@ Appl. Gen. Topol. 20, no. 1 (2019), 19-31doi:10.4995/agt.2019.8843
c© AGT, UPV, 2019

A note on measure and expansiveness on

uniform spaces

Pramod Das and Tarun Das

Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi-110007,

India (pramod.math.ju@gmail.com,tarukd@gmail.com)

Communicated by F. Balibrea

Abstract

We prove that the set of points doubly asymptotic to a point has mea-
sure zero with respect to any expansive outer regular measure for a
bi-measurable map on a separable uniform space. Consequently, we
give a class of measures which cannot be expansive for Denjoy home-
omorphisms on S1. We then investigate the existence of expansive
measures for maps with various dynamical notions. We further show
that measure expansive (strong measure expansive) homeomorphisms
with shadowing have periodic (strong periodic) shadowing. We relate
local weak specification and periodic shadowing for strong measure ex-
pansive systems.

2010 MSC: 54H20; 54E15.

Keywords: expansiveness; measure expansiveness; expansive measures;
equicontinuity; shadowing; specification.

1. Introduction

The fact that symbolic flows are expansive has been recognized for several
years and has provided the impetus for the study of expansive homeomorphisms
in a more general settings beginning with the Utz’s paper [17] in the middle of
the twentieth century. Evidently, mathematicians used the size of some parti-
cular subsets of the phase space to unleash many interesting phenomena of such
homeomorphisms. For instance, J. F. Jacobsen [10] proved that there exists no
expansive self-homeomorphism of a closed 2-cell by using the fact that the set

Received 29 October 2017 – Accepted 13 January 2019

http://dx.doi.org/10.4995/agt.2019.8843


P. Das and T. Das

of points doubly asymptotic to any fixed point of such a homeomorphism of a
compact metric space is at most countable. In recent years, A. Arbieto proved
[1] that the set of sinks of any homeomorphism with canonical coordinates has
measure zero with respect to any positively expansive measure by using the fact
that a set on which a measurable map of a separable metric space is Lyapunov
stable has measure zero with respect to any positively expansive measure. Fur-
ther, he proved that the set of heteroclinic points of a homeomorphism on a
compact metric space has measure zero with respect to any expansive measure
with the help of the fact that the set of points with converging semiorbits under
a homeomorphism of a compact metric space has measure zero with respect to
any expansive measure. Many of these results have been generalized [15] for
expansive measures for maps on non-compact, non-metrizable spaces.

Treading on the same path, in section 3, we conclude some results analogous
to that of expansive self-homeomorphisms on compact metric spaces by using
the fact that the set of points positively asymptotic to one point and negatively
asymptotic to another point has measure zero with respect to any expansive
outer regular measure for a bi-measurable map on a separable uniform space.
For instance, we give a class of measures which cannot be expansive for Denjoy
homeomorphisms on S1. We also prove that every stable class has measure zero
with respect to any positively expansive outer regular measure of a measurable
map on a separable uniform space. Then, we prove that the set of equicon-
tinuous homeomorphisms and the set of homeomorphisms admitting expansive
measure on a Lindelöf uniform space are disjoint. We prove that sink does not
exist for a bi-measurable map having canonical coordinates and strictly posi-
tive positively expansive measure. Finally, in section 4, we prove that measure
expansive (strong measure expansive) homeomorphisms with shadowing have
periodic (strong periodic) shadowing. We further relate local weak specification
and periodic shadowing for strong measure expansive systems.

2. Preliminaries

We denote the set of non-negative integers by N and the set of all integers by
Z. A point x ∈ X is called atom for a measure µ if µ({x}) > 0. A measure µ on
X is said to be non-atomic if it has no atom. Let us denote the set of all Borel
measures, the set of all non-atomic Borel measures and the set of all strictly
positive non-atomic Borel measures by M(X), NAM(X) and SPNAM(X)
respectively.

In 2013, A. Arbieto defined [1] the sets Φδ(x) = {y ∈ X | d(f
i(x), fi(y)) ≤ δ

for all i ∈ N} and Γδ(x) = {y ∈ X | d(f
i(x), fi(y)) ≤ δ for all i ∈ Z} for a

measurable and bi-measurable map respectively, on a metric space (X, d) to
introduce the following concepts.

Definition 2.1. An expansive (positively expansive) measure for a bi-measurable
(measurable) map f : X → X on a metric space (X, d) is a Borel measure µ
on X for which there is δ > 0 such that µ(Γδ(x)) (µ(Φδ(x))) = 0 for all x ∈ X.
In both the cases, such δ is called an expansive constant for µ.

