TOPOLOGICALLY POSITIVELY EXPANSIVE
DYNAMICAL SYSTEMS
DAVID RICHESON AND JIM WISEMAN
Abstract. We introduce the notion of topologically positively
expansive dynamical systems and compare this notion with the
well-known notion of positive expansiveness. We prove that the
two notions are identical on compact metric spaces, but topological
positive expansiveness is strictly stronger than positive expansiveness on noncompact spaces. Moreover, we show that the dynamics
for homeomorphisms possessing these properties is severely limited.
1. Introduction
Expansiveness is an important property in the study of dynamical
systems on metric spaces. A homeomorphism f is expansive if there
exists a constant ρ > 0 with the property that for any x != y there is
an integer n such that d(f n (x), f n (y)) > ρ. Roughly speaking, orbits
separate in either forward or backward time. In this paper, we investigate those dynamical systems with the property that orbits separate
in forward time. We make the following definitions.
Definition 1. Let (X, d) be a metric space and f : X → X a continuous map.
(1) f is positively expansive (PE) if there exists a real number ρ > 0
such that for any distinct x, y ∈ X there exists n ≥ 0 such that
d(f n (x), f n (y)) > ρ. ρ is called the expansive constant.
(2) f is topologically positively expansive (TPE) if for any open
neighborhood U ⊂ X×X of the diagonal, ∆ = {(x, x) : x ∈ X},
there exists a closed neighborhood N ⊂ U with the property
that N ⊂ Int F (N) and Inv N ⊂ ∆ (where F = f × f and
Inv N is the maximal invariant subset of N).
Perhaps a few words are in order to motivate these definitions. The
first definition, positive expansiveness, is the direct analogue of expansiveness. This property is well-known and has been extensively studied
Date: April 7, 2004.
1
2
DAVID RICHESON AND JIM WISEMAN
for continuous maps and compact spaces (see [RW3] for some references). It is easy to see that for a compact space X, a homeomorphism
f : X → X is expansive if and only if the diagonal ∆ is an isolated
invariant set for f × f ([A]). Analogously, for a positively separating
map we expect the diagonal to act like a repeller. Thus we have the
notion of topologically positively expansive maps.
In Section 2 we present definitions and theorems that we will need
later. In Section 3 we examine the relationship between these two
notions. We prove that for a compact space X they are equivalent
(Theorem 9). On the other hand, on a noncompact space the notions
are not equivalent. In Section 4 we restrict our attention to homeomorphisms. In this case we discover that the dynamics are severely
limited. We recall that finite spaces are the only compact spaces that
admit PE homeomorphisms (Theorem 11). We then describe the dynamics of PE (Theorem 14) and TPE (Theorem 17) homeomorphisms
on noncompact spaces.
2. Background information
Before proceeding with our discussion we define some technical terms
that we use throughout the paper. Suppose f : X → X is a continuous
map and x ∈ X. We define the ω-limit set of x to be
$
! "#
ω(x) =
cl
f n (x) .
N >0
n>N
If f is a homeomorphism we define the α-limit set of x to be
$
! "#
−n
α(x) =
cl
f (x) .
N >0
n>N
A set S is invariant if f (S) = S. Given a set N we let Inv N denote
the maximal invariant subset of N. For continuous maps we see that
Inv N = {x ∈ N : ∃ . . . x−1 , x0 , x1 , . . . ∈ N such that
x = x0 and f (xk ) = xk+1 for all k ∈ Z}.
An invariant set S is called an isolated invariant set provided that it
has a compact neighborhood N with the property that S = Inv N.
The set N is called an isolating neighborhood for S. The set S is an
attractor if there is an isolating neighborhood N for S with the property
that f (N) ⊂ Int N; in this case N is called an attracting neighborhood.
Likewise, S is a repeller if S has a repelling neighborhood, an isolating
neighborhood N with the property N ⊂ Int f (N).
We denote the open ε-ball about a%point x by Bε (x). The ε-neighborhood
about a set A is given by Bε (A) = x∈A Bε (x).
TOPOLOGICALLY POSITIVELY EXPANSIVE DYNAMICAL SYSTEMS
3
We also need the following definition and theorem from Conley index
theory. For more information about the Conley index see [FR].
