SOME COMMENTS ON SCHEGLOV'S UPPER
BOUND ON THE COMPLEXITY OF TRIANGULAR
BILLIARDS

Serge Troubetzkoy, Tyll Krueger

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Serge Troubetzkoy, Tyll Krueger. SOME COMMENTS ON SCHEGLOV'S UPPER BOUND ON THE COMPLEXITY OF TRIANGULAR BILLIARDS. 2024. hal-04695305

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<figure><img src="image_1.png" /></figure>
# SOME COMMENTS ON SCHEGLOV'S UPPER BOUND ON THE COMPLEXITY OF TRIANGULAR BILLIARDS

TYLL KRUEGER AND SERGE TROUBETZKOY

*Dedicated to Lyonia Bunimovich for his 75th birthday*

**ABSTRACT.** Scheglov showed that for almost every triangle the number of saddle connections of length at most $n$ grows slower than $e^{n^\varepsilon}$ for every $\varepsilon > 0$. We show that his result yields slightly stronger statements.

## 1. LYONIA

The three of us first met in 1988 in Moscow when Serge and Tyll were both visiting Sinai in Moscow. Tyll invited both Lyonia and Serge to work in Germany. Tyll and Lyonia next meet in Marseille in 1989, where much later Serge ended up working. Lyonia and Serge arrived in Bielefeld in 1990, Lyonia stayed for a year and Serge for 3 years. Interestingly Lyonia managed to arrive to West Germany from the Soviet Union without any border checks due to the recently fallen iron curtain.

Serge and Lyonia started a fruitful collaboration on a natural topic for us, we were studying billiards on a lattice. This topic has been studied independently in diverse fields: mathematics, physics (lattice gases), artificial life (the ant), ... Our best theorem has been discussed in the popular scientific literature (Scientific American, Math. Intelligencer, Science News, Pour la Science) and well as in several books, it became known as the Kong-Cohen theorem due to a careless reading of our article by the author of one of these expository articles.

Lyonia invited Grisha Galperin to Bielefeld. Eventually we were able to invite Grisha for a whole year which lead to our lifetime fascination and research on polygonal billiards, first with Grisha and then continuing without him when he left for the USA.

Another important aspect of Lyonia's visit was the fact that he explained to us his Markov partitions for billiards; this inspired Serge and Tyll to work on alternative constructions of metric Markov partitions for systems with singularities.

Beside mathematics we enjoyed exploring northern Germany together. Happily Lyonia's driving skills improved when Lyonia moved to the US, having survived Lyonia's early driving, we can wish Lyonia a very happy 75th birthday.
## 2. INTRODUCTION

The complexity of polygonal billiards has been the subject of many articles. The first result is the metric entropy of the billiard map with respect to the Liouville measure (and thus of the billiard flow) is zero, this was proven by Boldrighini, Keane and Marchetti, and independently by Sinai [BoKeMar, Si1, Si2]. Later Katok proved that the metric entropy of every invariant measure is zero [Ka].

To make more precise statements we are interested in the growth rate of saddle connections or of billiard words; we begin by defining them. A *saddle connection* is an orbit segment starting and ending at a corner of the polygon. Let $N_g(t)$ denote the number of saddle connections with geometric length at most $t$ and $N_c(n)$ denotes the number of saddle connections with combinatorial length at most $n$, here the number of links of a saddle connection is called its *combinatorial length*. Label the sides of the polygon by a finite alphabet $\mathcal{A}$ and code the forward orbit of a point by the sequence of sides it hits. Let $p(n)$ denote the number of distinct words of length $n$ which arise in this coding. Cassaigne, Hubert and Troubetzkoy showed that

$$ (1) \qquad p(n) = \sum_{j=0}^{n-1} N_c(j) $$

for all convex polygons [CaHuTr]. Their proof shows that this formula actually also holds for non-convex simply connected polygons since any $n$-cell is convex ([Ka]). A similar formula with possible low order correction terms was shown for all polygons by Bedaride [Be].

