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@ Appl. Gen. Topol. 19, no. 2 (2018), 239-244doi:10.4995/agt.2018.7962
c© AGT, UPV, 2018

Dynamics of real projective transformations

Sharan Gopal
a
and Srikanth Ravulapalli

b

a
BITS-Pilani, Hyderabad campus, Hyderabad, India. (sharanraghu@gmail.com)

b
School of Mathematics and Statistics, University of Hyderabad, Hyderabad, India.

(srikanth.hcu@gmail.com)

Communicated by F. Balibrea

Abstract

The dynamics of a projective transformation on a real projective space

are studied in this paper. The two main aspects of these transforma-

tions that are studied here are the topological entropy and the zeta

function. Topological entropy is an inherent property of a dynami-

cal system whereas the zeta function is a useful tool for the study

of periodic points. We find the zeta function for a general projective

transformation but entropy only for certain transformations on the real

projective line.

2010 MSC: 54H20; 37B40.

Keywords: topological entropy; zeta function; projective transformation.

1. Introduction

An n−dimensional real projective space, denoted by Pn(R) is the quotient
space Sn/∼, where the antipodal points are identified under the relation, ∼.
P1(R) is also called the real projective line. Let π : S

n → Pn(R) be the
quotient map. A projective transformation on Pn(R) associated to a matrix
A ∈ GLn+1(R), denoted by Ā is defined as Ā(π(x)) = π(Ax), for every x ∈ S

n.
A discrete dynamical system is, by definition, a pair (X, f), where X is

a topological space and f is a self map on X i.e., f : X → X. Though
f can be any map in a general setting, we need it to be a continuous map
in many cases. So, unless otherwise mentioned, we assume the map to be
continuous. Since we consider only discrete dynamical systems in this paper,

Received 26 August 2017 – Accepted 05 February 2018

http://dx.doi.org/10.4995/agt.2018.7962


S. Gopal and S. Ravulapalli

hereafter, we refer to them simply as dynamical systems. Given x ∈ X, the
sequence (x, f(x), f2(x), f3(x), ...) is called the trajectory of x, where fk(x) =
f ◦ f ◦ ... ◦ f(x) (k times) for k ∈ N and f0(x) = x. The set {fk(x) : k is a
non-negative integer} is called the orbit of x. The study of dynamics is mainly
about the eventual behavior of trajectories. A point x ∈ X is said to be periodic
if there is a k ∈ N such that fk(x) = x; any such k is called a period of x and
the least among them is called the least period of x. A periodic point x of
period 1 is called a fixed point i.e., f(x) = x and the set of fixed points of f is
denoted by Fix(f). We also use the notation |Y | to denote the cardinality of
any set Y .

In this paper, the typical dynamical system that we are going to consider
is (Pn(R), Ā). The periodic points of this system can be found very easily. If
v ∈ Sn is an eigenvector of A with eigenvalue λ, then Ā(π(v)) = π(Av) =
π(λv) = π(v); hence π(v) is a fixed point. Conversely, if π(v) is a periodic
point with period k, then it is a fixed point of Āk and thus π(Akv) = π(v) i.e.,
Akv = µv for some scalar µ. Hence, v is an eigenvector of Ak. To sum up,
we have shown that π(v) is periodic if and only if v is an eigenvector of Ak for
some k ∈ N.

The dynamics of projective transformations are well studied in the literature.
See for instance [4] and [6]. In this paper, we study the topological entropy
and the zeta function of projective transformations.

One of the best ways of measuring the complexity of a dynamical system is
finding its topological entropy. As stated in [3], topological entropy measures
the exponential growth rate of the number of essentially different orbit seg-
ments of length n. On the other hand, the zeta function collects combinatorial
information about the periodic points. In the next section, we calculate the en-
tropy of certain projective transformations on the real projective line, followed
by a section on finding the zeta function of a projective transformation on a
projective space of any dimension.

2. Topological entropy

Topological entropy was introduced by Adler, Konheim and McAndrew [1]
and here, we will use an equivalent definition for maps on compact metric
spaces given by Bowen [2]. Most of the basic facts about entropy, that we
mention here can be found in [3].