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Expansive measures on uniform spaces

Observe that a positively expansive measure for a bi-measurable map is also
an expansive measure.

In 2015, C. A. Morales generalized [15] these concepts to uniform spaces
with the sole difference being the role of spherical neighborhoods now played by
uniform neighborhoods. To understand his generalization we need the following
terminology.

Let X be a non-empty set. Then the diagonal of X × X is given by ∆(X) =
{(x, x)|x ∈ X}. For a subset R of X × X, we define R−1 = {(y, x)|(x, y) ∈ R}.
We say that R is symmetric if R = R−1. For two subsets U and V of X ×X, we
define their composition as U ◦ V = {(x, y) ∈ X × X| there is z ∈ X satisfying
(x, z) ∈ U and (z, y) ∈ V }. In this paper, we assume that the phase space of a
dynamical system is a uniform space (X, U), where U is a collection of subsets
of X × X satisfying the following properties ([11],[12]):

(1) Every D ∈ U contains ∆(X).
(2) If D ∈ U and E ⊃ D, then E ∈ U.
(3) If D, D′ ∈ U, then D ∩ D′ ∈ U.
(4) If D ∈ U, then D−1 ∈ U.
(5) For every D ∈ U there is symmetric D′ ∈ U such that D′ ◦ D′ ⊂ D.

The members of U are called entourages of X. If (X, U) is a uniform space,
then we can generate a topology on X by characterizing that a subset Y ⊂ X
is open if and only if there exists U ∈ U such that for each x ∈ Y the cross
section U[x] = {y ∈ X | (x, y) ∈ U} is contained in Y .

Morales introduced [15] ΦD(x) = {y ∈ X | (f
i(x), fi(y)) ∈ D for all i ∈ N}

and ΓD(x) = {y ∈ X | (f
i(x), fi(y)) ∈ D for all i ∈ Z } for some D ∈

U to generalize the respective concepts of expansive and positively expansive
measures.

Definition 2.2. An expansive (positively expansive) measure for bi-measurable
(measurable) map f : X → X on a uniform space is a Borel measure µ on X
for which there exists D ∈ U such that µ(ΓD(x))(µ(ΦD(x))) = 0 for all x ∈ X.
In both the cases, such D ∈ U is referred to as expansive entourage for µ.

Remark 2.3. One can verify that Remark 3.2 [8] holds for all dynamical notions
in the present paper.

Definition 2.4 ([15]). We say that a bi-measurable (measurable) map f :
X → X on a uniform space X is expansive (positively expansive) if there
exists D ∈ U such that for any distinct x, y ∈ X there exists n ∈ Z (n ∈ N)
such that (fn(x), fn(y)) /∈ D.

Remark 2.5. (i) Any non-atomic Borel measure on a uniform space X is ex-
pansive (positively expansive) measure for any expansive (positively expan-
sive) map on X. Recall from [13] that if X has no perfect subset and ev-
ery Borel probability measure on X is net additive, then a Borel probability
measure is atomic. So, there exists no expansive (positively expansive) map

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P. Das and T. Das

with expansive (positively expansive) probability measures on a Lindelöf uni-
form space without any perfect subset. For instance, (a) consider X = R and
U = {Ut | 0 < t < ∞}, where Ut = {(x, y) ∈ R

2 | x2+y2 < t}∪{(x, x) | x ∈ R}.
Then, f : R → R given by f(x) = 2x is expansive homeomorphism. (b) con-
sider X = {1/n, 1−1/n | n ∈ N}. Then f : X → X given by f(0) = 0, f(1) = 1
and f(x) = x+ for all x 6= 0, 1 where x+ denotes the point immediate right to
x, is an expansive homeomorphism. But both of the homeomorphism have no
expansive measure. Thus, expansiveness of a homeomorphism neither imply
nor is implied by the existence of an expansive measure for the map.

(ii) If f : X → X is an expansive homeomorphism with expansive entourage
E, then X must be T1 space. Indeed, if x 6= y in X, then there exists n ∈ Z
such that y /∈ f−n(E[fn(x)]). On the other hand, Proposition 2.18 [15] shows
that there are homeomorphisms on non-T1 uniform spaces admitting expansive
measure.

(iii) In [15], authors proved that any expansive probability measure for a bi-
measurable map on a Lindelöf uniform space is non-atomic. Thus, the measure
of a countable set under such a measure is equal to zero.

