I am at the Shanghai Center of Mathematical Sciences at the Jiangwan campus of Fudan University in Shanghai. There are two seminars here today.

Survivor set of Gauss map with a hole at 1

This was a talk by Lingmin Liau from the University of Wuhan. This is joint work with Cheng Liu.

First motivation & main results

In 2003, there was a paper by Jun Wu asking “How many points have the same Engel and Sylvester expansions?” Let us explain what these expansions are.

Engel’s expansion is \[\begin{equation} x= \frac{1}{d_{1}} + \frac{1}{d_{1} d_{2}} + \frac{1}{d_{1} d_{2} d_{3}} + \dots, \text{ for } 2 \leq d_{1} \leq d_{2} \leq \dots \end{equation}\]

Sylvester’s expansion is \[\begin{equation} x = \frac{1}{s_{1}} + \frac{1}{s_{2}} + \frac{1}{s_{3}} + \dots \text{ where } s_{k+1} \geq s_{j}(s_{j} - 1)+ 1 \end{equation}\]

This is related to dynamics via the following map. \[\begin{align} T(x) & = x - \tfrac{1}{ \lfloor \frac{1}{x}\rfloor + 1} \implies x \\ & = \frac{1}{ \lfloor \frac{1}{x}\rfloor + 1 } + Tx = \frac{1}{s_{1}} + \frac{1}{ \lfloor \frac{1}{ Tx} \rfloor + 1} + T^{2} x \dots \\ & = \frac{1}{s_{1}} + \frac{1}{ s_{2} } + \dots \end{align}\] There is a similar dynamics map for the first expansion which I did not understand.

Now we define the set \[\begin{equation} E = \{x : d_{1} = s_{1} , d_{2} = s_{2} , \dots\}. \end{equation}\] Then what is \(\dim_{H} E\)? The main result of Jun Wu is that this dimension is \(1/2\).

Before this Galambus showed in 1981, 1996 that the Lebesgue measure is 0.

We assume \(\alpha \in [0,1]\). For this, we must study the map \(T_{\alpha}\) which acts on \([\alpha - 1, \alpha)\) via \[\begin{equation} T_{\alpha}(x) = \begin{cases} |\tfrac{1}{x}| - \lfloor |\tfrac{1}{x}| + 1 - \alpha\rfloor & \text{ if }x \neq 0 \\ 0 & \text{ otherwise} \end{cases}. \end{equation}\]

For \(x \in \mathbb{R}\), let \(b_{0} \in \mathbb{Z}\) be such that \(x-b_{0} \in [\alpha-1,\alpha)\). For this

• \(T_{\alpha}^{n-1}( x - b_{0}) \neq 0\), define \(\varepsilon_{n}(x) = \mathrm{sgn}( T_{\alpha}^{n-1}(x - b_{0}))\), where \(b_{n}(x) = \lfloor |\frac{1}{T_{\alpha}^{n-1}(x-b_{0})}| + 1 - \alpha \rfloor\).
• We have \(T_{\alpha}^{n-1}(x-b_{0}) = 0, \varepsilon_{n}(x)=0, b_{n}(x)=0\), \(\forall n \geq n_{0} + 1\) \(\implies\) \[\begin{align} x & = b_{0}(x) + \frac{\varepsilon_{1}(x)}{b_{1}(x) + \frac{\varepsilon_{2}(x)}{ b_{2}(x) + \frac{\varepsilon_{3}(x)}{b_{3}(x) + \dots}}} \\ & =: [ b_{0}, \varepsilon_{1},b_{1},\varepsilon_{2},b_{2},\dots]_{ \alpha} \end{align}\] The same continued fraction and \(\alpha\)-CF means \[\begin{equation} a_{0} = b_{0}, \varepsilon_{n}=1 \forall n \geq 1, a_{n} = b_{n} \forall n \geq 1. \end{equation}\]

This is something similar to a phenomenon called Devil’s stairs (I don’t understand). The constant intervals of \(\alpha \mapsto S(\alpha)\) are

• \(a_{1} \geq 2, n \geq 1\) : \(([a_{1},a_{2},\dots, a_{2n},a_{1}], [a_{1},\dots,a_{2n},1])\)
• \(a_{1} \geq 2\) : \(([a_{1}\dots a_{2n+1},1), [a_{1} a_{2} \dots a_{2n+1} a_{1}]\)
• \(a_{2n+1} \geq 2\) \(( [1, a_2,\dots,a_{2n+1}], [1,\dots,a_{2n+1}-1, \overline{1 a_{2}}])\)

Relation with the survivor sets of open dynamical system

Define \[\begin{equation} E(\alpha) : = \{x \in [0,1) : T^{n} x \leq \alpha, \forall n \geq 0\}. \end{equation}\] This is called a “survivor set”.

We will prove that the \(S(\alpha)\) in our question satisfies \[\begin{equation} S(\alpha) = \{x \in [0,1] : T^{n} x < \alpha \forall n \geq 0\}. \end{equation}\] This is the same thing as \(E(\alpha)\) but with inequality turned into a strict inequality. We get \[\begin{equation} S(\alpha) \subseteq E(\alpha) \subseteq S(\alpha) \cup \mathbb{Q} \cup \bigsqcup_{n=0}^{ \infty } \{ x \in [0,1), T^{n} x = \alpha\}. \end{equation}\]

• Raith in 1994 studied \(f: [0,1] \rightarrow [0,1]\) in expanding monotone interval map with finitely many branches. We define \[\begin{equation} \mathrm{Hole} = \bigcup_{i=1}^{k} (a_{i} , b_{i}). \end{equation}\] Then we get that \(\dim_{H} E_{f}( \mathrm{Hole})\) is continuous with respect to the \(\mathrm{Hole}\).
• Carminati-Tiozzo studied in 2022 the Gauss map with a hole at \(0\).