Given a compact metric space (X, d) and a continuous map f : X → X, we
define a new metric dn, for every n ∈ N as dn(x, y) = max {d(f

i(x), fi(y)) :
0 ≤ i < n}. It can be shown that each of these metrics induces the same
topology on X as induced by d. A subset E ⊂ X is called an (n, ǫ)−separated
set if for any two distinct points x, y ∈ E, dn(x, y) ≥ ǫ. Since X is compact,
every (n, ǫ)-separated set is a finite set; otherwise, there will be a sequence (xk)
in E with no convergent subsequence, as d(xk, xk+1) ≥ ǫ for every k ∈ N, thus
contradicting the compactness of X. Now, let sep(n, ǫ, f) be the cardinality
of an (n, ǫ)−separated set with maximum cardinality.

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Dynamics of real projective transformations

Definition 2.1 (see [3]). The entropy h(f) of a system (X, f) is defined as

(2.1) h(f) = lim
ǫ→0+

lim sup
n→∞

1

n
log(sep(n, ǫ, f)).

Similar to sep(n, ǫ, f), two more numbers, namely span(n, ǫ, f) and cov(n, ǫ, f)
can be defined for every ǫ > 0 and n ∈ N. Here, cov(n, ǫ, f) is the cardinality
of a covering of X by least number of sets of dn-diameter less than ǫ. It is well
defined because, there does exist a finite cover of X by sets of dn-diameter less
than ǫ, as any cover of X with open sets of dn-diameter less than ǫ will have a
finite subcover for X. Finally, a subset A ⊂ X is called an (n, ǫ)-spanning set in
X, if for every x ∈ X, there is y ∈ A such that dn(x, y) < ǫ. As X is compact,
the open cover {Bdn(x, ǫ) : x ∈ X} (where Bdn(x, ǫ) is the open ball centered
at x and has radius ǫ with respect to the dn-diameter) has a finite subcover, say
{Bdn(x1, ǫ), Bdn(x2, ǫ), . . ., Bdn(xk, ǫ)}. Then, the set {x1, x2, . . , xk} is an
(n, ǫ)-spanning set. Since finite (n, ǫ)-spanning sets exist in a compact space,
we can find an (n, ǫ)-spanning set with minimum cardinality. This minimum
cardinality is called span(n, ǫ, f).

Lemma 2.2 (see [3]).

(2.2) cov(n, 2ǫ, f) ≤ span(n, ǫ, f) ≤ sep(n, ǫ, f) ≤ cov(n, ǫ, f).

Using this lemma, it follows easily that

h(f) = lim
ǫ→0+

lim sup
n→∞

1

n
log(sep(n, ǫ, f))(2.3)

= lim
ǫ→0+

lim sup
n→∞

1

n
log(span(n, ǫ, f))(2.4)

= lim
ǫ→0+

lim sup
n→∞

1

n
log(cov(n, ǫ, f))(2.5)

Proposition 2.3 (Proposition 2.5.3 in [3]). The topological entropy of a con-
tinuous map f : X → X does not depend on the choice of a particular metric
generating the topology of X.

Proposition 2.4 ([3]). The topological entropy of an isometry is zero.

In the following proposition, T 2 denotes the torus, R2/Z2. Any automor-
phism of this topological group, R2/Z2 which will be called a toral automor-
phism, is of the form π′(x) 7→ π′(Mx), where π′ : R2 → T 2 is the canonical
projection and M ∈ GL2(Z). We say that the automorphism is induced by the
matrix M and denote it by TM. If no eigenvalue of M has modulus 1, then TM
is called a hyperbolic toral automorphism.

Proposition 2.5 (Proposition 2.6.1 in [3]). The topological entropy of a hyper-
bolic toral automorphism TM : T

2 → T 2, with det(M) = 1 is equal to log |λ|,
where λ is the eigenvalue of M such that |λ| > 1.

All the above propositions can be found in [3]. Our proof of Theorem 2.6
relies mostly on the proof of Proposition 2.5, as given in [3].