(iv) Let f : X → X be a measurable map and µ an ergodic invariant proba-
bility measure for f. Then µ is positively expansive with expansive entourage
D if and only if µ(ΦD(x)) = 0 for all x ∈ E with µ(E) > 0. Indeed, if
ED = {x ∈ X | µ(ΦD(x)) = 0}, then by Lemma 2.10 [15] it is enough to
show that µ(ED) = 1. For fix x ∈ X, we have that ΦD(x) ⊂ f

−1(ΦD(f(x))).
Then by the fact that µ is invariant under f we have, µ(ΦD(x)) ≤ µ(ΦD(f(x)))
which implies that f−1(ED) ⊂ ED and hence µ(f

−1(ED) = µ(ED). Since µ is
ergodic, we have µ(ED) ∈ {0, 1}. But since E ⊂ ED we must have µ(ED) = 1.

Example 2.6. Let Z be the Sorgenfrey line, i.e. R equipped with the topology
based on the intervals of the form [x, y), where (x, y) ∈ R × Q with x < y.
(i) The map f : Z → Z given by f(x) = x for x ∈ Q and f(x) = 2x for
x ∈ R \ Q is not expansive but the Lebesgue measure is an expansive measure
for f.
(ii) The map f : Z → Z given by f(x) = 0 for x ∈ Q and f(x) = 2x for
x ∈ R \ Q is not positively expansive but the Lebesgue measure is a positively
expansive measure for f.

3. Expansive Measures

In [10], Jacobsen and Utz stated that the set of points doubly asymptotic
to a fixed point of an expansive homeomorphism on a compact metric space is
at most countable. In [18], R. Williams proved a stronger result which states
that the set of points doubly asymptotic to a given point of an expansive
homeomorphism on a compact metric space is at most countable. The purpose
of this section is to extend this result to expansive measures for bi-measurable

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Expansive measures on uniform spaces

maps on separable uniform space. We prove a stronger result and conclude our
result as a direct consequence.

Definition 3.1. Let f : X → X be a bi-measurable map on a uniform space
X and let x ∈ X be given. A point y ∈ X is said to be positively asymptotic to
x if for every E ∈ U there is a positive integer N such that (fn(x), fn(y)) ∈ E
for all n ≥ N. On the other hand, a point y ∈ X is said to be negatively
asymptotic to x if for every E ∈ U there is a positive integer M such that
(fn(x), fn(y)) ∈ E for all n ≤ −M. A point y ∈ X is said to be doubly
asymptotic to x if for every E ∈ U there is a positive integer N such that
(fn(x), fn(y)) ∈ E for all | n |≥ N.

Recall that a Borel measure µ on a topological space is outer regular if for
every measurable subset A and any ǫ > 0 there is an open subset O ⊃ A such
that µ(O \ A) < ǫ. The following lemma was proved in [15]. For the sake of
completeness we present it here.

Lemma 3.2. Let µ be a Borel measure on a topological space. Then for every
measurable Lindelöf subset K with µ(K) > 0 there are z ∈ K and an open
neighborhood U of z such that µ(K ∩ W) > 0 for every open neighborhood
W ⊂ U of z.

Proof. Otherwise, for every z ∈ K there is an open neighborhood Uz ⊂ U
satisfying µ(K ∩ Uz) = 0. Since K is Lindelöf, the open cover {K ∩ Uz :
z ∈ K} of K admits a countable sub-cover, i.e. there is a sequence {zl}l∈N
in K satisfying K =

⋃
l∈N(K ∩ Uzl). So, µ(K) =

∑
l∈N µ(K ∩ Uzl) = 0, a

contradiction. �

Theorem 3.3. The set of points positively asymptotic to p and negatively
asymptotic to q for given p, q ∈ X for a bi-measurable map f : X → X on a
separable uniform space has measure zero with respect to any expansive outer
regular measure µ of f.

Proof. Let p, q ∈ X be given. Let D be an expansive entourage for µ and D′ ∈
U be symmetric such that D′ ◦D′ ⊂ D. Let A be the set of all points positively
asymptotic and negatively asymptotic to p and q respectively. If AN = {x ∈
X | (fn(x), fn(p)) ∈ D′ for all n ≥ N and (fn(x), fn(q)) ∈ D′ for all n ≤ −N}.
It is easy to verify that A ⊂

⋃
N≥0 AN and each AN is measurable. We show

that µ(
⋃

N≥1 AN) = 0. If possible, suppose µ(
⋃

N≥1 AN) > 0. So, there is

M ≥ 1 such that µ(AM ) > 0.

