##### Document Text Contents

Page 1

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

Page 9

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

Page 17

ARITHMETIC CODING AND ENTROPY 17

[13] C. Liverani, Decay of correlations, Ann. of Math. (2) 142 (1995), 239–301

[14] M. Morse, Symbolic dynamics, Institute for Advanced Study Notes, Princeton

(unpublished), 1996.

[15] W. Parry and M. Pollicott, Zeta functions and periodic orbit structure of

hyperbolic dynamics, Astérisque, 187–188, 1990.

[16] Ya. B. Pesin and B.S. Pitskel, Topological pressure and variational principle

for noncompact sets (in Russian), Funkcional. Anal. Prilozh. 18 (1984), 50–

63.

[17] A.B. Polyakov, On a measure with maximal entropy for a special flow over a

local perturbation of a topological Bernoulli scheme (in Russian). Mathematics

of the USSR, Sbornik, 30 (2001).

[18] P. Sarnak, Prime geodesic theorem, Ph.D. Thesis, Stanford, 1980.

[19] Ya. G. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surveys, 27

(1972), no. 4, 21–69.

[20] S.V. Savchenko, Special flows constructed by countable topological Markov

chains (in Russian), Funkcional. Anal. Prilozh. 32 (1998), 40–53; translation

in Funct. Anal. Appl. 32 (1998), no. 1, 32–41.

[21] D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geo-

metrically finite Kleinian groups, Acta Math. 153 (1984), 259–277.

[22] D. Zagier, Zetafunktionen und quadratische Körper: eine Einführung in die

höhere Zahlentheorie, Hochschultext, Springer, Berlin, Heidelberg, New York,

1982.

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]

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

Page 9

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

Page 17

ARITHMETIC CODING AND ENTROPY 17

[13] C. Liverani, Decay of correlations, Ann. of Math. (2) 142 (1995), 239–301

[14] M. Morse, Symbolic dynamics, Institute for Advanced Study Notes, Princeton

(unpublished), 1996.

[15] W. Parry and M. Pollicott, Zeta functions and periodic orbit structure of

hyperbolic dynamics, Astérisque, 187–188, 1990.

[16] Ya. B. Pesin and B.S. Pitskel, Topological pressure and variational principle

for noncompact sets (in Russian), Funkcional. Anal. Prilozh. 18 (1984), 50–

63.

[17] A.B. Polyakov, On a measure with maximal entropy for a special flow over a

local perturbation of a topological Bernoulli scheme (in Russian). Mathematics

of the USSR, Sbornik, 30 (2001).

[18] P. Sarnak, Prime geodesic theorem, Ph.D. Thesis, Stanford, 1980.

[19] Ya. G. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surveys, 27

(1972), no. 4, 21–69.

[20] S.V. Savchenko, Special flows constructed by countable topological Markov

chains (in Russian), Funkcional. Anal. Prilozh. 32 (1998), 40–53; translation

in Funct. Anal. Appl. 32 (1998), no. 1, 32–41.

[21] D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geo-

metrically finite Kleinian groups, Acta Math. 153 (1984), 259–277.

[22] D. Zagier, Zetafunktionen und quadratische Körper: eine Einführung in die

höhere Zahlentheorie, Hochschultext, Springer, Berlin, Heidelberg, New York,

1982.

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]