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ARITHMETIC CODING AND ENTROPY FOR THE
POSITIVE GEODESIC FLOW ON THE MODULAR

SURFACE

BORIS GUREVICH AND SVETLANA KATOK

This article, whose authors had the privilege and good fortune of studying at
Moscow University when I.G. Petrovsky was its Rector, is dedicated to his

memory.

Abstract. In this article we study geodesics on the modular sur-
face by means of their arithmetic codes. Closed geodesics for which
arithmetic and geometric codes coincide were identified in [8]. Here
they are described as periodic orbits of a special flow over a topo-
logical Markov chain with countable alphabet which we call the
positive geodesic flow. We obtain an explicit formula for the ceil-
ing function and two-sided estimates for the topological entropy of
the positive geodesic flow, which turns out to be separated from
one, the topological entropy of the geodesic flow on the modular
surface.

Introduction

Let H = {z ∈ C | Im z > 0} be the upper half-plane endowed
with the hyperbolic metric. Geodesics on the modular surface M =
PSL(2,Z)\H can be coded in two different ways. Let

F = {z ∈ H | |z| ≥ 1, |Re z| ≤
1

2
}

be the standard fundamental region for PSL(2,Z) whose sides are iden-
tified by the generators of PSL(2,Z), T (z) = z + 1 and S(z) = −1

z
(see Figure 1). In this article we will consider only oriented geodesics
which do not go to the cusp of M in either direction (the corresponding
geodesics on F contain no vertical segments), and often refer to them
simply as geodesics. Notice that all closed geodesics belong to this set,

Date: October 5, 2001.
1991 Mathematics Subject Classification. 37D40, 37B40, 20H05.
Key words and phrases. Geodesic flow, modular surface, Fuchsian group, en-

tropy, topological entropy.
The first author was partially supported by RFBS, grants 99-01-00284 and 99-

01-00314.
1

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ARITHMETIC CODING AND ENTROPY 9

and therefore the error term can be written in the form

log g(x)− log g(σx),
where

g(x) =
(w(x)− u(x))


w(x)2 − 1

w(x)2


1− u(x)2
,

and

(2) f(x) = 2 logw(x) + log g(x)− log g(σx).


Corollary 5. The length of a closed geodesic with the arithmetic code
(n1, . . . , nm) is equal to

2 log
m∏
i=1

wi,

where w1, . . . , wm are the attractive fixed points of all reduced matrices
corresponding to this closed geodesic.

2. Special representation of the positive geodesic flow

2.1. Characterization of positive geodesics.

Definition 6. We call a geodesic γ positive if all segments of the
double–infinite sequence comprising it begin and end in the set P .

Since a positive geodesic avoids the set Q, its arithmetic code counts
the number of times it hits the side v2 of F in the positive direction.
The following result gives a characterization of positive geodesics in
terms of their arithmetic code and is a generalization of Theorems 1
and 2 of [8].

Theorem 7. The following are equivalent:

(1) γ is positive;
(2) (γ) = (. . . , n−1, n0, n1, n2, . . . ) is subject to the Platonic restric-

tions, i.e. 1
ni

+ 1
ni+1
≤ 1

2
for all i;

(3) the geometric and arithmetic codes of γ coincide, i.e. (γ) = [γ];
(4) all segments comprising the geodesic γ in F are positively (clock-

wise) oriented.

Proof. It is clear from the definition and the discussion preceding The-
orem 3 that (1), (3) and (4) are equivalent. It remains to prove that
(1) and (2) are equivalent.

(2)⇒ (1). Assume that (2) holds, but γ is not positive. This means
that it crosses Q. The geodesic γ can be approximated by a sequence

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ARITHMETIC CODING AND ENTROPY 17

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Faculty of Mechanics & Mathematics, Moscow State University,

Vorobyovy Gory, Moscow 119899, Russia

E-mail address: [email protected]

Department of Mathematics, Pennsylvania State University, Uni-

versity Park, PA 16802

E-mail address: katok [email protected]

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