Since X is separable uniform space, it is second countable and since µ is outer
regular, Lusin Theorem [9] implies that for every ǫ > 0 there is a measurable
subset Cǫ with µ(X \Cǫ) < ǫ such that f

i |Cǫ is continuous for all integer i with
| i |≤ M. Taking ǫ = µ(AM )/2 we obtain a measurable subset C = Cµ(AM )/2
such that fi |C continuous for all integer i with | i |≤ M and µ(AM ∩ C) > 0.

Since K = AM ∩C is Lindelöf, we can apply Lemma 3.2 to obtain z ∈ AM ∩C
and D0 ∈ U satisfying µ(AM ∩ C ∩ D1[z]) > 0 for all entourage D1 ⊂ D0...(*).

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P. Das and T. Das

Fix such a z and D0. Since z ∈ C and f
i |C is continuous for all | i |≤ M, we

can fix D∗ ∈ U with D∗ ⊂ D0 such that f
i(w) ∈ D[fi(z)] whenever | i |≤ M

and w ∈ C ∩ D∗[z].

Now, take w ∈ AM ∩ C ∩ D
∗[z]. Since w ∈ C ∩ D∗[z], we already have

fi(w) ∈ D[fi(z)] for all | i |≤ M. Since z ∈ AM ∩C, we have w, z ∈ AM . Hence,
(fi(w), fi(p)) ∈ D′ and (fi(z), fi(q)) ∈ D′ for all i ≥ M and (fi(w), fi(p)) ∈
D′ and (fi(z), fi(q)) ∈ D′ for all i ≤ −M. Since D′ is symmetric and D′ ◦D′ ⊂
D, we get (fi(z), fi(w)) ∈ D for all | i |≥ M, i.e., fi(w) ∈ D[fi(z)] for all | i |≥
M. Thus fi(w) ∈ D[fi(z)] for all i ∈ Z which implies AM ∩C ∩D

∗[z] ⊂ φD(z).
Since D is an expansive entourage for µ, we have µ(AM ∩ C ∩ D

∗[z]) = 0.
On the other hand, since D∗ ⊂ D0, we can take D1 = D

∗ in (*) to obtain
µ(AM ∩ C ∩ D

∗[z]) > 0, a contradiction. This completes the proof. �

Lemma 1 [2] shows that if f : X → X is an expansive homeomorphism on
a compact metric space, then A =

⋃
N≥0 AN but it is not true in general for

bi-measurable map admitting an expansive measure.

Example 3.4. Consider f : R → R given by f(x) = x for all x ∈ Q and
f(x) = 2x for all x ∈ R \ Q. The Lebesgue measure is an expansive measure
for f with any δ > 0 as expansive constant. Let p, q ∈ Q with d(p, q) < δ and
choose x = (p + q)/2. Then, x ∈ AN for all N ≥ 0 but x /∈ A.

We have the following corollary to Theorem 3.3 and the well-known fact that
every Borel probability measure on a metric space is outer regular.

Corollary 3.5. The set of points positively asymptotic and negatively asymp-
totic to two given points under a homeomorphism f : X → X of a separable
metric space has measure zero with respect to any expansive probability measure.

Theorem 3.6 ([14]).

(i) There exists no self-homeomorphism on [0, 1] admitting an expansive
measure.

(ii) A self-homeomorphism of S1 admits expansive measure if and only if
it is Denjoy.

The following corollary significantly cuts down the collection of expansive
measures for Denjoy homeomorphisms on S1.

Corollary 3.7. An outer regular measure µ ∈ SPNAM(X) cannot be expan-
sive for a Denjoy homeomorphism on S1.

Proof. Suppose by contradiction that an outer regular measure µ ∈ SPNAM(X)
is expansive for a Denjoy homeomorphism f on S1 with an expansive constant
δ. As is well known, f exhibits unique minimal set M which is infinite, totally
disconnected and has no isolated point. Since S1 \ M is an open subset of S1,
it can be written as disjoint union of countably many open arcs, say {Ij} so

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Expansive measures on uniform spaces

that diam(Ij) → 0 as j → ∞ and for fixed j, f
n(Ij) 6= Ij for all n 6= 0 because

of Theorem 3.6(i). Thus, we have diam(fn(Ij)) → 0 as | n |→ ∞. For a, b ∈ Ij,

let fn0(ab) be the longest length in {fn(ab) | n ∈ Z}. Take a, b ∈ Ij such that

the length between fn0(a) and fn0(b) is less than δ. Then, diam(fn(ab)) < δ

for all n ∈ Z. Since µ(ab) > 0, it follows that δ is not expansive constant for
µ, a contradiction. �

As in [7], let the stable set of x ∈ X is

W s(x) = {y ∈ X | ∀B ∈ U, ∃n ∈ N such that (fi(x), fi(y)) ∈ B for all i ≥ n}

and in the invertible case, the unstable set of x ∈ X is

W u(x) ={y ∈ X | ∀B ∈ U, ∃n ∈ N such that (fi(x), fi(y)) ∈ B for all

i ≤ −n}.