Definition 2. Let S be an isolated invariant set and suppose L ⊂
N are compact sets. The pair (N, L) is called a filtration pair for S
provided N and L are each the closures of their interiors and
(1) cl(N\L) is an isolating neighborhood of S,
/ Int N} in N,
(2) L is a neighborhood of N − = {x ∈ N : f (x) ∈
and
(3) f (L) ∩ cl(N\L) = ∅.
Theorem 3 ([FR]). Let X be a locally compact space, f : X → X be a
continuous map and S ⊂ X be an isolated invariant set with isolating
neighborhood M. Then there exists a filtration pair (N, L) for S with
N\L ⊂ M. Moreover, if we let NL denote the quotient space N/L
where the collapsed set L is denoted [L] and is taken as the base-point,
then f induces a continuous base-point preserving map fP : NL → NL
with the property [L] ⊂ Int fP−1 ([L]).
We will also need the notion of bounded dynamical systems. A
continuous map f : X → X is bounded if there exists a compact set
W with the property that for any x ∈ X there is an n ≥ 0 such
that f n (x) ∈ W . Such a set, W , is called a window. Obviously every
dynamical system on a compact space X is bounded, but this is not so
for noncompact spaces.
Below we state several properties that are equivalent to boundedness;
the theorem was first proved, as far as we can ascertain, in [Fo], and
reproved in [RW] (see [RW], [RW2], [RW3] for applications of boundedness).
Theorem 4 ([Fo, RW]). If X is a locally compact metric space and
f : X → X is a continuous map, then the following are equivalent.
(1) f is bounded.
(2) There is a compact set V such that ∅ =
' ω(x) ⊂ V for all x ∈ X.
(3) There exists a forward invariant window.
(4) There is a compact global attractor Λ (that is, there is an attractor Λ with the property that ∅ =
' ω(x) ⊂ Λ for every x ∈ X).
A set S ⊂ X is called forward minimal if it is a closed set with
f (S) ⊂ S, such that no proper closed subset of S has this property.
Notice that S is a forward minimal set if and only if ω(x) = S for every
x ∈ S. A dynamical system on a space X is forward minimal if X is a
forward minimal set. A short proof of the following corollary is found
in [RW]. This corollary will be useful in Section 4.
4
DAVID RICHESON AND JIM WISEMAN
Corollary 5 ([B, G, HK, RW]). A noncompact, locally compact space
does not admit a forward minimal dynamical system.
3. Continuous maps
In the introduction we discussed two dynamical notions for continuous maps: positive expansiveness and topological positive expansiveness. In this section we will show how these two properties are related.
We will see that on a compact space the notions are the same, but that
they are different on a noncompact space.
First we present a proposition detailing some basic properties of positively expansive maps on compact spaces (see [AH] for details). We
will see that this theorem fails to hold for noncompact spaces.
Proposition 6. Let X be a compact metric space and f : X → X be
a positively expansive continuous map. Then the following are true.
(1) f is positively expansive with respect to any metric compatible
with the topology.
(2) f n is positively expansive for all n > 0.
Proposition 6 gives some useful properties for PE maps on compact
spaces. The following examples show that properties (1) and (2) may
not hold on noncompact spaces.
Example 7. Consider the homeomorphism f : R → R given by f (x) =
2x. Clearly f is PE with respect to the usual metric. However, if we
give R the metric inherited from S 1 \{north pole}, then f is no longer
PE.
Example 8. Consider the following subsets of R2 (with the usual metric).
X1 = {(n, 0) : n ∈ Z, n #= 0},
X2 = {(n, n) : n ∈ Z, n odd} ∪ {(n, 1/n) : 0 #= n ∈ Z, n even}.
Let X = X1 ∪ X2 and define f : X → X to be the function that shifts
elements to the right. That is, for (n, 0) ∈ X1 , f (n, 0) = (n + 1, 0),
except f (−1, 0) = (1, 0). In X2 we have f (n, n) = (n+1, 1/(n+1)) for n
odd, except f (−1, −1) = (1, 1), and f (n, 1/n) = (n+1, n+1) for n even.
Then f is PE, but f 2 is not (for n even d(f 2i (n, 1/n), f 2i (n, 0)) → 0 as
i → ∞).
Better still, Bryant and Coleman give a pathological example of a
homeomorphism f : [0, ∞) → [0, ∞) with the property that f is PE,
but f k is not PE for any k > 1 ([BC]). Moreover, by acting by f on
each ray from the origin we obtain a homeomorphism g : Rn → Rn
with the same property.
