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13Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 2, pp. 297-307. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220210
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 37C50, 28C15
On Almost Shadowable Measures
K. Lee, A. Rojas
In this paper we study the almost shadowable measures for homeomorphisms on compact metric spaces. First, we give examples of measures that are not shadowable. Next, we show that almost shadowable measures are weakly shadowable, namely, that there are Borelians with a measure close to 1 such that every pseudo-orbit through it can be shadowed. Afterwards, the set of weakly shadowable measures is shown to be an FaS subset of the space of Borel probability measures. Also, we show that the weakly shadowable measures can be weakly* approximated by shadowable ones. Furthermore, the closure of the set of shadowable points has full measure with respect to any weakly shadowable measure. We show that the notions of shadowableness, almost shadowableness and weak shadowableness coincide for finitely supported measures, or, for every measure when the set of shadowable points is closed. We investigate the stability of weakly shadowable expansive measures for homeomorphisms on compact metric spaces.
Keywords: Shadowable measures, Shadowable points, Homeomorphism.
1. Introduction
The shadowable measures (or measures with the pseudo-orbit tracing property) were introduced in [6] as measure-theoretical counterpart of the classical shadowing property [3]. See also [2, 9, 10, 14, 16] for further notions of measure shadowableness. Some properties were obtained, for example, the property that every shadowable expansive measure is topologically
Received March 25, 2022 Accepted April 24, 2022
Work supported by the Basic Science Research Program through the NRF founded by the Ministry of Education (Grant Number: 2022R1l1A3053628).
Keonhee Lee klee@cnu.ac.kr
Department of Mathematics, Chungnam National University Daejeon 305-764, Republic of Korea.
Arnoldo Rojas tchivatze@gmail.com
Instituto de Matemática, Universidade Federal do Rio de Janeiro P.O. Box 68530 21945-970, Rio de Janeiro, Brazil
stable [15]. In [4] it was proved that a homeomorphism of a compact metric space has the shadowing property if and only if every ergodic invariant measure satisfies the property of having full measure at the set of shadowable points. Measures with the later property were termed almost shadowable in [13]. It was proved that the space of shadowable (resp. almost shadowable) measures is an FaS (resp. Gs) subset of the space of Borel probability measures. Furthermore, every shadowable measure is almost shadowable and every almost shadowable measure can be weakly* approximated by shadowable ones.
In this paper we first show that there are homeomorphisms of compact metric spaces exhibiting almost shadowable measures which are not shadowable. Afterwards, we show that the almost shadowable measures are close to being shadowable in the following sense: For every e > 0 there are 5 > 0 and a Borelian B with measure ^ 1 — e such that every ¿-pseudo-orbit through B can be e-shadowed. We show that this property (throughout called weak shadowableness) implies that the closure of the shadowable points has full measure but not conversely. We show that the set of weakly shadowable measures form together a FaS subset of the space of Borel probability measures. Likewise, the almost shadowable measures, the weakly shadowable measures can be weakly* approximated by shadowable measures. We also show that shadowableness, almost shadowableness and weak shadowableness coincide for finitely supported measures or even for every measure as soon as the set of shadowable points is closed. Finally, we investigate the stability of weakly shadowable expansive measures on compact metric spaces. Let us state these results in a precise way.
Consider a homeomorphism of a compact metric space f: X ^ X. Given 5 > 0, we say that a bi-infinite sequence (xk)keR is a 5-pseudo-orbit if
d(f (xk), Xk+i) <5, Vk e Z,
and that it can be 5-shadowed if there is x e X such that
d(fk(x), xk) <5, Vk e Z.
We say that the sequence (xk)keZ of X is through B c X if x0 e B. A homeomorphism f has the shadowing property if for every e > 0 there is 5 > 0 such that every 5-pseudo-orbit can be e-shadowed. Some authors do not use strict inequalities in these definitions as we have used above. However, the definitions with nonstrict inequalities are equivalent to the above ones.
Definition 1 (see [7]). We say that x e X is a shadowable point of a homeomorphism f: X ^ X if for every e > 0 there is 5 > 0 such that every 5-pseudo-orbit through x can be e-shadowed. Denote by Sh(f) the set of shadowable points of f.