If x ∈ X, then for E ∈ U, we define the local stable set

W s(x, E) = {y ∈ X | (fn(x), fn(y)) ∈ E, ∀n ∈ N}

and in the invertible case, the local unstable set

W u(x, E) = {y ∈ X | (f−n(x), f−n(y)) ∈ E, ∀n ∈ N}.

Definition 3.8. A point x ∈ X is called heteroclinic point for a map f : X →
X if x ∈ W s(O(p)) ∩ W u(O(q)) for some periodic points p, q ∈ X.

Corollary 3.9. The set of heteroclinic points of a bi-measurable map f : X →
X of separable uniform space, with at most countably many periodic points has
measure zero with respect to any expansive outer regular measure µ of f.

Proof. From Theorem 3.3, it follows that W s(x) ∩ W u(y) for any x, y ∈ X has
measure zero and then the result follows from the fact that the countable union
of measure zero set has measure zero. �

Next example shows that the set of heteroclinic points of a bi-measurable
map with more than countably many periodic points and an expansive measure
on a separable uniform space may have measure zero.

Example 3.10. Let C be the Cantor set in [−1, 0] and X = C ∪(0, ∞) and let
f : X → X be given by f(x) = x for all x ∈ C and f(x) = 2x for all x ∈ (0, ∞).
The Lebesgue measure is expansive measure for f.

Theorem 3.11. The set of points doubly asymptotic to a given point for a
bi-measurable map f : X → X on a separable uniform space has measure zero
with respect to any expansive outer regular measure µ of f.

Proof. It follows in the same fashion if we take p = q in Theorem 3.3. �

Note that any non-atomic Borel probability measure is an expansive measure
for an expansive homeomorphism of a compact metric space. So, Theorem 3.11
implies that the set of points doubly asymptotic to a given point under an ex-
pansive homeomorphism has measure zero with respect to any non-atomic Borel

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P. Das and T. Das

probability measure. From this and well-known measure theoretical results [16],
we obtain that the set of points doubly asymptotic to a given point under an
expansive homeomorphism is at most countable, which is the fact used by R.
Williams [18] to prove that there exists no expansive self-homeomorphisms on
a closed 2-cell.

Corollary 3.12. Let X be a separable uniform space and f : X → X be a
bi-measurable map with an expansive measure µ ∈ SPNAM(X). Then, for
each x ∈ X and each neighbourhood N of x there exists y ∈ N such that x and
y are not doubly asymptotic.

One can prove the following theorem in same fashion as in the proof of
Theorem 3.3.

Theorem 3.13. Every stable class of a measurable map f on a separable uni-
form space has measure zero with respect to any positively expansive outer reg-
ular measure µ of f.

Corollary 3.14. Let X be a separable uniform space and µ ∈ SPNAM(X)
be a positively expansive measure of a measurable map f on X. Then, for each
x ∈ X and each neighbourhood N of x there exists y ∈ X such that x and y
are not asymptotic.

Definition 3.15. Let X be a uniform space. A homeomorphism (continuous
map) f : X → X is called equicontinuous (positively equicontinuous) if for
each x ∈ X and every E ∈ U there exists Dx ∈ U (depends on x) such that
(x, y) ∈ Dx implies (f

n(x), fn(y)) ∈ E for all n ∈ Z (n ∈ N).

Remark 3.16. Observe that Corollary 3.11 and Corollary 3.14 are obvious gen-
eralizations of Theorem 3 [2] of B. F. Bryant. From these corollaries it is clear
that an equicontinuous (positively equicontinuous) map on a separable uniform
space cannot have any expansive (positively expansive) measure. We now give
a direct proof of this result for maps on Lindelöf uniform spaces.

Theorem 3.17. Let f : X → X be equicontinuous (positively equicontinu-
ous) map on a Lindelöf uniform space X. Then, f cannot have any expansive
(positively expansive) measure.

