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@ Appl. Gen. Topol. 17, no. 2(2016), 123-127doi:10.4995/agt.2016.4555
c© AGT, UPV, 2016

A note on uniform entropy for maps having

topological specification property

Sejal Shah a, Ruchi Das b and Tarun Das b

a
Department of Mathematics, Faculty of Science, The Maharaja Sayajirao University of Baroda,

Vadodara - 390002, Gujarat, India. (sks1010@gmail.com)
b

Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi -

110007, India. (rdasmsu@gmail.com, tarukd@gmail.com)

Abstract

We prove that if a uniformly continuous self-map f of a uniform space

has topological specification property then the map f has positive uni-

form entropy, which extends the similar known result for homeomor-

phisms on compact metric spaces having specification property. An

example is also provided to justify that the converse is not true.

2010 MSC: 37B40; 37B20.

Keywords: topological specification property; uniform entropy; uniform
spaces.

1. Introduction

In [1], authors have defined and studied the notion of topological entropy
of a continuous self-map f of a compact topological space as an analogue of
the measure theoretic entropy. The relation between topological entropy and
measure theoretic entropy is established by the variational principle, which
asserts that h(T ) = sup{hµ(T )|µ ∈ PT (X)}, i.e., topological entropy equals
the supremum of the measure theoretic entropies hµ(T ), where µ ranges over
all T -invariant Borel probability measures on X [10].

A dynamical system is called deterministic if its topological entropy vanishes
[9]. One may argue that the future of a deterministic dynamical system can
be predicted if its past is known [16]. In a similar way positive entropy may

Received 18 January 2016 – Accepted 21 July 2016

http://dx.doi.org/10.4995/agt.2016.4555


S. Shah, R. Das and T. Das

be related to randomness and chaos. Topological entropy also plays an impor-
tant role as an invariant for classification of continuous maps up to conjugacy.
A remarkable contribution of Bowen is the definition of entropy for uniformly
continuous self-maps on metric spaces [4]. Authors in ([11], [12], [14]) have
extended Bowen’s definition of entropy to uniformly continuous self-maps on
uniform spaces. The uniform entropy considered in this paper is the one intro-
duced in [12]. For details on uniform entropy on a uniform space one can also
refer [8]. The survey [8] is also devoted to the study of entropy for topological
groups wherein the uniform entropy is used to define topological entropy for
continuous endomorphisms of locally compact groups. For a review of entropy
in various fields of mathematics and science one can refer [2].

Recently in [7], authors have extended Smale’s spectral decomposition the-
orem for Anosov diffeomorphisms of compact manifolds to general topological
spaces wherein they have extended the notions of expansivity and shadowing
for general topological spaces. Sensitivity and Devaney chaos are also studied
for continuous group actions on uniform spaces in [6].

Let X be a non-empty set and let △X = {(x, x)|x ∈ X}, called as the
diagonal of X × X. A subset M of X × X is said to be symmetric if M = MT ,
where MT = {(y, x)|(x, y) ∈ M}. The composite U ◦V of two subsets U and V
of X × X is defined to be the set {(x, y) ∈ X × X| there exists z ∈ X satisfying
(x, z) ∈ U and (z, y) ∈ V }.

Definition 1.1 ([13]). Let X be a non-empty set. A uniform structure on X
is a non-empty set U of subsets of X × X satisfying the following conditions:

i: if U ∈ U then △X ⊂ U,
ii: if U ∈ U and U ⊂ V ⊂ X × X then V ∈ U,
iii: if U ∈ U and V ∈ U then U ∩ V ∈ U,
iv: if U ∈ U then UT ∈ U,
v: if U ∈ U then there exists V ∈ U such that V ◦ V ⊂ U.

The elements of U are then called the entourages of the uniform structure and
the pair (X, U) is called a uniform space.