Recall that a Borel probability measure of a compact metric space X is a a-additive measure i defined in the Borel sigma-algebra B(X) of X (the sigma-algebra generated by the open sets) such that ¡(X) = 1. The elements of B(X) will be referred to as Borelians. The support of i is defined by supp(i) = {x e X: ¡(U) > 0 for every neighborhood U of x}. We say that i is supported on B c X if supp(i) c B. The set of Borel probability measures of X is denoted by M(X). The weak* topology on M(X) is the topology generated by the convergence
¡n ^ i if and only if J 0din ^ J 0di
for every continuous map 0: X ^ R. It is known that under this topology M(X) is a compact metrizable space [12].
Definition 2 (see [6]). We say that / £ M(X) is a shadowable measure of a homeomor-phism f: X — X if for every e > 0 there are 5 > 0 and a Borel set B C X with /(B) = 1 such that every 5-pseudo-orbit through B can be e-shadowed.
The following connection between these two concepts was given in [13]:
Definition 3. We say that / £ M(X) is an almost shadowable measure of a homeomor-phism f: X — X if /(Sh(f)) = 1.
As already mentioned, every shadowable measure is almost shadowable and every almost shadowable measure can be weakly* approximated by shadowable ones (see Lemma 2.4 and Theorem 1.8 in [13], respectively). The question arises if every almost shadowable measure is indeed shadowable. But the answer is negative by the following result.
Theorem 1. There are a homeomorphism of a compact metric space f: X — X and a Borel probability measure / of X which is almost shadowable, but not shadoviable.
Nevertheless, we will prove that the almost shadowable measures are close to being shad-owable. To explain it better, we introduce the following definition.
Definition 4. We say that / £ M(X) is a weakly shadowable measure of a homeomorphism f: X — X if for every e > 0 there are 5 > 0 and a Borel set B C X with /(B) ^ 1 — e such that every 5-pseudo-orbit through B can be e-shadowed.
The only difference between this and the definition of shadowable measure is the measure of B which is not 1, but ^ 1 — e. In particular, every shadowable measure is weakly shadow-able. In the following result we will prove that every almost shadowable measure is also weakly measurable.
Theorem 2. Every almost shadowable measure of a homeomorphism of a compact metric space is weakly shadowable.
The next result proves a kind of converse of the above theorem. Denote by B the closure of a subset B C X.
Theorem 3. Every weakly shadowable measure of a homeomorphism of a compact metric space f: X — X satisfies /(Sh(f)) = 1. There are a homeomorphism of compact metric space f: X —> X and a Borel pivbability measure fi of X which is not weakly shadowable, but satisfies i(Sh(f)) = 1.
Next, we consider the Borel hierarchy [5] of the set of weakly shadowable measures. Remember that a subset of a topological space is a Gs subset if it is the intersection of countably many open sets. We say that it is an Fa subset if it is the union of countably many closed sets. Also, an FaS subset is the intersection of countably many Fa subsets. The following result can be seen as an improvement of Theorem 1.8 in [13].
Theorem 4. The set of weakly shadowable measures of a homeomorphism of a compact metric space f: X — X is an FaS subset of M.(X). Moreover, every weakly shadowable measure can be weakly* approximated by shadowable ones.
Finally, we compare the above notions of measure shadowableness in some special cases. We call a measure / finitely supported if it is supported on a finite set.
Theorem 5. Let f: X — X be a homeomorphism of a compact metric space. Then the following properties are equivalent for every finitely supported Borel probability measure or, for every Borel probability measure / as soon as Sh(f) is closed:
1) n is shadowable;
2) n is almost shadowable;
3) n is weakly shadowable.
Notice that this equivalence fails in general (by Theorem 1).
This paper is organized as follows. In Section 2 we prove the above theorems. The stability of weakly shadowable expansive measures is discussed in Appendix A.
2. Proof of the theorems
We will need a useful lemma giving a sufficient condition for the closeness of the set of shadowable points.
Lemma 1. Let f: X ^ X be a homeomorphism of a compact metric space. Suppose that the following property holds:
(Q) There is a sequence of positive numbers (5kn such that, for every k e N, every 5k -pseudoorbit, through Sh(f) can be ^-shadowed.