Proof. Suppose by contradiction that f has an expansive measure µ with ex-
pansive entourage D. Then, for each x ∈ X there exists Dx ∈ U such that
(x, y) ∈ Dx implies (f

n(x), fn(y)) ∈ D for all n ∈ Z. Thus, Dx[x] ⊂ ΓD(x).
Since D is expansive entourage for µ, µ(Dx[x]) ≤ µ(ΓD(x)) = 0 for all x ∈ X.
Now, {Dx[x] | x ∈ X} is an open cover for X and since X is Lindelöf there
is {xi}i∈N such that {Dxi[xi] | i ∈ N} is an open covering for X. Therefore,
µ(X) ≤ Σi∈Nµ(Dxi[xi]) which implies µ(X) = 0, which is not the case. The
proof for positively equicontinuous map is similar. �

Example 3.18. The map f : (0, 1) → (0, 1) given by f(x) = xn for some
n ∈ N+, the rotation of the unit circle, a contraction [15] on any Lindelöf

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Expansive measures on uniform spaces

uniform space are equicontinuous and therefore, they cannot have expansive
measure.

Thus, the two classes of homeomorphisms important in topological dynamics
namely equicontinuous homeomorphisms and homeomorphisms admitting an
expansive measure are disjoint. On the other hand, the identity map on a
discrete uniform space is both expansive and equicontinuous.

Definition 3.19. Let f : X → X be a map on a uniform space X and let
D, E ∈ U be given. Then, a sequence {xi}i∈N is said to be D-pseudo orbit if
(f(xi), xi+1) ∈ D for all i ∈ N. A sequence {xi}i∈N is said to be E-shadowed
by some point x ∈ X if (fi(x), xi) ∈ E for all i ∈ N.

(i) f is said to have shadowing if for every E ∈ U there exists D ∈ U such
that every D-pseudo orbit is E-shadowed by some point x ∈ X.

(ii) f is said to have h-shadowing if for every E ∈ U there exists D ∈ U such
that if {x0, x1, ..., xm} satisfy (f(xi), xi+1) ∈ D for 0 ≤ i ≤ (m − 1),
then there exists x ∈ X such that (fi(x), xi) ∈ E for 0 ≤ i ≤ (m − 1)
and fm(x) = xm.

Corollary 3.20. Let X be a uniform space and let f : X → X be a bi-
measurable map which satisfies either one of the following.

(i) f has h-shadowing.
(ii) f is positively expansive and has shadowing.

Then f−1 cannot have positively expansive measure.

Proof. (i) Let E ∈ U be given and let D ∈ U be given for E by the h-shadowing
of f. Let x ∈ X be fixed and y be such that (x, y) ∈ D. For any fixed natural
number m > 0 the finite sequence {f−m(x), f−m+1(x), ..., f−1(x), y} is a D-
chain. Hence, by h-shadowing there is z ∈ X such that (fi(z), f−m+i(x)) ∈ E
for 0 ≤ i ≤ (m − 1) and fm(z) = y which implies ((f−1)m(y), (f−1)m(x)) =
(f−m(x), z) ∈ E. This shows that f−1 is positively equicontinuous and hence,
by Theorem 3.17 it cannot have positively expansive measure.

(ii) Let E ∈ U be expansive entourage for f and D ∈ U with D ⊂ E be given
for E by the shadowing of f. Fix a finite D-chain {x0, x1, x2, x3, ..., xm} and
extend to a D-pseudo orbit {x0, x1, x2, x3, ..., xm, f(xm), f

2(xm), f
3(xm), ...}.

If z ∈ X is a point which E-shadows the above D-pseudo orbit then (fi(z), xi) ∈
E for i < m and (fj+m(z), fj(xm)) ∈ E for all j ≥ 0 which implies (f

i(z), xi) ∈
E for i < m and (fj(fm(z)), fj(xm)) ∈ E for all j ≥ 0. Hence, (f

i(z), xi) ∈ E
for i < m and fm(z) = xm. This shows that f has h-shadowing. Then by (i),
f−1 cannot have positively expansive measure. �

Definition 3.21. Let f : X → X be a bijective map on a uniform space
X. Then, a point x ∈ X is called sink [15] (source) for f if W u(x, D) =
{x} (W s(x, D) = {x}) for some entourage D. We say that f has canonical
coordinates if for every entourage E there is an entourage D such that (x, y) ∈
D implies W s(x, E) ∩ W u(y, E) 6= φ.

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P. Das and T. Das

Proposition 3.22. Sink does not exist for a bi-measurable map f : X →
X with canonical coordinates admitting a positively expansive measure µ ∈
SPNAM(X).

Proof. Let x ∈ X be a sink and let E ∈ U be an expansive entourage for the
positively expansive measure µ. Then by Lemma 4.6 [15], there exists D ∈ U
such that D[x] ⊂ W s(x, E). But W s(x, E) has measure zero and hence, D[x]
must have measure zero. This is a contradiction to the fact that the measure
is strictly positive. Thus, there does not exist any sink for such f. �

4. Measure Expansive Systems

In this section, we study measure expansive and strong measure expansive
homeomorphisms on uniform spaces. For the metric definitions of the following
notions reader may see in [6].

