(T, x,y) = \frac{v'_a(x)}{v'_a(y)} \end{aligned}$$

where $v'_a$ is the Radon–Nikodym derivative of the non-singular inverse branches $v_a$ of $T^n$ with respect to Lebesgue measure.

Strong bounded distortion property means that there exists a constant $C>1$ such that, for all Markov cells a of positive measure, ${\mathcal {G}}(T, x,y)\le C$ for almost every $(x,y)\in Dv_a\times DV_a$.

Schweiger collection is a special subset of the partition $\tau \subset \alpha$ such that the strong bounded distortion property with some $C>1$ holds for all cylinders in $\tau$ and for any two cylinders of positive measure $[b]\in \tau$ and $[a]\in \alpha$ $[a,b]\in \tau$ and $\cup _{B\in \tau }B = X.$

Weak bounded distortion property means that for a given Markov map there exists a Schweiger collection.

Lemma 17

Schweiger jump transformation of $F_C$ has weak bounded distortion property.

Proof

It is enough to check that the statement holds for almost all points in the absorbing set. The proof is based on a direct calculation and is organized as follows. First we check that the condition holds for one step of our accelerated algorithm; then we use it to prove that a stronger condition holds and this condition implies the statement for all the iterations of the accelerated algorithm.

Lets consider the cylinder $B_n = (1\ldots 12)$ (the opposite case of $C_n = (2\ldots 2 1)$ is similar). The index n refers to the number of 1 in the cylinder coding. The projective version of $F_C$ is one of the two maps: either if $n = 2k$,

$$\begin{aligned} {\hat{F}}_C(x_1, x_3) = \left( \frac{1 - x_1 - x_3}{1-x_1}, \frac{k+x_3 - (k+1)x_1}{1-x_1}\right) , \end{aligned}$$

either if $n = 2k+1$,

$$\begin{aligned} {\hat{F}}_C(x_1, x_3) = \left( \frac{x_3}{1-x_1}, \frac{1+k-(k+2)x_1}{1-x_1}\right) . \end{aligned}$$

The Jacobian of the map in both cases is $J = \frac{1}{(1-x_1)^3}.$

We treat here only the situation $n = 2k$ (the other one is completely similar). In this case the cylinder identifies the triangle in the parameter space with the following vertices:

$$\begin{aligned} \left( \frac{k}{k+1}, \frac{1}{k+1} \right) ; \quad \left( \frac{k}{k+1}, 0 \right) ; \quad \left( \frac{k+1}{k+2}, \frac{1}{k+2} \right) . \end{aligned}$$

Hence we know that $x_1\in [k/k+1, (k+1)/(k+2)].$

Therefore, the ratio of two Jacobian’s taken at points a and b in the same cylinder satisfies $\frac{(k+1)^3}{(k+2)^3}< \frac{J(a)}{J(b)}<\frac{(k+2)^3}{(k+1)^3}$. It follows that this ratio is uniformly bounded (e.g. by $\frac{1}{8}$ and 8).

Now we need to check that the ration of two Jacobians remains uniformly bounded after any number of steps of the accelerated algorithm. In order to do it, we introduce another bounded distortion property.

Definition 9

We say that the map $T:X \rightarrow X$ satisfies Renyi condition if two following properties hold:

• for two points a, b from the same element of the partition we have

  $$\begin{aligned} \left| \frac{J_T(a)}{J_T(b)} - 1 \right| \le C \Vert T(a) - T(b)\Vert \end{aligned}$$

  (1)

  with some uniform constant C ($J_T$ is the Jacobian map of T);

• $|J_{T^n}|\ge K\lambda ^n$ for some uniform constants $K>0$ and $\lambda >1$.

This condition is a classical version of the bounded distortion property widely used in the study of ergodic properties of different interval maps (see, for example, [12]). It is easy to check that if T satisfies Renyi condition, then any iteration $T^{k}$ of the map T also satisfies the above conditions with some uniform constant $C'$ (that is the same for all k) (see Theorem 1.2 in Chapter 3 in [22]). This fact implies, in particular, that the ratio of Jacobians of the iterated map at any two points from the same elements in the corresponding partition is uniformly bounded. Therefore, if one checks that our jump transformation (a.k.a. acceleration of Cassaigne algorithm) $F_C$ satisfies Renyi condition for the partition given by $B_n$ and $C_n$ for the first step of the algorithm, it is enough to conclude that the union of $B_n$ and $C_n$ gives us a Schweiger collection. And thus the map $F_C$ satisfies weak distortion property.

Now it remains to check that Renyi condition is satisfied for $F_C$. We consider the case of $B_n$ (the other case is very similar).

As we explained above, for any $a = (a_1, a_3)$ and $b = (b_1,b_3)$ from the same cell of the partition $|\frac{1-a_1}{1-b_1}|$ is bounded by some uniform constants from above and from below (we took 1/8 and 8). So

$$\begin{aligned} \left| \frac{J(b)}{J(a)} -1 \right|&= \left| \left( \frac{1-a_1}{1-b_1}\right) ^2 + \frac{1-a_1}{1-b_1} + 1 \right| \cdot \left| \frac{1-a_1}{1-b_1}-1 \right| \\&\le (2^2 + 2 + 1) \left| \frac{1-a_1}{1-b_1}-1 \right| . \end{aligned}$$

Now we note that

$$\begin{aligned} {\hat{F}}_{C}(a) - {\hat{F}}_{C}(b) = (A,B), \end{aligned}$$

where $A = \frac{b_3}{1-b_1} - \frac{a_3}{1-a_1}$ and $B = \frac{1-b_3}{1-b_1} - \frac{1-a_3}{1-a_1}$. Since $A^2 + B^2 \ge \frac{(A+B)^2}{2},$ we conclude that

$$\begin{aligned} \Vert {\hat{F}}_{C}(a) - {\hat{F}}_{C}(b)\Vert \ge \frac{1}{\sqrt{2} } \cdot \frac{\left| b_1 - a_1 \right| }{(1-a_1)(1-b_1)} \ge \frac{1}{\sqrt{2} } \cdot \frac{\left| b_1 - a_1 \right| }{1-b_1} = \frac{1}{\sqrt{2}} \cdot \left| \frac{1-a_1}{1-b_1} - 1 \right| . \end{aligned}$$

Therefore the condition (1) is satisfied with constant $C = 7 \sqrt{2}$.

Also $J_T = \frac{1}{(1-a_1)^3}\ge 8$ because as we know $a_1 \ge \left( \frac{k}{k+1}\right) .$ It implies the second part of Renyi condition. $\square$

To conclude we use the following famous result on ergodicity of Markov maps whose proof can be found in [1] (Theorem 4.6.3).

Theorem 18

(Aaronson–Denker–Urbanski) A topologically transitive Markov map with weak bounded distortion property is either conservative and ergodic or totally dissipative.

Since the Cassaigne algorithm (and therefore its acceleration) can not be totally dissipative (the density of its invariant measure was calculated in [3]), Schweiger jump transformation of $F_C$ is ergodic.

Remark 5

The same conclusion can be obtained from the so called Adler Folklore theorem (see discussion after Theorem 1.2 in [22] where the argument can be done for $\gamma = 1$) using the Renyi property we established above.

1.6 A.6 Ergodicity of the original algorithm

The following theorem proved by Schweiger (see [27], Theorem 11, p 19) now implies ergodicity of the normalized Cassaigne algorithm:

Theorem 19

If Schweiger jump transformation of the Markovian algorithm is ergodic, then the original algorithm is also ergodic.