Then Sh(f) is closed.
Proof. Take x € Sh(f) and e > 0. Fix k e N such that | < e and let 8 = 6k be given by (Q)- Let. (xn)neZ be a ¿-pseudo-orbit, through x. Since x G Sh(f), there is y e Sh(f) such that
d{x, y) < min 6 - d{f{x_1), ar0)} and d{f{x), f{y)) <5- d{f{x0), xj. Define the bi-infinite sequence (yn)neZ by
{x0 if n = 0, y if n = 0.
Since
d(f (y-i), yo) < d(f (x-i), xo) + d(x, y) <5
and
d(f (yo), yi) < d(f (y), f (x)) + d(f(x), xi) < 5,
we have that (yn)^Z is a 5k-pseudo-orbit through y e Sh(f). It then follows from (Q) that there is 2 e X such that
d(fn(z),yn)<^, VneZ. Since | < e, we have d(fn(z), xn) = d(fn(z), yn) < e (for n / 0) and
d(z, x0) = d(z, x) < d(z, y) + d(x, y) <\ + \<\ + \ = e-
Then (xn)^z can be e-shadowed (by z) and so x e Sh(f). Therefore, Sh(f) is closed and the proof follows. □
Proof of Theorem 1. Define X = [0, 1] U (an)n&j+ where an = 1 + ^ equipped with the Euclidean metric from R. Define f = Idx the identity of X. By Theorem 1.9 in [7] we have that Sh(f) = {an: n e N}. Define n = ^ . Then ^i(Sh(f)) = 1 and so n is almost
n£N+ n
shadowable.
To finish we prove that i is not shadowable. Suppose by contradiction that it is. Then there are a sequence 5k > 0 and Borelians Bk with ¡(X \ Bk) = 0 such that every 5k-pseudo-orbit through Bk can be ^-shadowed (Vfc e N). Since n{X \ Bk) = 0, the definition of n implies {an: n e N} C Bk for every k e N. Since Sh(f) = {an: n e N}, we get Sh(f) C Bk for every k e N. It follows that, every ¿fc-pseudo-orbit through Sh(f) can be ^-shadowed. Therefore, Property (Q) in Lemma 1 holds and so Sh(f) is closed. But since 1 e Sh(f) \ Sh(f), Sh(f) cannot be closed, so we get a contradiction. This completes the proof. □
Proof of Theorem 2. Fix e > 0. Since i is regular, there is B* C Sh(f) compact such that ¡(B*) ^ 1 — e. Now fix x e B*. Then x e Sh(f) and so f has the shadowing property through {x} (for the corresponding definition see [7]). Then, by Lemma 2.2 in [7], there is 5x > 0 such that every 5x-pseudo-orbit through the closed ball B[x, 5x] can be e-shadowed.
Next, note that the open ball collection {B(x, Sx): x e B*} is an open covering of B* which
is compact. Hence, there are finitely many points xl
x, e B* such that B* c B, where
B = [J B[Xi, SXi].
i=1
It follows that
¡(B) ^ ¡(B*) ^ 1 — e.
Now define 5 = min(5xi, ..., 5xi). If (xk)keZ is a ¿-pseudo-orbit through B, then there is 1 ^ i ^ l such that (xk)keZ is a 5xi-pseudo-orbit through B^. Therefore, (xk)keZ can be e-shadowed, proving the result. □
Proof of Theorem 3. It suffices to prove supp(i) C Sh(f). Fix x e supp(i) and e > 0. Since
¿t is weakly shadowable, there are sequences Sn > 0 and Bn of Borelians with f.i(X\Bn) ^ such that every ¿„-pseudo-orbit, through Bn can be -shadowed. We claim that.
p| Bn \ n B[x, e] = 0, VN ^ 2.
\n>N
Otherwise,
(2.1)
1 ^ ^ f| Bn U B[x, e] = ^bJ + ß(B[x, e])
Kn>N
Kn>N
1 - (1\X \ [J B^ + ß(B[x, e]) ^ 1 -Y, v(x \ Bn) + ß(B[x, e]) ^
n>N
n>N
> 1 - E ™ • + > yN >2'
which is absurd. Hence, (2.1) holds.