Definition 4.1. Let X be a uniform space and f : X → X be a homeomor-
phism. Let D, E ∈ U be a given entourage. Then,

(i) A sequence {xi}i∈Z is said to be D-pseudo orbit if (f(xi), xi+1) ∈ D for
all i ∈ Z. A D-pseudo orbit is said to be periodic with period N ≥ 1 if
xi+N = xi for all i ∈ Z. A sequence {xi}i∈Z is said to be E-shadowed
by some point x ∈ X if (fi(x), xi) ∈ E for all i ∈ Z.

(ii) f is said to have shadowing if for every E ∈ U there exists D ∈ U such
that every D-pseudo orbit is E-shadowed by some point in X.

(iii) f is said to have periodic shadowing if for every E ∈ U there exists
D ∈ U such that every periodic D-pseudo orbit is E-shadowed by some
periodic point in X.

(iv) f is said to have strong periodic shadowing if for every E ∈ U there
exists D ∈ U such that every periodic D-pseudo orbit with period N
is E-shadowed by some periodic point with period N.

(v) f is said to have local weak specification if for every E ∈ U there exists
M ≥ 1 and D ∈ U such that if {x0, x1, x2, ..., xk} satisfy (f

n(xi), xi+1) ∈
D for some n ≥ M, then there exists x ∈ X such that (fj+in(x), fj(xi)) ∈
E for 0 ≤ i ≤ (k − 1) and 0 ≤ j < n.

Definition 4.2. Let X be a uniform space and f : X → X be a homeomor-
phism. Then,

(i) f is said to be measure expansive if there exists D ∈ U such that
for every invariant non-atomic probability measure µ on X, we have
µ(ΓD(x)) = 0 for all x ∈ X. Such D is called measure expansive
entourage.

(ii) f is said to be strong measure expansive if there exists D ∈ U such that
for every invariant probability measure µ on X, we have µ(ΓD(x)) =
µ({x}) for all x ∈ X. Such D is called strong measure expansive
entourage.

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Expansive measures on uniform spaces

Theorem 4.3. Let X be a locally compact, para-compact, Hausdorff uniform
space and let f : X → X be an uniform equivalence. Then, f has local weak
specification if and only if it has shadowing.

Proof. Suppose that f has local weak specification. Let E ∈ U be symmetric
and let M ≥ 1, D ∈ U be given by the local weak specification of f. By para-
compactness, we can assume that E be such that E[K] is compact for compact
subset K ⊂ X. Let {xi}i∈Z be a sequence such that (f

n(xi), xi+1) ∈ D for
some n ≥ M and for all i ∈ Z. Then for each k ≥ 1, there exists yk such that
(fj+in(yk), f

j(xi)) ∈ E for 0 ≤| i |≤ (k −1) and 0 ≤ j ≤ (n−1). Thus for each
k ≥ 1, yk ∈ E[x0]. By local compactness, there exists a compact neighborhood
K of x0 such that yk ∈ E[K], which is compact. Therefore, {yk}k∈N has a
convergent subsequence converging to some point, say y in X. By continuity
of f, we have (fj+in(y), fj(xi)) ∈ E for all i ∈ Z and 0 ≤ j ≤ (n − 1). This
means that fn has shadowing for each n ≥ M and similarly as in Proposition
3.4(a) [8], we conclude that f has shadowing. On the other hand, by definition
shadowing implies local weak specification with N = 1. �

Theorem 4.4. Let X be a first countable, Hausdorff uniform space and f :
X → X a measure expansive homeomorphism. If f has shadowing, then it has
periodic shadowing.

Proof. Let E′′ ∈ U be a measure expansive entourage for f. Let E ∈ U be
closed, symmetric such that E2 ⊂ E′′. Further, let E′ ∈ U be symmetric such
that E′2 ⊂ E. Let D be given for E′ by the shadowing of f. If {xi}i∈Z is
a periodic D-pseudo orbit with period N, then there exists x ∈ X such that
(fi(x), xi) ∈ E

′ for all i ∈ Z. If x is periodic, then nothing to show. Suppose
that x is not periodic.

Let us fix 0 ≤ l < N and k ∈ Z.

Since xi+l+kN = xi+l, (f
i(fl(x)), xi+l) ∈ E

′ and (xi+l, f
i(fkN (fl(x))) ∈ E′

for all i ∈ Z
we have (fi(fl(x)), fi(fkN(fl(x)))) ∈ E′2 ⊂ E for all i ∈ Z.

