Posted by: matheuscmss | August 22, 2017

HD(M\L) < 0.986927

My friend Gugu and I have just uploaded to the arXiv our paper {HD(M\setminus L) < 0.986927}. This article continues our investigations of the Hausdorff dimension {HD(M\setminus L)} of the complement of the Lagrange spectrum {L} in the Markov spectrum {M}. More precisely, we showed in a previous paper (see also this blog post here) that {HD(M\setminus L) > 0} and we prove now that {HD(M\setminus L)<1}.

The key dynamical idea to give upper bounds on {HD(M\setminus L)} is to show that any sufficiently large element {m\in M\setminus L} is realized by a sequence {\underline{\theta}\in(\mathbb{N}^*)^{\mathbb{Z}}} whose past or future dynamics lies in the gaps of an appropriate horseshoe.

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

Lemma 1 Fix {\Lambda}horseshoe of a surface diffeomorphism {\varphi} and {f} a height function. For simplicity, let us denote the orbits of {\varphi} by {x_n:=\varphi^n(x)}. Denote by

\displaystyle M=\{\sup\limits_{n\in\mathbb{Z}}f(x_n): x_0\in\Lambda\} \quad \textrm{ and } \quad L=\{\limsup\limits_{n\rightarrow\infty}f(x_n): x_0\in\Lambda\}

the corresponding Markov and Lagrange spectra.Let {\widetilde{\Lambda}} a subhorseshoe of {\Lambda} and set

\displaystyle m(\widetilde{\Lambda}) = \max\limits_{y\in\widetilde{\Lambda}} m(y) \quad (= \max\limits_{z\in\widetilde{\Lambda}} f(z) )

where {m(a):=\sup\limits_{n\in\mathbb{Z}}f(a_n)} is the Markov value of {a}. Consider {m\in M\setminus L} such that {m>m(\widetilde{\Lambda})}, and denote by {x_0\in\Lambda} a point with

\displaystyle m=\sup\limits_{n\in\mathbb{Z}} f(x_n) = f(x_0)

Then, either {\alpha(x_0)\cap \overline{\Lambda}=\emptyset} or {\omega(x_0)\cap \overline{\Lambda}=\emptyset} (where {\alpha(x)} and {\omega(x)} denote the {\alpha} and {\omega} limit sets of the orbit of {x}).

Proof: By contradiction, suppose that {z\in\alpha(x)\cap\widetilde{\Lambda}} and {w\in\omega(x)\cap\widetilde{\Lambda}}.

Since {m\in M\setminus L} and {m>m(\widetilde{\Lambda})}, we can select {\varepsilon>0} and {N\in\mathbb{N}} such that {f(x_n)<m-\varepsilon} for all {|n|\geq N}, and {m(\widetilde{\Lambda})<m-\varepsilon}. Also, the definitions allow us to take {m_k\rightarrow-\infty} and {n_k\rightarrow\infty} such that {x_{m_k}\rightarrow z} and {x_{n_k}\rightarrow w}.

Fix {y\in\widetilde{\Lambda}} with dense orbit and consider pieces {y_{r_k}\dots y_{s_k}} of the orbit of {y} with {y_{r_k}\rightarrow w} and {y_{s_k}\rightarrow z}.

Consider the pseudo-orbits {x_0\dots x_{n_k} y_{r_k}\dots y_{s_k}x_{m_k}\dots x_0}. By the shadowing lemma, we obtain a sequence of periodic orbits accumulating {x_0} whose Markov values converge to {m}. In particular, {m\in L}, a contradiction. \Box

In simple terms, this lemma says that an element {m\in M\setminus L} with {m>m(\widetilde{\Lambda})} is associated to an orbit {(x_n)_{n\in\mathbb{Z}}} whose past dynamics (described by {\alpha(x_0)}) or future dynamics (described by {\omega(x_0)}) avoids {\widetilde{\Lambda}}. Thus, there exists {k\in\mathbb{N}} such that either the piece {(x_n)_{n\leq -k}} of past orbit or the piece {(x_n)_{n\geq k}} of future orbit avoids a neighborhood of {\widetilde{\Lambda}} in {\Lambda} (i.e., one of these pieces of orbit lives in the gaps of {\Lambda\setminus\widetilde{\Lambda}}).