\n>N
Next, we prove
f]Bn c Sh(f), VN G N. (2.2)
"n
n>N
Fix N e N+ and y e f| Bn. Take n > N such that
KB[x, e])
2 n < e
and define 5 = Sn. Then every ¿-pseudo-orbit through y is a 5n-pseudo-orbit through Bn
(because y G Bn for all n ^ N) and so it can be M-BJg. -shadowed. As < e, such
a 5-pseudo-orbit can be e-shadowed, yielding (2.2). Combining (2.1) and (2.2), we obtain Sh(f )n flB[x, e] / 0. As e is arbitrary, x G Sh(f) and the result follows.
Next, we construct a compact metric space X and a homeomorphism /: X —> X exhibiting a Borel probability measure n which is not weakly shadowable though ^i(Sh(f)) = 1. Define
1
X = S (1) U ( J S (1 +
V .n v
n ,
n&N
where S(r) is the circle of radius r of R2 centered at (0, 0). Define f: X ^ X by setting fn = = f ls(i+i/n) to be a Morse-Smale diffeomorphism with 2 + 4(n — 1) alternated hyperbolic fixed points and let f |S(1) be the identity of S(1) (a reference for the definition and the main properties of these diffeomorphisms is [11]). Equipping X with the Euclidean metric, we find that X is a compact metric space and that f: X ^ X is a homeomorphism. Since the Morse-Smale diffeomorphisms (but not the identity) have the shadowing property, one has
Sh(f)= |J 5 (l + -Y (2-3)
In particular, Sh(f) = X, hence j(Sh(f)) = 1 for every Borel probability measure f.
Now take j as the Lebesgue measure of the inner circle S(1) and suppose by contradiction that it is weakly shadowable. Then there would exist a sequence of numbers ôn > 0 and Bore-lian Bn with n(Bn) ^ 1 — such that, every ¿„-pseudo-orbit, through Bn can be ^--shadowed. Defining B = H Bn, we get
n^2
<x
n=2
so ¡(B) > 0, thus BnS(1) = 0. However, B c Sh(f) (see the proof of (2.2)) contradicting (2.3). Then i is not weakly shadowable. □
Let i be a Borel probability measure of a metric space X. We say that x e X is an atom of i if ¡({x}) > 0. Denote by A(i) the set of atoms of ¡.
Lemma 2. If f: X ^ X is a homeomorphism of a compact metric space, then A(i) c Sh(f) for every weakly shadowable measure
Proof. Take x G A{n) and e > 0. Define e' = min(e'^({'T})). Then e' > 0 and so, by the weak shadowableness of ¡, there are 5 > 0 and a Borelian B with ¡(X \ B) ^ e' such that every 5-pseudo-orbit through B can be e-shadowded.
We claim that x E B. Otherwise, B П {x} = 0 and so 1 ^ /(B U {x}) = /(B) + /({x}) = 1 - /(X \ B) + /({x}) ^ 1 - e' + /({x}) ^
which is absurd. Therefore, x E B .It follows that every ¿-pseudo-orbit through x is a ¿-pseudo-orbit through B and so it can be e-shadowed. Then x E Sh(f ), proving A(/) с Sh(f ). □
Proof of Theorem 5. First we assume that / is finitely supported. By Theorems 1 and 2 we only have to prove that, if / is weakly shadowable, then / is shadowable. Since / is finitely supported, supp(/) = A(/). Since / is weakly shadowable, A(/) с Sh(f ) by Lemma 2, so supp(/) с Sh(f ). Since supp(/) is finite and contained in Sh(f ), / is shadowable.
Now we assume that Sh(f ) is closed. Again by Theorems 1 and 2 it suffices to prove that (3) implies (1). Assume that ц is weakly shadowable. By Theorem 3 we have Sh(f)) = 1 and so supp(^i) С Sh(f). But Sh(f) is closed, so supp(^) С Sh(f), thus ц is shadowable by Lemma 2.5 in [13]. □
To prove Theorem 4, we will use techniques from [13]. Let f : X — X be a homeomorphism of a compact metric space. Given e, ¿ > 0, we say that a Borelian B с X is (e, ¿)-shadowable if every ¿-pseudo-orbit through B can be e-shadowed. We denote
D(e, ¿) = {/ E M(X): 3(e, ¿)-shadowable B such that /(B) ^ 1 - e}.