Thus, fkN(fl(x)) ⊂ ΓE(f
l(x)) for every k ∈ Z.

Thus, the closure of the orbit of x is contained in
⋃N−1

l=0 ΓE(f
l(x)). If µ is a

weak accumulation point of the uniform distribution supported on the finite or-

bit {x, f(x), f2(x), ..., fN−1(x)}, then it is invariant and µ(
⋃N−1

l=0 ΓE(f
l(x))) =

1. Hence, µ(ΓE(f
l(x))) > 0 for some 0 ≤ l < N. Since f is measure expansive,

µ must be atomic. Thus there exists z ∈
⋃N−1

l=0 ΓE(f
l(x)) such that µ({z}) > 0.

Since µ is invariant probability measure, z must be periodic.

Let z ∈ ΓE(f
l(x)) for some 0 ≤ l < N. Then,

(fi(x), fi(f−l(z))) = (fi(x), fi−l(z)) ∈ E and also, (fi(x), xi) ∈ E
′ ⊂ E for

all i ∈ Z.
Therefore, (fi(f−l(z)), xi) ∈ E

2 for all i ∈ Z.

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P. Das and T. Das

This completes the proof. �

Example 4.5.

(i) Similarly as in Theorem A [5], we can produce N-expansive homeomor-
phisms with shadowing. Such homeomorphisms are clearly measure
expansive and therefore, has periodic shadowing.

(ii) Let f be an expansive homeomorphism with shadowing. Then, one can
verify that the N-expansive homeomorphisms constructed in Proposi-
tion 2.18 [15] have shadowing. Since such homeomorphisms are mea-
sure expansive, they have periodic shadowing.

Corollary 4.6. Let X be a first countable, locally compact, para-compact,
Hausdorff uniform space and f : X → X a strong measure expansive uni-
form equivalence with Per(f) 6= φ. If f has local weak specification then it has
periodic shadowing.

Proof. It follows from Theorem 4.3 and Theorem 4.4. �

Lemma 4.7. Let X be a uniform space and f : X → X a strong measure
expansive homeomorphism such that Per(f) 6= φ. Then, f |P er(f) is expansive.

Proof. Let D ∈ U be a strong measure expansive entourage for f. If Per(f) is
a single point, then we have nothing to show. Therefore, assume that Per(f)
contains more than one point. If possible, suppose there exists x 6= y in Per(f)
such that y ∈ ΓD(x). Let µ be an invariant probability measure such that
µ(x) > 0 and µ(y) > 0. Thus, µ(ΓD(x)) ≥ µ(x)+µ(y) > µ(x), a contradiction.
So, f |P er(f) is expansive with expansive entourage D. �

Lemma 4.8. Let X be a uniform space and f : X → X a homeomorphism
such that Per(f) = φ. Then, f is measure expansive if and only if it is strong
measure expansive.

Hint: An invariant atomic probability measure for f must be supported on
periodic points.

Theorem 4.9. Let X be a first countable, Hausdorff uniform space and f :
X → X a strong measure expansive homeomorphism with Per(f) 6= φ. If f
has shadowing, then it has strong periodic shadowing.

Proof. By Theorem 4.4, f has periodic shadowing. Let E′ ∈ U be a strong
measure expansive entourage for f. Let E ∈ U be symmetric such that E2 ⊂ E′

and let D ∈ U be given for E by shadowing of f. Let {xi}i∈Z be a D-pseudo
orbit with period N ≥ 1. Then, by periodic shadowing there exists x ∈ Per(f)
such that

(fi(x), xi) ∈ E for all i ∈ Z.

Since xi+N = xi for all i ∈ Z, we have (xi, f
i(fN(x))) = (xi+N , f

i+N(x)) ∈
E for all i ∈ Z. Therefore, (fi(x), fi(fN(x))) ∈ E2 ⊂ E′ for all i ∈ Z. Since

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Expansive measures on uniform spaces

by Lemma 4.7, f/Per(f) is expansive with expansive entourage E′, we must
have fN(x) = x. This completes the proof. �

Example 4.10. Let Z be the Sorgenfrey line. Then, the map f : Z → Z given
by f(x) = 2x is strong measure expansive homeomorphism with shadowing.
Thus, f has strong periodic shadowing.

Acknowledgements. The first author is supported by the Department of Sci-
ence and Technology, Government of India, under INSPIRE Fellowship (Reg-
istration No- IF150210) Program since March 2015.

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