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S. Gopal and S. Ravulapalli

Theorem 2.6. Let A be a 2 × 2 matrix of determinant 1 with eigenvalues
λ and 1

λ
, where |λ| > 1. Then the entropy of the corresponding projective

transformation Ā is log |λ|.

Proof. Let v1 and v2 be unit eigenvectors of A corresponding to λ and
1
λ

respectively. Now, for u, v ∈ S1, define d̃(u, v) = max(|a1|, |a2|), where

u − v = a1v1 + a2v2. d̃ is a metric on S
1 and it induces the metric d on P1(R),

where d(x, y) is the d̃-distance between the sets π−1(x) and π−1(y).

Note that the definition of d̃ can be extended to a metric on R2 and an open
ball of radius ǫ centered at (u0, v0) ∈ R

2 under d̃ is a parallelogram centered at
(u0, v0) with its sides parallel to v1 and v2 and each having a length 2ǫ. Then
an open ball of radius ǫ in S1 centered at (u0, v0) ∈ S

1 is an arc centered at
(u0, v0), which is formed by the intersection of S

1 with the above parallelogram.

Further, an ǫ d̃n-ball in R
2, with respect to the map induced by A is again a

parallelogram with sides of lengths 2ǫ and 2ǫ
|λ|n

which are parallel to v1 and v2

respectively. Now, an ǫ d̃n-ball in S
1 is thus an arc passing through the center

of a parallelogram with the above dimensions. So, its length is at least the
smaller side of the parallelogram i.e., 2ǫ

|λ|n
. On the other hand, its length is at

most the perimeter of the parallelogram, which is equal to 4ǫ + 4ǫ
|λ|n

(See the

figure). It follows from Archimedean property of real numbers that, if a and b
are any two positive real numbers, then there is a positive integer k such that
(a + b) ≤ kab. Thus, we can find a positive integer depending on ǫ, say k(ǫ),

such that 4ǫ + 4ǫ
|λ|n

≤
k(ǫ)ǫ2

|λ|n
. Since π is a local isometry, for sufficiently small ǫ,

we can assume that ǫ dn-balls in P1(R) have the same dimensions.

Since the diameter of an ǫ dn-ball in P1(R) is at most
k(ǫ)ǫ2

|λ|n
, the minimum

number of such balls that are required to cover P1(R) is
π

k(ǫ)ǫ2

|λ|n

, as the Euclidean

length of P1(R) is π. Since a set of diameter 2ǫ is contained in an open ball of

radius ǫ, we have, cov(n, 2ǫ, Ā) ≥
π|λ|n

k(ǫ)ǫ2
. Thus, we have h(Ā) ≥ log |λ|.

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Dynamics of real projective transformations

Similarly, since the diameter of an ǫ dn-ball in P1(R) is at least
2ǫ
|λ|n

, P1(R)

can be covered by π2ǫ
|λ|n

number of arcs. Hence, cov(n, 2ǫ, Ā) ≤
π|λ|n

2ǫ
. So,

h(Ā) ≤ log |λ|.
Thus, we conclude that h(Ā) = log |λ|. �

In [3], the authors have also given a proof of the fact that the entropy of a
hyperbolic toral automorphism induced by a matrix A on an n−dimensional

torus, T n is equal to
m∑

i=1

log|αi|, where α1, α2, . . . , αm are those eigenvalues

of A that have modulus strictly larger than 1 (See Proposition 2.6.4 in [3]).
The proof relies on the idea of decomposing Rn in to generalised eigenspaces
of A and is similar to the proof of the corresponding result on 2−dimensional
torus, mentioned above (Proposition 2.5). On the same lines, it is hoped that
a result similar to Theorem 2.6 can be obtained for projective transformations
on higher dimensional projective spaces also.