Denote by Mwsh(f) the set of weakly shadowable measures of f. The following lemma corresponds to Lemma 2.2 in [13].
Lemma 3. If f : X — X is a homeomorphism of a compact metric space, then
<x <x <x
Mwsh(f ) = П U П D(n-1 + l-1, m-1).
n=1 m=11=1
Proof. As in the aforementioned lemma in [13], one has
ro ro
Mwsh(f) C n u D(n-1, m-1).
n=1 m=1
On the other hand, if i e D(e, 5) (for given e, 5 > 0), there exists an (e, 5)-shadowable set B with ¡(B) ^ 1 — e. Then, if l e N, B is still (e + l-^-shadowable and ¡(B) ^ 1 — (e + l-1).
ro
Hence, ¡i e H D(e + l-1, 5), proving 1=1
ro
D(e, 5) C p| D(e + l-1,5), Ve, 5 > 0. 1=1
Applying this inclusion with (e, 5) = (n-1, m-1) to the claim, we get
ro ro ro
Mwsh(f) C p U 0 D(n-1 + l-1,m-1).
n=1 m=11=1
For the converse inclusion take e > 0 and
^ ^ ^
I e f| U f|D(n-1 + l-1 ,m-1).
n=1 m=11=1
Fix n0, l0 e N with n-1 +1-1 < e. We have
I e U (^^(no"1 + l-1, m-1)
m=11=1
<x
and so there is m0 e N such that i e H D(n-1 +1-1, m0). It follows that
1=1
I e D(n-1 + l-1, m-1).
Taking 5 = m-1, we have that there is an (n-1 + l-1, 5)-shadowable set B with
I(B) ^ 1 — (n-1 + l-1).
Since n-1 + l-1 < e, B is still (e, 5)-shadowable with ¡(B) ^ 1 — e. Since e is arbitrary, I e Mwsh(f), proving the desired converse inclusion. This completes the proof. □
Proof of Theorem 4. Let f: X — X be a homeomorphism of a compact metric space. By Lemma 3 we have
^ ^ ^
Mwsh(f) = n U D D(n-1 + l-1, m-1). (2.4)
n=1 m=11=1
On the other hand, it is not difficult to see that
<x
p| D(n-1 + l-1, m-1) = p| D(n-1 + Y, m-1), Vn, m e N.
l=1 Y>0
As in Lemma 2.3 of [13], we have that P| D(e + y, 5) is closed in M(X) for every e, 5 > 0.
Y>0
<x
Then f| D(n-1 +1-1, m-1) is closed in M(X). Hence, l=1
|J P| D(n-1 + l-1, m-1)
m=11=1
is Fa for every n e N. Therefore, (2.4) implies that Mwsh(f) is an FaS subset of M(X).
To finish we will prove that every weakly shadowable measure ¿t of / can be weakly* approximated by shadowable measures. We have n(Sh(f)) = 1 by Theorem 3. Then supp(^) C Sh(f) and so Lemma 2.8 in [13] provides a sequence vn e M(X) supported on Sh(f) such that vn — ¡. Since vn is supported on Sh(f), vn is shadowable (c.f. Lemma 2.5 in [13]), proving the result. □
A. Stability of weakly shadowable measures
In this appendix we discuss the stability of the weakly shadowable measures which are expansive according to the following definition [8].
Definition 5. Let f: X ^ X be a homeomorphism of a metric space X. A Borel probability measure / is expansive if there is 5 > 0 (called the expansivity constant) such that /(Ts(x)) = 0 for every x £ X, where
Following [1], we denote by 2X the set formed by the subsets of a metric space X. A map H: X — 2X will be referred to as a set-valued map of X. Define the domain of H by Dom(H) = = {x e X: H(x) = 0}. We say that H is compact-valued if H(x) is compact for every x e X. We write d(H, Idx) ^ e for some e > 0 if H(x) C B[x, e] for every x e X. We say that H is upper semicontinuous if for every x e Dom(H) and every neighborhood O of H(x) there is n > 0 such that H(y) C O whenever y e X satisfies d(x, y) < n.