Main argument

For our interest, we study \[\begin{align} S(\tfrac{1}{n}) & = \{x \mid T^{k} x < \tfrac{1}{n} \forall k \geq 0\} \\ & = \{x \mid a_{k} > n \forall k \geq 0\}. \end{align}\] We have \[\begin{equation} \lim_{n \rightarrow \infty} \dim_{H} S(\tfrac{1}{n}) = \frac{1}{2}. \end{equation}\] This is due to Good from 1941. Then \(S(0) = ?\). This is the set \(\{T^{n} x < 0\} = \{\}\)

Intermediate step

Final remarks

We have \(F(\alpha):= S(\alpha) \cap (\text{Bounded})\). We can prove \(\alpha \mapsto \dim_{H} F(\alpha)\) is continuous on \((0,1)\). This is related to some IFS (I don’t understand).

Density of shapes of periodic tori: Part I (the cubic case)

The speaker is Nguyen-Thi Dang. She is from Université Paris-Saclay.

This is joint work with myself and J. Li. See also my notes here.

A dynamical problem

Consider \(G = \mathrm{SL}(3,\mathbb{R})\) and \(\Gamma = \mathrm{SL}(3,\mathbb{Z})\). We have \[\begin{equation} A = \{ \begin{bmatrix} a_{1} & & \\ & a_{2} & \\ & & a_{3} \end{bmatrix} \mid a_{i} > 0, a_{1}a_{2}a_{3} = 1 \}. \end{equation}\] We also have \[\begin{equation} M = \{ \begin{bmatrix} \pm 1 & & \\ & \pm 1 & \\ & & \pm 1 \end{bmatrix}, \det( \cdot ) = 1 \} \end{equation}\] We have \(\mathfrak{a} = \log A = \{ (t_{1},t_{2},t_{3}) \mid \sum_{{i}}^{} t_{i}= 0\}.\)

Then \(P(F) \subseteq \mathfrak{a}\) is a \(2\)-lattice in \(\mathfrak{a} \simeq \mathbb{R}^{2}\). Fix an isomorphism \(\mathfrak{a} \simeq \mathbb{R}^{2}\). We then have \({P}(F) = \sqrt{\mathrm{covol}( P(F))} g_{F} \mathbb{Z}^{2}\) for some \(g_{F} \in \mathrm{SL}(2,\mathbb{R})\)

A construction

We will look at \(0 < a_{1} < a_{2}\) in \(\mathbb{N}\) such that \(a_{1} \gg 1\) and \(a_{2} - a_{1} \gg 1\). Then \(f(X) = X(X-a_{1})(X-a_{2}) - 1\) is irreducible over \(\mathbb{Q}\).

We consider the field \(K = \mathbb{Q}[X]/ \langle f(X)\rangle\) This is isomorphic to \(\mathbb{Q}^{3}\) additively.

Ordered embeddings

For \(\sigma = (\sigma_{1},\sigma_{2},\sigma_{3}) : K \rightarrow \mathbb{R}\) are given by \(X \mapsto \alpha_{i}\) where \(\alpha_{1},\alpha_{2},\alpha_{3}\) are the roots of the polynomial.

Dirichlet’s unit theorem

Using this, one can construct a periodic torus. We consider the action of \(K^{\times}\) on \(K\) and get \[\begin{align} K & \xrightarrow[]{} \mathbb{R}^{3} \\ x & \mapsto \mathrm{diag}( \sigma_{1}(x),\sigma_{2}(x), \sigma_{3}(x) ). \end{align}\] Then we look at \(\mathcal{O}^{\times}\) acting on \(\mathcal{O}\) and write \(\mathcal{O} = h_{\mathcal{O}} \mathbb{Z}^{3} \subseteq \mathbb{R}^{3}\).

Indeed, observe that \(\alpha, \alpha-a_{1},\alpha - a_{2} \in \mathcal{O}^{\times}\). This lemma is due to an optimal inequality for cubic fields by Cusick.

But we want density of shapes!

Congruence conditions

What we will try is \[\begin{align} (a_{1},a_{2}) & \simeq (0,1)\pmod{2^{c}}, \text{ for }c \in \mathbb{Z}_{\geq 0} \\ (a_{1},a_{2}) & \simeq (1,0)\pmod{3^{d}}, \text{ for }d \in \mathbb{Z}_{\geq 0} \\ (a_{1},a_{2}) & \simeq (1,1)\pmod{5^{r}}, \text{ for }r \in \mathbb{Z}_{\geq 0} \\ \end{align}\] Then we have that \(\mathcal{O}_{(c,d,r)} = \mathbb{Z} + 2^{c} 3^{d} 5^{r} \mathcal{O}\) satisfies \[\begin{equation} \mathcal{O}_{(c,d,r)}^{\times} = \{ \pm \alpha^{m} (\alpha - a_{1})^{n} \mid \underbrace{ \begin{cases} \varphi_{(0,1)}(m,n) \simeq 0 \pmod{ 7\cdot 2^{c-1}} \\ \varphi_{(1,0)}(m,n) \simeq 0 \pmod{ 8\cdot 3^{d-1}} \\ \varphi_{(1,1)}(m,n) \simeq 0 \pmod{ 24\cdot 5^{r-1}} \\ \end{cases}}_{\text{ defines a lattice }\Lambda_{(c,d,r)} \subseteq \mathbb{Z}^{2}} \} \end{equation}\] where \(\varphi_{(0,1)}, \varphi_{(1,0)}, \varphi_{(1,1)} \in (\mathbb{Z}^{2})^{*}\).

Parameters