TOPOLOGICALLY POSITIVELY EXPANSIVE DYNAMICAL SYSTEMS
5
For compact spaces we have the following theorem.
Theorem 9. Let (X, d) be a compact metric space and let f : X → X
be a continuous map. Then f is positively expansive iff f is topologically
positively expansive.
Proof. Suppose X is a compact space and f : X → X is a positively
expansive map with expansive constant ρ. If X is a one-point space,
then f is clearly TPE. Thus we may assume that X has at least two
points. Consider the space X × X and the map F = f × f . We give
X × X the taxicab metric, that is, D((x, y), (z, w)) = d(x, z) + d(y, w).
The set M = cl(Bρ (∆)) is an isolating neighborhood (for ∆ if f is
surjective, and for a subset of ∆ otherwise); indeed, for each z ∈ M\∆
there exists n > 0 such that F n (z) $∈ M. Thus, there exists a filtration
pair containing ∆, (N, L), with N\L ⊂ M. (Actually, Theorem 3
applies only in the case that Inv M = ∆; however, the proof is still
valid in the case that Inv M ! ∆ and ∆ is only positively invariant.)
Let NL denote the associated pointed space and let Y = NL \∆. By
construction, for each z ∈ Y there exists n ≥ 0 such that F n (z) = [L].
Thus [L] is a window for F : Y → Y and we conclude that this map
is bounded. By Theorem 4 there exists a window W ⊂ Y for F with
F (W ) ⊂ Int W . Let U be the set in X × X corresponding to the set
cl(Y \W ). U has the property that F −1 (U) ⊂ Int U and Inv U ⊂ ∆.
Thus f is TPE.
Now suppose that f is TPE. Then there exists a compact neighborhood N of ∆ such that N ⊂ Int F (N) and Inv N ⊂ ∆. Since
N is a compact neighborhood of ∆, there exists ρ > 0 such that
Bρ (∆) ⊂ N. For any y, z ∈ X (y $= z) there is n ≥ 0 such that
F n (y, z) = (f n (y), f n(z)) $∈ N. In particular d(f n (y), f n (z)) > ρ.
Thus f is PE.
!
Notice that if f is a surjective map satisfying the properties of Theorem 9 then ∆ is a repeller for f × f .
Theorem 9 does not hold for noncompact spaces; it is not the case
that the two notions are equivalent. For instance, we will see that
the homeomorphism in Example 18 is PE, but is not TPE. Also, in
Theorem 9 we saw that every TPE map is PE with respect to any
compatible metric. This may not be the case for noncompact spaces,
but we can show that every TPE map is PE with respect to some
compatible metric.
Theorem 10. If X is a locally compact metric space and f : X → X is
a topologically positively expansive continuous map, then f is positively
6
DAVID RICHESON AND JIM WISEMAN
expansive with respect to some metric compatible with the topology of
X.
Proof. Suppose f : X → X is a topologically positively expansive
continuous map and (X, d) is a locally compact metric space. Then
there exists a closed neighborhood N of the diagonal ∆ ⊂ X × X such
that N ⊂ Int F (N) and Inv(N, F ) ⊂ ∆, where F = f × f . Pick ε > 0.
We will create a metric δ on X compatible with the topology such that
for (x, y) $∈ N, δ(x, y) > ε. Then, since for every distinct pair of points
x, y ∈ X, there is an n ≥ 0 such that (f n (x), f n (y)) $∈ N, we can
conclude that f is positively expansive with expansive constant ε.
We may assume that N has compact cross sections (if not, since X is
locally compact we may replace N by a subset of N with this property).
Also, we may assume that N is symmetric about the diagonal; that is,
(x, y) ∈ N iff (y, x) ∈ N (if not, replace N by N ∩ N ! , where N ! =
{(x, y) ∈ X × X : (y, x) ∈ N}). Let g : X × X → R be a continuous
function that is equal to ε on ∆ and equal to 0 on (X × X)\ Int N.
Define the function h : X × X → R by h(x, y) = g(x, y) + g(y, x). Then
h is a continuous function satisfying the following properties
(1) h(x, x) = 2ε for all (x, x) ∈ X × X,
(2) h(x, y) = 0 for all (x, y) $∈ N, and
(3) h(x, y) = h(y, x) for all (x, y) ∈ X × X.