For any rational polygon Masur showed that $0 < \liminf \frac{N_g(t)}{t^2} \le \limsup \frac{N_g(t)}{t^2} < \infty$ [Mas1, Mas2], similar inequalities hold for $N_c(n)$, thus (1) implies that $0 < \liminf \frac{p(n)}{n^3} \le \limsup \frac{p(n)}{n^3} < \infty$ for any rational polygon.

For each $m \ge 3$ and each $k \ge 1$ Hooper constructed an open set of $m$-gons for which $\liminf_{t \to \infty} N_g(t)/(t \log^k(t)) > 0$ [Ho]. Troubetzkoy showed that $p(n)$ grows at least quadratically in any polygon [Tr]. Krueger, Nogueira and Troubetzkoy showed that for a generic polygon $N_c(n)$ grows quadratically along a subsequence [KrNoTr].

Scheglov showed that for almost every triangle $N_c(n)$ grows slower than $e^{n\epsilon}$ for every $\epsilon > 0$ [Sc1, Sc2], and then he showed that for almost every right triangle $N_c(n) \ge n^{\frac{2}{\sqrt{3}}-\epsilon}$ infinitely often for every $\epsilon > 0$ [Sc3]. In this article we will give some complementary results to Scheglov's upper bound.

## 3. RESULTS

Billiard dynamics commutes with planar similarities, therefore we suppose our triangle has longest side of length 1, and place it so that
the longest side connects the origin to the point (1, 0), and the smallest
angle is at the origin. The set $\Delta$ of such triangles is then parameterized
by two angles between each of the shorter sides with the x-axis, i.e.,
$\Delta = \{(\alpha, \beta) : 0 < \alpha \le \beta \le \pi - \alpha - \beta < \pi\}$. We will consider the
Lebesgue measure on the set $\Delta$.

The *billiard flow* $\phi_t$ in a polygon $Q$ is the free motion of a point particle with unit speed and specular reflections from the boundary; the reflections are not defined at corners of the polygon.

The *billiard map* $S$ is the first return map of the billiard flow to the boundary $\partial Q$, it is not defined for points for which the next arrival to the boundary is a vertex.

A *saddle connection* is an orbit segment which starts at a vertex and end at a vertex (not necessarily the same one). The combinatorial length of a saddle connection is the number of links in the trajectory. Let $N_c(n)$ count the number of saddle connections of combinatorial length at most $n$. Scheglov [Sc1] showed

**Theorem A.** For every $\varepsilon > 0$ for a.e. triangle

$$
(2) \qquad \limsup_{n \to \infty} \frac{N_c(n)}{e^{n\varepsilon}} < \infty.
$$

We do not know if the complementary set of triangles not satisfying (2) is empty or not, but have the following structural result.

**Proposition 1.** For every $\varepsilon > 0$, the set of triangles for which

$$
(3) \qquad \limsup_{n \to \infty} \frac{N_c(n)}{e^{n\varepsilon}} = \infty
$$

is a $G_\delta$ set (possibly empty). Equivalently the set of triangles satisfying
(2) is a $F_\sigma$ set.

**Proof.** Fix $\varepsilon \in (0,1)$. Let $I$ denote the set of triangles satisfying (3). If $I$ is empty or finite then we are done, so suppose $I$ is infinite.

For each $k \in \mathbb{N}$ and for each $Q \in I$ we choose $m = m(k, Q)$ such that $N_c(m) \ge k e^{m\varepsilon}$. By [KrNoTr, Lemma 7] we can choose $\delta_{k,Q} > 0$ so small that for any $P \in B(Q, \delta_{k,Q})$ we have $N_c^P(m) \ge N_c^Q(m)$. Consider the set

$$
G := \bigcap_{K \ge 1} \bigcup_{k \ge K} \bigcup_{Q \in I} B(Q, \delta_{k,Q}).
$$

Clearly $G$ is a $G_\delta$ set and $I \subset G$. Furthermore for any $Q \in G$ we have a
sequence $i_j$ such that $Q \in B(Q_{i_j}, \delta_{i_j})$ and thus $N_c^Q(m_{i_j}) \ge N_c^{Q_{i_j}}(m_{i_j}) \ge$
$i_j e^{(m_{i_j})^\epsilon}$, and so

$$
\[
\limsup_{n \to \infty} \frac{N_c^Q(n)}{e^{n\epsilon}} = \infty,
\]
so $Q \in I$.
$$