We work here with uniform space (X, U). Note that the fact that points
x and y are close (in terms of distance) in a metric space X is equivalent to
the fact that point (x, y) is close to the diagonal △X of X × X in a uniform
space (X, U). If (X, U) is a uniform space, then there is an induced topology
on X characterized by the fact that the neighborhoods of an arbitrary point
x ∈ X consists of the sets U[x], where U varies over all entourages of X. The
set U[x] = {y ∈ X|(x, y) ∈ U} is called the cross section of U at x ∈ X. We

denote the subsets of X × X by U and the subsets of X by U̇.
The specification property for homeomorphisms on a compact metric space

has turned out to be an important notion in the study of dynamical systems.
It was first introduced by Bowen to give the distribution of periodic points for
Axiom A diffeomorphisms [5]. Informally the specification property means that
it is possible to shadow two distinct pieces of orbits which are sufficiently apart
in time by a single orbit. Let f be a self homeomorphism on a compact metric

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Topological specification property and uniform entropy

space (X, d). The map f is said to satisfy specification property if for every
ǫ > 0 there exists a positive integer M(ǫ) such that for any finite sequence
x1, x2, ..., xk in X, any integers a1 ≤ b1 < a2 ≤ b2 < ... < ak ≤ bk with
aj − bj−1 ≥ M (2 ≤ j ≤ k) and p > M + (bk − a1), there exists x ∈ X such
that fp(x) = x and d(fi(x), fi(xj)) < ǫ (aj ≤ i ≤ bj, 1 ≤ j ≤ k).

In [3], it has been proved that on a compact metric space X if f : X → X is a
homeomorphism satisfying specification property then the topological entropy
of f is positive. In section 2, we extend this result to uniformly continuous
self-maps defined on uniform spaces. In particular, we prove that a uniformly
continuous map defined on a uniform space having topological specification
property has positive uniform entropy.

2. Main Result

In [15], we have introduced the notion of topological specification property
for a homeomorphism on a uniform space however the notion of topological
specification property can be defined for continuous maps as follows:

Definition 2.1. A continuous self-map f is said to have topological specifi-
cation property if for every symmetric neighborhood U of the diagonal △X
there exists a positive integer M such that for any finite sequence of points
x1, x2, ..., xk in X, any integers 0 = a1 ≤ b1 < a2 ≤ b2 < ... < ak ≤ bk with
aj − bj−1 ≥ M (2 ≤ j ≤ k) and any p > M + (bk − a1), there exists x ∈ X such
that fp(x) = x and F i(x, xj) ∈ U, aj ≤ i ≤ bj, 1 ≤ j ≤ k.

Remark 2.2. If (X, d) is a compact metric space, then for any neighborhood U
of △X, we can find ǫ > 0 such that Uǫ = d

−1(0, ǫ) ⊂ U. On the other hand,
every Uǫ is a neighborhood of △X. Thus for compact metric spaces, definitions
of topological specification property and of specification property coincide.

Let f : X → X be a uniformly continuous map, Us denote the set of all
symmetric elements of U, Uo denote the set of open elements of U, n be a
positive integer and let U ∈ Us. A subset Ė of X is said to be (n, U)-separated

with respect to f if for each pair of distinct points x, y in Ė there exists j such
that 0 ≤ j < n and F j(x, y) 6∈ U, where F = f × f. For a subset Ż of X,

a subset Ȧ of X is said to be an (n, U)-spanning set for Ż with respect to f

if for each x ∈ Ż there exists y ∈ Ȧ such that for all j with 0 ≤ j < n, we
have F j(x, y) ∈ U. Let K(X) denote the set of all compact subsets of X. For

K̇ ∈ K(X), let sn(U, K̇, f) denote the maximal cardinality of (n, U)-separated

sets contained in K̇ and let rn(U, K̇, f) denote the minimal cardinality of (n, U)-

spanning sets for K̇. Note that a maximal (n, U)-separated subset of K̇ is an

(n, U)-spanning set for K̇. Define

rf (U, K̇) = lim sup
n→∞

1
n
logrn(U, K̇, f)

sf (U, K̇) = lim sup
n→∞

1
n
logsn(U, K̇, f)

and

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S. Shah, R. Das and T. Das

h(f, K̇, U) = lim{rf (U, K̇)|U ∈ U
s}

= lim{rf (U, K̇)|U ∈ U
o}

= lim{sf (U, K̇)|U ∈ U
s}

= lim{sf (U, K̇)|U ∈ U
o} .

For the proofs of above equalities one can refer Lemma 1 in [12].

Definition 2.3 ([12]). The number h(f, U) defined by sup{h(f, K̇, U)|K̇ ∈
K(X)} is called the uniform entropy of f with respect to the uniformity U.

Remark 2.4. For a uniformly continuous self-map of a complete metric space,
the uniform entropy is equal to the Bowen’s entropy [14].

Theorem 2.5. Let (X, U) be a uniform space and let f : X → X be a uniformly
continuous map having topological specification property then h(f, U) is positive.