It is easy to check that D((x, y), (z, w)) = d(x, z) + d(y, w) + |h(x, y) −
h(z, w)| is a metric compatible with the topology of X × X. Furthermore, for (x, x) ∈ ∆ and (z, w) $∈ N we have D((x, x), (z, w)) =
d(x, z) + d(x, w) + |h(x, x) − h(z, w)| = d(x, z) + d(x, w) + 2ε > ε.
Now, define a metric δ on X by δ(x, y) = supa∈X D((a, x), (a, y)).
Note that δ(x, y) is actually maxa∈S D((a, x), (a, y)), where the maximum is over the set S = {a ∈ X : (a, x) ∈ N or (a, y) ∈ N}, since
if neither (a, x) nor (a, y) is in N, then D((a, x), (a, y)) = d(a, a) +
d(x, y) + |h(a, x) + h(a, y)| = d(x, y) ≤ D((b, x), (b, y)) for any b ∈ N.
Since N has compact cross sections, S is compact, so δ is a well-defined
metric compatible with the topology of X.
Finally, for any (x, y) $∈ N we have δ(x, y) = supa∈X d((a, x), (a, y)) ≥
d((x, x), (x, y)) > ε.
!
4. Homeomorphisms
In this section we restrict our attention to homeomorphisms. From
Theorem 9 we know that on a compact space both notions, PE and
TPE, are equivalent. In fact, as the following theorem states, most
compact spaces do not admit an invertible, positively separating dynamical system (see [RW3] for a proof and for further references).
TOPOLOGICALLY POSITIVELY EXPANSIVE DYNAMICAL SYSTEMS
7
Theorem 11 ([KR],[H],[RW3]). Let X be a compact metric space. A
homeomorphism f : X → X is positively expansive (or topologically
positively expansive) if and only if X is finite.
Before investigating noncompact spaces we prove the following corollary which will be useful to us later.
Corollary 12. Suppose X is compact and f : X → f (X) ⊂ X is a
positively expansive homeomorphism onto its image. Then X is finite
and hence f : X → X is a homeomorphism.
Proof. Since X is compact, f is bounded, and by Theorem 4 there
exists a compact global attractor, Λ. Then f : Λ → Λ is a PE homeomorphism of a compact set. By Theorem 11, Λ is finite. Taking some
large m, each point of Λ is a fixed point for g = f m . Since X is compact, g : X → X is PE. Let x ∈ X. Since ω(x, g) ⊂ Λ, and Λ consists
of a finite number of fixed points, ω(x, g) is a single fixed point. Since
g is PE, x = ω(x, g), and it follows that X = Λ.
!
Proposition 13. Let X be a locally compact metric space and suppose
f : X → X is a homeomorphism that is positively expansive with
respect to some compatible metric. Then every compact invariant set
is a finite collection of repelling periodic orbits.
Proof. Let f : X → X be PE with expansive constant ρ and let Λ ⊂ X
be a compact, invariant set. Since f : Λ → Λ is PE, Theorem 11 states
that Λ is finite. Thus, to prove the proposition it suffices to show that
each periodic orbit is repelling.
Suppose Λ = {x1 , . . . , xn } is a periodic orbit with f (xi ) = xi+1 for
i = 1, . . . , n−1 and f (xn ) = x1 . Let ε be a constant such that 0 < ε < ρ
and such that f (Bε (xi )) ∩ Bε (xj ) = ∅ for j '= i!
+ 1 (i = 1, . . . , n − 1)
and f (Bε (xn )) ∩ Bε (xj ) = ∅ for j '= 1. Let U = ni=1 Bε (xi ). Since f is
positively expansive, the only orbit that remains in U for all forward
time is the periodic orbit. That is, for each x ∈ U\Λ there is an m > 0
such that f m (x) '∈ U.
Let X ∗ be the one-point compactification of X and let Y = X ∗ \Λ.
Then f extends to a homeomorphism on the noncompact space Y and,
by the discussion above, W = Y \U is a window for f : Y → Y . By
Theorem 4, there is a window W0 such that f (W0 ) ⊂ Int W0 . Thus,
the set N = X ∗ \W0 is a repelling neighborhood for Λ.
!
Many noncompact spaces admit PE dynamical systems but, as the
following theorem states, the dynamics are very restricted.
