My friend Gugu and I have just uploaded to the arXiv our paper . This article continues our investigations of the Hausdorff dimension of the complement of the Lagrange spectrum in the Markov spectrum . More precisely, we showed in a previous paper (see also this blog post here) that and we prove now that .

The key *dynamical* idea to give upper bounds on is to show that any sufficiently large element is realized by a sequence whose past or future dynamics lies in the *gaps* of an appropriate horseshoe.

*Qualitately* speaking, this idea is explained by the following lemma.

Lemma 1Fix a horseshoe of a surface diffeomorphism and a height function. For simplicity, let us denote the orbits of by . Denote by

the corresponding Markov and Lagrange spectra.Let a subhorseshoe of and set

where is the Markov value of . Consider such that , and denote by a point with

Then, either or (where and denote the and limit sets of the orbit of ).

*Proof:* By contradiction, suppose that and .

Since and , we can select and such that for all , and . Also, the definitions allow us to take and such that and .

Fix with dense orbit and consider pieces of the orbit of with and .

Consider the pseudo-orbits . By the shadowing lemma, we obtain a sequence of periodic orbits accumulating whose Markov values converge to . In particular, , a contradiction.

In simple terms, this lemma says that an element with is associated to an orbit whose past dynamics (described by ) or future dynamics (described by ) avoids . Thus, there exists such that either the piece of past orbit or the piece of future orbit avoids a neighborhood of in (i.e., one of these pieces of orbit lives in the gaps of ).

Remark 1As it turns out, this qualitative lemma is not sufficient for our purposes and, for this reason, Gugu and I end up using a quantitative version of this lemma (called Lemma 3.1) in our paper.

Once we got some constraints on the dynamics of orbits generating elements of , our strategy to estimate consists in careful choices of and .

For the sake of exposition, let us explain how our strategy yields some bounds for .

Perron proved that any has the form where .

Consider the subhorseshoe (of sequences formed by concatenations of two consecutive 1’s and two consecutive 2’s).

By applying (a quantitative version of) Lemma 1 (cf. Remark 1), one concludes that if , then the past or future dynamics of lives in the *gaps* of .

This means that, up to replacing by , for all sufficiently large:

- either there is an
*unique*extension of giving a sequence whose Markov value in ; - or there are two continuations and of so that the interval is
*disjoint*from the Cantor setassociated to .

(Here, denotes continued fraction expansions.)

Note that this dichotomy imposes *severe* restrictions on the future of because there are not many ways to build sequences associated to . More precisely, we claim that at each sufficiently large step ,

- either we get a forced continuation ;
- or our possible continuations are and .

Indeed, suppose that we have two possible continuations and . If denotes the interval of numbers in whose continued fraction expansion starts by , then the intervals , appear in the following order on the real line:

Thus:

- the continuation is not possible (otherwise would contain and,
*a fortiori*, intersect ; - the continuation is not possible (otherwise would contain and,
*a fortiori*, intersect

so that our continuations are and . Now, we observe that the intervals and , , appear in the following order on the real line:

Hence:

- the continuation is not possible (otherwise would contain and,
*a fortiori*, intersect ; - the continuation is not possible (otherwise would contain and,
*a fortiori*, intersect

so that our claim is proved.

This claim allows us to bound the Hausdorff dimension of

In fact, the claim says that we refine the natural cover of by the intervals by replacing it by a “forced” or by the couple of intervals

Therefore, it follows from the definition of Hausdorff dimension that

Because we can assume that with and , our discussion so far can be summarized by following proposition:

Proposition 2is contained in the arithmetic sum

where for any parameter satisfying (1).

Since the arithmetic sum is the projection , , of the product set , this proposition implies the following result:

Corollary 3where satisfies (1).

The Hausdorff dimension of was computed with high accuracy by Hensley among other authors: one has . In particular,

where verifies (1).

Closing this post, let us show that (1) holds for and, consequently,

For this sake, recall that

where is the denominator of .

Hence, if we set

then the recurrence formula implies that

where .

Because and for all , we have

This completes the argument because .

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