Proof. Let x, y ∈ X, x 6= y, U be a symmetric neighborhood of △X such that
(x, y) 6∈ U2 = U ◦ U and M be a number as in the definition of topological
specification property.

Consider two distinct (n + 1)-tuples, (z0, z1, z2, ..., zn) and (z
′
0, z

′
1, z

′
2, ..., z

′
n)

with z0 = x, z
′
0 = y, zi, z

′
i ∈ {x, y}(1 ≤ i ≤ n) and integers a0 = b0 = 0,

a1 = b1 = M, a2 = b2 = 2M, ..., an = bn = nM. Since f has topological spec-
ification property, there exist z, z′ ∈ X satisfying the definition of topological
specification property. Note that z 6= z′. For if z = z′ then

F i(z, zi) ∈ U, aj ≤ i ≤ bj and 0 ≤ i, j ≤ n and
F i(z, z′i) ∈ U, aj ≤ i ≤ bj and 0 ≤ i, j ≤ n.

Therefore F iM (z, zi) ∈ U, 0 ≤ i ≤ n and F
iM (z, z′i) ∈ U, 0 ≤ i ≤ n. For i = 0,

(z, z0) ∈ U and (z, z
′
0) ∈ U implying that (x, y) ∈ U

2, which contradicts our
choice of U.

Using similar arguments, one can prove that for distinct (n + 1)-tuples
(z0, z1, z2, ..., zn) with zi ∈ {x, y}(0 ≤ i ≤ n) associated z are different. Thus
there are at least 2n+1 points which are (nM, U)-separated which implies

h(f, U) = sup{h(f, K̇, U)|K̇ ∈ K(X)}

≥ lim
U∈Us

sf (U, K̇)

= lim
n→∞

sup 1
n
logsn(U, K̇, f)

= lim
n→∞

sup 1
nM

log2n+1

= log2
M

> 0.

�

The following example justifies that the converse of the above result is not
true.

Example 2.6. Let f : R → R be defined by f(x) = 2x, for all x ∈ R, where
the set of real numbers R has the usual uniformity U. Note that the Bowen’s
entropy of f is log2 [11]. By Remark 2.4, h(f, U) = log2. It is known that a

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Topological specification property and uniform entropy

map f having topological specification property has the dense set of periodic
points [15]. Since f(x) = 2x does not have dense set of periodic points therefore
it does not have the topological specification property.

Acknowledgements. The second author is supported by UGC Major Re-
search Project F.N. 42-25/2013(SR)

References

[1] R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc.
114 (1965), 309–319 .

[2] J. Amigó, K. Keller and V. Unakafova, On entropy, entropy-like quantities, and appli-
cations, Discrete Contin. Dyn. Syst. Ser. B 20 (2015), 3301–3343.

[3] N. Aoki, Topological dynamics, Topics in general topology. Amsterdam, North-Holland
Publishing Co. 41 (1989), 625–740.

[4] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer.
Math. Soc. 153 (1971), 401–414.

[5] R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer.
Math. Soc. 154 (1971) 377–397.

[6] T. Ceccherini-Silberstein and M. Coornaert, Sensitivity and Devaney’s chaos in uniform
spaces, J. Dyn. Control Syst. 19 (2013), 349–357.

[7] T. Das, K. Lee, D. Richeson and J. Wiseman, Spectral decomposition for topologically
Anosov homeomorphisms on noncompact and non-metrizable spaces, Topology Appl.
160 (2013), 149–158.

[8] D. Dikranjan, M. Sanchis and S. Virili, New and old facts about entropy in uniform
spaces and topological groups, Topology Appl. 159 (2012), 1916–1942.

[9] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophan-
tine approximation, Math. Systems Theory 1 (1967), 1–49.

[10] T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London

Math. Soc. 3 (1971), 176–180.
[11] J. Hofer, Topological entropy for noncompact spaces, Michigan Math. J. 21 (1974),

235–242.
[12] B. Hood, Topological entropy and uniform spaces, J. London Math. Soc. 8 (1974), 633–

641.
[13] J. Kelley, General Topology, D. Van Nostrand Company (1955).
[14] T. Kimura, Completion theorem for uniform entropy, Comment. Math. Univ. Carolin.

39 (1998), 389–399.
[15] S. Shah, R. Das and T. Das, Specification property for topological spaces, J. Dyn.

Control Syst. 22 (2016), 615–622.
[16] B. Weiss, Single Orbit Dynamics, CBMS Regional Conference Series in Mathematics,

American Mathematical Society, Providence, RI 95 (2000).

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