Remark 1 As 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 {M\setminus L}, our strategy to estimate {HD(M\setminus L)} consists in careful choices of {\widetilde{\Lambda}} and {\Lambda}.

For the sake of exposition, let us explain how our strategy yields some bounds for {HD((M\setminus L)\cap [3.06, \sqrt{12}])}.

Perron proved that any {m\in M\cap(-\infty, \sqrt{12}]} has the form {m=m(\underline{\theta})} where {\underline{\theta}\in\{1,2\}^{\mathbb{Z}}=\Lambda}.

Consider the subhorseshoe {\widetilde{\Lambda} = \{11,22\}^{\mathbb{Z}}} (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 {m=m(\underline{\theta})\in (M\setminus L)\cap [3.06, \sqrt{12}]}, then the past or future dynamics of {\underline{\theta}} lives in the gaps of {\widetilde{\Lambda}}.

This means that, up to replacing {\underline{\theta}=(\theta_n)_{n\in\mathbb{Z}}} by {(\theta_{-n})_{n\in\mathbb{Z}}}, for all {n\in\mathbb{N}} sufficiently large:

  • either there is an unique extension of {\dots\theta_0\dots\theta_n} giving a sequence whose Markov value in {(M\setminus L)\cap [3.06,\sqrt{12}]};
  • or there are two continuations {\dots\theta_0\dots\theta_n1\alpha_{n+2}} and {\dots\theta_0\dots\theta_n2\beta_{n+2}} of {\dots\theta_0\dots\theta_n} so that the interval {[[0;2\beta_{n+1}], [0;1\alpha_{n+1}]]} is disjoint from the Cantor set

    \displaystyle K(\{11,22\}):=\{[0;\gamma]:\gamma\in\{11, 22\}^{\mathbb{N}}\}

    associated to {\widetilde{\Lambda} = \{11,22\}^{\mathbb{Z}}}.

(Here, {[a_0;a_1,\dots] = a_0+\frac{1}{a_1+\frac{1}{\ddots}}} denotes continued fraction expansions.)

Note that this dichotomy imposes severe restrictions on the future {(\theta_n)_{n\geq 0}} of {\underline{\theta}} because there are not many ways to build sequences associated to {(M\setminus L)\cap[3.06,\sqrt{12}]}. More precisely, we claim that at each sufficiently large step {n},

  • either we get a forced continuation {\dots\theta_0\dots\theta_n\rightarrow \dots\theta_0\dots\theta_n\theta_{n+1}};
  • or our possible continuations are {\dots\theta_0\dots\theta_n112\alpha_{n+4}} and {\dots\theta_0\dots\theta_n221\beta_{n+4}}.

Indeed, suppose that we have two possible continuations {\dots\theta_0\dots\theta_n1\alpha_{n+2}} and {\dots\theta_0\dots\theta_n2\beta_{n+2}}. If {I(a_1,\dots,a_n)} denotes the interval of numbers in {[0,1]} whose continued fraction expansion starts by {[0;a_1,\dots, a_n,\dots]}, then the intervals {I(a_1a_2)}, {(a_1,a_2)\in\{1,2\}^2} appear in the following order on the real line:

\displaystyle I(21), I(22), I(11), I(12)

Thus:

  • the continuation {\dots\theta_0\dots\theta_n12\alpha_{n+3}} is not possible (otherwise {[[0;2\beta_{n+2}]], [0;12\alpha_{n+3}]]} would contain {I(11)} and, a fortiori, intersect {K(\{11,22\})};
  • the continuation {\dots\theta_0\dots\theta_n21\beta_{n+3}} is not possible (otherwise {[[0;21\beta_{n+3}]], [0;1\alpha_{n+2}]]} would contain {I(22)} and, a fortiori, intersect {K(\{11,22\})}

so that our continuations are {\dots\theta_0\dots\theta_n11\alpha_{n+3}} and {\dots\theta_0\dots\theta_n22\beta_{n+3}}. Now, we observe that the intervals {I(11a_3)} and {I(22a_3)}, {a_3\in\{1,2\}}, appear in the following order on the real line:

\displaystyle I(222), I(221), I(112), I(111)

Hence:

  • the continuation {\dots\theta_0\dots\theta_n111\alpha_{n+4}} is not possible (otherwise {[[0;22\beta_{n+3}]], [0;111\alpha_{n+4}]]} would contain {I(112)} and, a fortiori, intersect {K(\{11,22\})};
  • the continuation {\dots\theta_0\dots\theta_n222\beta_{n+3}} is not possible (otherwise {[[0;222\beta_{n+4}]], [0;11\alpha_{n+3}]]} would contain {I(221)} and, a fortiori, intersect {K(\{11,22\})}

so that our claim is proved.