<figure><img src="image_2.png" /></figure>

Theorem A has the following simple corollary.
**Corollary 2.** For every $\varepsilon > 0$ for a.e. triangle

$$ (4) \qquad \limsup_{n \to \infty} \frac{N_c(n)}{e^{n\varepsilon}} = 0. $$

**Proof.** For fixed $\varepsilon > 0$ consider the set of triangles $F_{\varepsilon} := \{Q : 0 < \limsup_{n \to \infty} \frac{N_c(n)}{e^{n\varepsilon}} < \infty\}$. Clearly if $Q \in F_{\varepsilon}$ then $\limsup_{n \to \infty} \frac{N_c(n)}{e^{n\varepsilon}} = \infty$ for all $\varepsilon' < \varepsilon$. Thus if a set of positive measure of $Q$ was in $F_{\varepsilon}$ for some $\varepsilon$ this would contradict Scheglov's theorem for all $\varepsilon' < \varepsilon$. Thus the complement of $F_{\varepsilon}$ has full measure, again applying Scheglov's theorem we conclude that (4) holds. $\square$

In Corollary 2 it is not necessary to fix $\varepsilon$, we have the following strengthening of this result.

**Proposition 3.** There exists a monotonically decreasing sequence $\varepsilon_n \to 0$ of positive numbers such that for a.e. triangle we have

$$ (5) \qquad \limsup_{n \to \infty} \frac{N_c(n)}{e^{n\varepsilon_n}} = 0. $$

**Proof.** For each $k \ge 1$ consider the set of triangles

$$ A_k := \{Q : \limsup_{n \to \infty} \frac{N_c(n)}{e^{n^{1/k}}} = 0\}. $$

This set is of full measure by Corollary 2, thus the set $A := \cap_{k \ge 1} A_k$ has full measure as well.

For each $Q \in A$ we have an infinite monotonically increasing sequence $m_k = m_k(Q)$ such that

$$ \frac{N_c(n)}{e^{n^{1/k}}} < 1/k $$

for all $n \ge m_k$ and $k \ge 1$.

We choose a strictly increasing sequence $(a_k)_k$ growing so quickly that the set $G_k := \{Q \in G : m_i \le a_i \text{ for } 1 \le i \le k\}$ has measure at least $1 - 1/k$. Notice that by definition $G_k \subset G_{k+1}$ and the set $G := \cup_{k \ge 1} G_k$ is a full measure set. For any $Q \in G$ there exists an $\ell = \ell(Q)$ such that $Q \in G_k$ for all $k \ge \ell$, thus

$$ (6) \qquad \frac{N_c(n)}{e^{n^{1/m_k}}} < 1/m_k $$

for all $n \ge m_k = m_k(Q) \le a_k$ with $k \ge \ell$. We conclude that for each $Q \in G$ that for each $n \ge a_k$ with $k \ge \ell$ Equation (6) holds and thus setting $\varepsilon_n := 1/a_n$ Equation (5) holds as well. $\square$

Remember that a subset of a topological space is called *meager* if it is the countable union of nowhere dense sets, or equivalently if its complement is *residual*, i.e., it is the countable intersection of sets, each of whose interior is dense.
In the previous result we have no control over the decay rate of $\epsilon_n$, if it decays extremely quickly then the complexity of a.e. triangle is quite small. We know from [KrNoTr, Theorem 1] that the complexity of a generic triangle grows quadratically along subsequences, of course this set of generic triangles could be of measure zero. Thus to state the next result we fix $\delta > 0$ and let $b_n$ be such that $e^{n^{b_n}} = n^{2+\delta}$. Define $\bar{\epsilon}_n := \max(\epsilon_n, b_n)$ and the set

$$B := \{Q : \liminf_{n \to \infty} \frac{N_c(n)}{e^{n\bar{\epsilon}_n}} > 0\}.$$