The C0-distance between the homeomorphisms f, g: X — X is defined by
Definition 6 (see [6]). Let f: X — X be a homeomorphism of a compact metric space. A Borel probability measure ¡i is topologically stable if for every e > 0 there is 5 > 0 such that for every homeomorphism g with d(f, g) ^ 5 there is an upper semicontinuous compact-valued map with measurable domain H of X such that
• ¡(Dom(H)) = 1;
• ¡i o H = 0;
• d(H, Idx) < e;
• f o H = H o g.
It was proved in Theorem 3.1 of [6] that every shadowable expansive measure of a homeomor-phism of a compact metric space is topologically stable. The natural question is if the conclusion of this theorem holds for almost shadowable or weakly shadowable measures. Although we do not have an answer for this question yet, it is possible to prove that every weakly shadowable expansive measure satisfies some kind of stability. More precisely, the following result holds.
Theorem 6. Let f: X — X be a homeomorphism of a compact metric space X. Then every weakly shadowable expansive measure ¡i satisfies the following stability property: For every e > 0 there is 5 > 0 such that for every homeomorphism g with d(f, g) ^ 5 there is an upper semicontinuous compact-valued map with measurable domain H of X such that
• ¡(Dom(H)) ^ 1 — e;
• n o H = 0;
• d(H, Idx) ^ e;
• f o H = H o g.
r(x) = {y £ X: d(fn(x), fn(y)) < 5, Vn £ Z}.
d(f, g) = sup d(f (x), g(x)).
xex
Proof. Let I be a weakly shadowable expansive measure of a homeomorphism f : X - X of a compact metric space X. Let e be the expansivity constant of ¡. Take e > 0 and 0 < e' < < min (|, e). For this e' we let 8 and B be given by the weak shadowableness of ¿t. Fix a homeomorphism g with d(f, g) ^ 5 and define the set-valued map H of X by
H(x) = H f~n(B[gn(x), e']), x e X. (A.1)
This is clearly a compact-valued map of X. As in [6], it follows that Dom(H) is closed (hence measurable), and that B c Dom(H), so
¡(X \ Dom(H)) < ¡(X \ B) < e' < e,
hence ¡(X\Dom(H)) ^ e. Similarly, H is upper semicontinuous and ¡oH = 0, i.e., ¡(H(x)) = 0 for every x e X. It follows from the definition of H in (A.1) that H(x) c B[x, e']. Since e' < e, we also have d(H, IdX) ^ e. That f o H = H o g can be proved as in [6], proving the result. □
Remark 1. The only difference between the stability property in this theorem and the definition of topologically stable measure is the measure of B.
This remark motivates the study of Borel probability measures satisfying the stability property in Theorem 6. We single them out in the following definition.
Definition 7. Let f: X — X be a homeomorphism of a compact metric space X. A Borel probability measure ! is called weakly topologically stable if for every e > 0 there is 5 > 0 such that for every homeomorphism g with d(f, g) ^ 5 there is an upper semicontinuous compact-valued map with measurable domain H of X such that
• ¡(Dom(H)) ^ 1 — e;
• ! o H = 0;
• d(H, IdX) < e;
• f o H = H o g.
It is possible to prove that the dynamics of the weakly topologically stable measures is as rich as that of the topologically stable ones.
Recall that a periodic point of a homeomorphism f: X — X is a point x satisfying fn(x) = = x for some positive integer n. The minimal of such integers is called the period of x. The set of periodic points is denoted by Per(f) (this is clearly a Borelian of X). The omega-limit set of x with respect to f is defined by
u(x) = {y e X: y = lim fnk (x) for some sequence nk — to}.
A periodic point x is a sink if there is a neighborhood U of x such that w(y) is the f -orbit Of (x) = = {f n(x): n e Z} of x for every y e U .A source is a sink for the inverse f-1.