Theorem 14. Let X be a noncompact, locally compact metric space
and let f : X → X be a homeomorphism that is positively expansive
8
DAVID RICHESON AND JIM WISEMAN
with respect to some compatible metric. Then, for each x ∈ X one of
the following holds:
(1) x is periodic,
(2) ω(x) = ∅, or
(3) ω(x) is noncompact.
Moreover, every periodic orbit is repelling, thus the set of periodic points
in X is nowhere dense.
Proof. Let f : X → X be a PE homeomorphism of a noncompact,
locally compact space. To prove the first half of the theorem it will
suffice to prove that if ω(x) is a nonempty, compact set then x is
periodic. Indeed, suppose ω(x) is nonempty and compact for some
x ∈ X. Then consider the set
Y = {f n (x) : n ≥ 0} ∪ ω(x).
Then Y is compact and f : Y → f (Y ) ⊂ Y is a PE homeomorphism
onto its image. By Corollary 12, Y is finite, thus x is periodic.
Lastly, by Proposition 13 every periodic orbit is repelling. Thus, the
set of periodic points is nowhere dense in X.
!
The following corollary follows from Theorem 11, Theorem 4 and
Theorem 14.
Corollary 15. Let X be an infinite, locally compact metric space. A
homeomorphism f : X → X cannot be both bounded and positively
expansive.
We now investigate TPE homeomorphisms.
Lemma 16. Let f : X → X be a topologically positively expansive homeomorphism of a locally compact metric space X. For each
x ∈ X there exists δ = δ(x) > 0 such that if d(x, y) < δ, then
lim d(f −n (x), f −n (y)) = 0.
n→∞
Proof. Let f : X → X be a TPE homeomorphism. Consider the onepoint compactification X ∗ = X ∪ {∗} of X. Let g : X ∗ → [0, ∞) be
any continuous
function such that g −1(0) = {∗}. Let U ⊂ X × X be
!
the set x∈X ({x} × Bg(x) (x)). Clearly U is an open neighborhood of
the diagonal ∆. Since f is TPE, there is a closed neighborhood N of
the diagonal such that N ⊂ U, N ⊂ Int F (N), and Inv N = ∆ (where
F = f × f ). Every (y, z) ∈ N has the property that F −n (y, z) ∈ N
for all n ≥ 0. Moreover, as n → ∞ the orbit limits on ∆ ∪ {∗}, so
d(f −n (y), f −n(z)) → 0 as n → 0. So, taking any δ(x) > 0 such that
{x} × Bδ(x) (x) ⊂ N the lemma follows.
!
TOPOLOGICALLY POSITIVELY EXPANSIVE DYNAMICAL SYSTEMS
9
Theorem 17. Suppose X is a noncompact, locally compact metric
space, f : X → X is a topologically positively expansive homeomorphism, and x ∈ X. Then the following are true.
(1) α(x) = ∅ or α(x) is a periodic orbit.
(2) Either
(a) x is periodic, or
(b) ω(x) = ∅, or
(c) ω(x) is not compact. In this case α(y) = ∅ and ω(y) = ∅
for every y ∈ ω(x).
(3) If α(x) $= ∅ and ω(x) $= ∅, then x is periodic.
Moreover, every periodic orbit is repelling, and thus the set of periodic
points in X is nowhere dense.
Proof. Let f be TPE. Then there exists a closed neighborhood N of
the diagonal ∆ with ∆ = Inv N and with N ⊂ Int(f × f )(N).
(1) For the sake of contradiction, suppose that α(x) is infinite for
some x ∈ X. Since α(x) is closed, it is locally compact and, by Theorem
11, it is noncompact. We will prove that α(x) is backward minimal
(α(x) is forward minimal for f −1 ), thereby contradicting Corollary 5.
Let y, z ∈ α(x), and let ε > 0. It suffices to show that there is an n > 0
such that d(f −n (y), z) < ε. By Lemma 16 there exists δ > 0 such that
for any w ∈ X, if d(w, y) < δ, then d(f −i(w), f −i(y)) → 0 as i → ∞.
Since y ∈ α(x) there exist p > 0 such that d(f −p (x), y) < δ. Since
z ∈ α(x) and since d(f −i(f −p (x)), f −i (y)) → 0 as i → ∞, there exists
r > p such that d(f −r (x), z) < ε/2 and d(f −r (x), f −r+p (y)) < ε/2.
Thus, by the triangle inequality d(f −n (y), z) < ε, where n = r − p, and
we obtain the contradiction.