Finally we show

**Proposition 4.** *The set B is a zero measure meager set.*

**Proof.** Proposition 3 implies that B has zero measure.

Theorem 1 from [KrNoTr] tells us that there is a dense $G_\delta$ set G of triangles such that for each $Q \in G$ there is a monotonically increasing sequence $n_k$ such that $\frac{\log(N_c(n_k))}{\log(n_k)} \to 2$. If $Q \in G$ then

$$\frac{N_c(n_k)}{e^{n_k}} \le \frac{N_c(n_k)}{n_k^{2+\delta_0}} \to 0$$

and thus $Q \notin B$. We have shown that the complement of B contains the dense $G_\delta$ set G, thus it is residual and so B is meager. □

In the same vein we have

**Proposition 5.** *There is a full measure dense $G_\delta$ set set S of triangles such that for each triangle $Q \in S$ for any $\delta > 0$ we have $\liminf_{n \to \infty} N_c(n)/e^{n\delta} = 0$.*

We only give a sketch of the proof since the proof essentially repeats the proof of Theorem 1 from [KrNoTr]. For any fixed $\delta > 0$ we repeat the proof of Theorem 1 in [KrNoTr] replacing $\log(N_c(n))/\log(n^2)$ by the term $N_c(n)/e^{n\delta}$, Lemma 8 of [KrNoTr] becomes much simpler since we are not aiming for polynomial growth.

The $G_\delta$ set in the original proof started with taking a union of a small neighborhood of each element of a countable dense set of rational polygons; we replace this by taking the countable union of a small neighborhood of each triangle verifying the Scheglov theorem. This yields a dense $G_\delta$ set $S_\delta$ of full measure verifying $\lim_{k \to \infty} N_c(n_k)/e^{n_k} = 0$. Intersecting the sets $S_{\delta_k}$ for any sequence $\delta_k \to 0$ yields the desired result.

## REFERENCES

[Be] N. Bedaride, *Billiard complexity in rational polyhedra*, Regul. Chaotic Dyn. 8 (2003) 97–104.

[BoKeMar] C. Boldrighini, M. Keane and F. Marchetti, *Billiards in polygons*, Ann. Probab. 6 (1978) 532–540.

[CaHuTr] J. Cassaigne, P. Hubert and S. Troubetzkoy, *Complexity and growth for polygonal billiards*, Ann. I. Fourier 52 (2002) 835–847.
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[Ho] W.P. Hooper, *Lower bounds on growth rates of periodic billiard trajectories in some irrational polygons*, J. Mod. Dyn. 1 (2007) 649–663.

[Ka] A. Katok, *The growth rate for the number of singular and periodic orbits for a polygonal billiard*, Commun. Math. Phys. 111 (1987) 151–160.

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[Mas1] H. Masur, *Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential*, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), 215–228, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988.

[Mas2] H. Masur, *The growth rate of trajectories of a quadratic differential*, Ergod. Theory Dyn. Syst. 10 (1990) 151–176.

[Sc1] D. Scheglov, *Complexity growth of a typical triangular billiard is weakly exponential*, J. Anal. Math. 142 (2020) 105–124.

[Sc2] D. Scheglov, *Growth of periodic orbits and generalized diagonals for typical triangular billiards*, J. Mod. Dyn. 7 (2013) 31–44.

[Sc3] D. Scheglov, *Complexity lower bound for typical right triangular billiards*, arXiv:2207.00910 [cs.LG].

[Si1] Ya. G. Sinai, *Introduction to ergodic theory*, Translated by V. Scheffer. Mathematical Notes, 18, Princeton University Press, Princeton, N.J., 1976. 144 pp.

[Si2] Ya. G. Sinai, *Topics in ergodic theory*, Princeton Mathematical Series, 44, Princeton University Press, Princeton, NJ, 1994. 218 pp.

[Tr] S. Troubetzkoy, *Complexity lower bounds for polygonal billiards*, Chaos 8 (1998) 242–244.

WROCLAW UNIVERSITY OF TECHNOLOGY, FACULTY OF COMMUNICATION AND INFORMATION TECHNOLOGY, UL. JANISZEWSKIEGO 11/17, 50-372 WROCLAW, POLAND

Email address: tyll.krueger@pwr.edu.pl

AIX MARSEILLE UNIV, CNRS, I2M, MARSEILLE, FRANCE

Email address: serge.troubetzkoy@univ-amu.fr