This claim allows us to bound the Hausdorff dimension of

\displaystyle K:=\{[\theta_0;\theta_1,\dots]: 3.06<m(\underline{\theta})<\sqrt{12} \textrm{ as above}\}

In fact, the claim says that we refine the natural cover of {K} by the intervals {I(\theta_1,\dots, \theta_n)} by replacing it by a “forced” {I(\theta_1,\dots, \theta_n,\theta_{n+1})} or by the couple of intervals

\displaystyle I(\theta_1,\dots, \theta_n, 1, 1, 2) \quad \textrm{ and } \quad I(\theta_1,\dots, \theta_n, 2, 2, 1)

Therefore, it follows from the definition of Hausdorff dimension that

\displaystyle HD(K)\leq s_0

for any parameter {0\leq s_0\leq 1} such that

\displaystyle |I(\theta_1,\dots,\theta_n,1,1,2)|^{s_0} + |I(\theta_1,\dots,\theta_n,2,2,1)|^{s_0} \leq |I(\theta_1,\dots,\theta_n)|^{s_0} \ \ \ \ \ (1)

Because we can assume that {m=m(\underline{\theta}) = [\theta_0;\theta_1,\dots]+[0;\theta_{-1},\dots]} with {[\theta_0;\theta_1,\dots]\in K} and {[0;\theta_{-1},\dots]\in C(2) := \{[0;\gamma]: \gamma\in\{1,2\}^{\mathbb{N}}\}}, our discussion so far can be summarized by following proposition:

Proposition 2 {(M\setminus L)\cap[3.06,\sqrt{12}]} is contained in the arithmetic sum

\displaystyle C(2)+K

where {HD(K)\leq s_0} for any parameter {0\leq s_0\leq 1} satisfying (1).

Since the arithmetic sum {C(2)+K} is the projection {\pi(C(2)\times K)}, {\pi(x,y)=x+y}, of the product set {C(2)\times K}, this proposition implies the following result:

Corollary 3 {HD((M\setminus L)\cap[3.06,\sqrt{12}])\leq HD(C(2))+s_0} where {0\leq s_0\leq 1} satisfies (1).

The Hausdorff dimension of {C(2)} was computed with high accuracy by Hensley among other authors: one has {HD(C(2))<0.531291}. In particular,

\displaystyle HD((M\setminus L)\cap[3.06,\sqrt{12}])<0.531291+s_0

where {s_0} verifies (1).

Closing this post, let us show that (1) holds for {s_0=0.174813} and, consequently,

\displaystyle HD((M\setminus L)\cap[3.06,\sqrt{12}])<0.706104

For this sake, recall that

\displaystyle |I(b_1,\dots,b_n)|=\frac{1}{q_n(q_n+q_{n-1})},

where {q_j} is the denominator of {[0;b_1,\dots,b_j]}.

Hence, if we set

\displaystyle g(s) := \frac{|I(a_1,\dots,a_n,1,1,2)|^s+|I(a_1,\dots,a_n,2,2,1)|^s}{|I(a_1,\dots,a_n)|^s}

then the recurrence formula {q_{j+2}=a_{j+2}q_{j+1}+q_j} implies that

\displaystyle g(s) = \left(\frac{r+1}{(3r+5)(4r+7)}\right)^s + \left(\frac{r+1}{(3r+7)(5r+12)}\right)^s

where {r=q_{n-1}/q_n\in (0,1)}.

Because {\frac{r+1}{(3r+5)(4r+7)}\leq \frac{1}{35}} and {\frac{r+1}{(3r+7)(5r+12)}<\frac{1}{81.98}} for all {0\leq r\leq 1}, we have

\displaystyle g(s)<\left(\frac{1}{35}\right)^s + \left(\frac{1}{81.98}\right)^s

This completes the argument because {\left(\frac{1}{35}\right)^{0.174813} + \left(\frac{1}{81.98}\right)^{0.174813} < 1}.

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