(2) Let x ∈ X be a point that is not periodic and for which ω(x) $= ∅.
Theorem 11 implies that ω(x) is noncompact. From (1) we know that
for each y ∈ ω(x), either α(y) = ∅ or α(y) is finite. By Proposition
13 we know that every periodic orbit is repelling, thus ω(x) cannot
contain a periodic orbit. So it must be the case that α(y) = ∅ for every
y ∈ ω(x).
Next, we must show that ω(y) = ∅ for every y ∈ ω(x). Consider the
point (x, f k (x)) ∈ X × X where k > 0. Then
ω((x, f k (x)), f × f ) = {(y, f k (y)) : y ∈ ω(x)}.
Since N is a repelling neighborhood, we know that this set does not
intersect N, and that this is true for all k > 0. Thus, for each y ∈ ω(x)
there are a δ = δ(y) > 0 and an ε = ε(y) > 0 such that d(w, f k (w)) > δ
for all k > 0 and all w ∈ ω(x) with d(w, y) ≤ ε (choose ε and δ so that
the set {(x, z) : d(x, y) ≤ ε, d(x, z) ≤ δ} ⊂ Int N). It follows that if
10
DAVID RICHESON AND JIM WISEMAN
y ∈ ω(x), then the forward orbit of y can have no limit points, so ω(y)
is empty.
(3) Suppose α(x) "= ∅ and ω(x) "= ∅ for some x ∈ X. Let y ∈ ω(x).
By Lemma 16 there exists δ > 0 such that for any w ∈ X, if d(w, y) < δ,
then d(f −i(w), f −i(y)) → 0 as i → ∞. Since y ∈ ω(x), there exists
n > 0 such that d(f n (x), y) < δ. Thus d(f −i (f n (x)), f −i (y)) → 0 as
i → ∞. In particular, α(y) = α(x), and thus α(y) is a finite, nonempty
set. From (2) we conclude that x is periodic.
!
We conclude with two examples of PE homeomorphisms. We note
that both homeomorphisms display the properties given in Theorem
14. We see that since they don’t satisfy the properties in Theorem 17
they cannot be TPE.
p
l
Figure 1. Nonempty, noncompact omega limit set.
Example 18. Let f : R2 → R2 be the time-one map of the flow shown
in Figure 1. The map has a repelling fixed point, p, and an invariant
line, l. We may assume that the homeomorphism restricted to l moves
points to the right at an exponential rate. Thus, f is PE with respect
to the usual metric.
Notice that ω(x) ⊂ l is infinite and noncompact for any point in
the open upper half plane except x = p (in fact, we may arrange the
spiralling so that ω(x) = l) and ω(x) = ∅ for any point in the closed
lower half plane. Also α(x) = ∅ or α(x) = {p} for every x ∈ R2 .
Also, notice that although f is PE, f is not TPE. This fact follows
from Theorem 17 and the observation that there are many points with
nonempty alpha and omega limit sets.
Example 19. In this example we construct a homeomorphism on the
punctured torus. We begin with a vector v ∈ R2 with irrational slope
and let h : R2 → R2 be translation by v. Let g : T → T be the
TOPOLOGICALLY POSITIVELY EXPANSIVE DYNAMICAL SYSTEMS
11
associated map on the torus; that is, g(x) = π ◦ h(y) where π : R2 → T
is the usual projection and y ∈ π −1 (x). Modify g by slowing down
the rotation near the origin so that the origin becomes a fixed point.
We may now remove this fixed point to obtain the noncompact space
X = T \{0} and a homeomorphism f : X → X. Finally, we may alter
the metric on X by allowing it to “blow up” at the puncture. That is,
distances on the rays from emanating from the puncture go to infinity
as one approaches the hole. Visually we may think of this as the metric
inherited from R3 if the puncture is pulled off to infinity. With this
metric f is PE. Observe that f satisfies all of the properties given in
Theorem 14. Every point x ∈ T has a dense orbit, however, there are
three different behaviors. If x $= π(λv) for any λ ∈ R then α(x) = X
and ω(x) = X. If x = π(λv) and λ > 0 then α(x) = ∅ and ω(x) = X,
and if x = π(λv) with λ < 0 then α = X and ω(x) = ∅. Since f does
not satisfy the properties given in Theorem 17 we know that it is not
TPE.
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