3. Zeta function

The zeta function collects combinatorial information about the periodic
points. We follow [3] for the definition and other basic facts of the zeta
function. For a dynamical system (X, f), if |Fix(fk)| is finite for every k,
we define the zeta function ζf (z) of f to be the formal power series ζf (z) =
exp(

∑∞
k=1

1
k
|Fix(fk)|zk). The zeta function can also be expressed by a product

formula. Let P(f) denote the collection of all periodic orbits of f i.e., a typical
element of P(f) will be {x0, f(x0), . . ., f

k−1(x0)}, where k is the least period
of x0. Now, the zeta function of f can be written as ζf (z) =

∏

γ∈P(f)

(1 − z|γ|)−1

where |γ| is the number of elements in γ.
We use the following lemma in proving Theorem 3.2.

Lemma 3.1. If µ is a non-zero eigenvalue of Ak for some k ∈ N such that
there is a unique eigenvalue λ of A with λk = µ, then the eigenspaces of Ak

and A corresponding to µ and λ respectively, are same.

The lemma follows easily from the facts that, under the assumptions of
the hypothesis, the number of Jordan blocks in the Jordan normal form of
A corresponding to λ is same as the number of Jordan blocks in the Jordan
normal form of Ak corresponding to µ.

Theorem 3.2. Let Ā be a projective transformation on Pn(R) induced by a
matrix A ∈ GLn+1(R). Ā possesses zeta function if and only if each eigenspace
of A is one-dimensional and no two eigenvalues have same absolute value. In
such case, the zeta function is given by ζf (z) =

1
(1−z)l

.

Proof. Suppose that each eigenspace of A is one-dimensional and no two eigen-
values of A have same absolute value. If π(v) is a periodic point of Ā where
v ∈ S1, then v is an eigenvector of Ak for some k ∈ N, say Akv = µv, µ ∈ R.
Then there is an eigenvalue λ of A such that λk = µ. If λ1 and λ2 are two

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S. Gopal and S. Ravulapalli

different eigenvalues of A such that λk1 = λ
k
2 = µ, then |λ1| = |λ2|, contrary

to the hypothesis. Thus, λ is unique. Hence, by the above lemma, v lies in
the eigenspace of A corresponding to λ. Thus v̄ is a fixed point of Ā i.e., fixed
points are the only periodic points. In other words, Fix(Āk) = Fix(Ā) for any
k. Further, since each eigenspace of A is one-dimensional, there are as many

fixed points as the eigenvalues. Thus, ζf (z) = exp(l
∑∞

k=1
z
k

k
) = 1

(1−z)l
, where

l is the number of eigenvalues of A.
Conversely, suppose Ā possesses zeta function. Then there should be finitely

many fixed points and thus each eigenspace should be one dimensional. If
possible, suppose there are two different eigenvalues λ1 and λ2 such that |λ1| =
|λ2|. Since λ1 and λ2 are real, λ

2
1 = λ

2
2; say µ = λ

2
1. Then µ is an eigenvalue of

A2. If v1 and v2 are eigenvectors of A corresponding to λ1 and λ2 respectively,
then v1 and v2 are eigenvectors of A

2 corresponding to the same eigenvalue µ.
So, the dimension of eigenspace of A2 corresponding to µ is greater than 1 and
thus there are infinitely many periodic points of Ā with period 2, implying that
the zeta function doesn’t exist, contradicting the hypothesis. �

Acknowledgements. The authors thank the referee for his suggestions. The

first author acknowledges the financial support received under the Research

Initiation Grant provided by BITS-Pilani. The second author thanks UGC,

India for receiving the financial support as a UGC - Senior Research Fellow.

References

[1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Transactions
of the American Mathematical Society 114 (1965), 309–319.

[2] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Transactions of
the American Mathematical Society 153 (1971), 401–414.

[3] M. Brin and G. Stuck, Introduction to dynamical systems, Cambridge University Press
(2004).

[4] S. G. Dani, Dynamical properties of linear and projective transformations and their
applications, Indian J. Pure Appl. Math. 35 (2004), 1365–1394.

[5] R. Devaney, An introduction to chaotic dynamical systems, Second edition, Addison-
Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1989.

[6] N. H. Kuiper, Topological conjugacy of real projective transformations, Topology 15
(1976), 13–22.

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