Recall that a homeomorphism f: X — X of a metric space X is expansive if there is e > 0 (called the expansivity constant) such that re(x) = {x} for every x e X. Recall that f is minimal whenever every f-orbit is dense in X.
Recall that a homeomorphism is periodic if every point is a periodic point. We say that a homeomorphism can be C0-approximated by a class of homeomorphism C if p belongs to the closure of C with respect to the weak* topology (namely, for every 5 > 0 there is g e C such that d(f, g) ^ 5).
Given a continuous map h : X — X and a Borel measure i of X, we define the pullback measure h*(i) = i o h-1. The next result was proved for topologically stable measures in [6]. The proof in the weakly topological case is similar.
Theorem 7. The following properties hold for every homeomorphism of a compact metric space f : X — X :
1) A weakly topologically stable measure i of f has neither isolated points of X in its support nor atoms among the sinks or sources of f.
2) If f is expansive, every weakly topologically stable measure of f is nonatomic (hence expansive ). In particular, if X is countable, then f has no weakly topologically stable measures.
3) If f is minimal and C0-approximated by periodic homeomorphisms, the f has no weakly topologically stable measures.
4) If h : X — X is a homeomorphisms and i is a weakly topologically stable measure of f, then h-1(i) is a weakly topologically stable measure of h-1 o f o h.
References
[1] Aubin, J.-P. and Frankowska, H., Set-Valued Analysis, Boston, Mass.: Birkhâuser, 2009.
[2] Bagherzad, M.R. and Kim, S.J., Homeomorphisms with Various Measure Shadowing, J. Chungcheong Math. Soc, 2015, vol. 28, no. 3, pp. 483-489.
[3] Bowen, R., w-Limit Sets for Axiom A Diffeomorphisms, J. Differential Equations, 1975, vol. 18, no. 2, pp. 333-339.
[4] Kawaguchi, N., Quantitative Shadowable Points, Dyn. Syst, 2017, vol. 32, no. 4, pp. 504-518.
[5] Kechris, A. S., Classical Descriptive Set Theory, Grad. Texts in Math., vol. 156, New York: Springer,
[6] Lee, K. and Morales, C. A., Topological Stability and Pseudo-Orbit Tracing Property for Expansive Measures, J. Differential Equations, 2017, vol. 262, no. 6, pp. 3467-3487.
[7] Morales, C. A., Shadowable Points, Dyn. Syst., 2016, vol. 31, no. 3, pp. 347-356.
[8] Morales, C.A. and Sirvent, V.F., Expansive Measures, Rio de Janeiro: IMPA, 2013.
[9] Moriyasu, K., Sakai, K., and Sumi, N., Diffeomorphisms with Shadowable Measures, Axioms, 2018, vol. 7, no. 4, Art. 93, 10 pp.
[10] Moriyasu, K., Sakai, K., and Sumi, N., Shadowing Property and Invariant Measures Having Full Supports, Qual. Theory Dyn. Syst., 2020, vol. 19, no. 1, Paper No. 3, 7 pp.
[11] Palis, J. Jr. and de Melo, W., Geometric Theory of Dynamical Systems: An Introduction, New York: Springer, 1982.
[12] Parthasarathy, K.R., Probability Measures on Metric Spaces, Probability and Mathematical Statistics, No. 3, New York: Acad. Press, 1967.
[13] Shin, B., On the Set of Shadowable Measures, J. Math. Anal. Appl., 2019, vol. 469, no. 2, pp. 872-
[14] Tikhomirov, S., On Absolute Nonshadowability of Transitive Maps, Differ. Uravn. Protsessy Upr, 2016, no. 3, pp. 57-65.
[15] Walters, P., On the Pseudo-Orbit Tracing Property and Its Relationship to Stability, in The Structure of Attractors in Dynamical Systems: Proc. Conf. (North Dakota State Univ., Fargo, N. D., 1977), Lecture Notes in Math., vol. 668, Berlin: Springer, 1978, pp. 231-244.
[16] Yuan, G.-Ch. and Yorke, J. A., An Open Set of Maps for Which Every Point Is Absolutely Non-shadowable, Proc. Amer. Math. Soc.., 2000, vol. 128, no. 3, pp. 909-918.
1995.
881.
