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On Uniqueness of Conjugacy of Continuous and Piecewise Monotone Functions
Fixed Point Theory and Applications volume 2009, Article number: 230414 (2009)
Abstract
We investigate the existence and uniqueness of solutions of the functional equation
,
, where
are closed intervals, and
,
are some continuous piecewise monotone functions. A fixed point principle plays a crucial role in the proof of our main result.
1. Introduction
Let be closed, bounded, and nondegenerate (i.e., neither of them consists of a single point) real intervals, and let
be continuous functions. The aim of this paper is to discuss, under some additional assumptions on the maps
and
, the problem of (topological) conjugacy of
and
. More precisely, we investigate the existence and uniqueness of solutions
of the following functional equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ1_HTML.gif)
Let us recall that a homeomorphism satisfying (1.1) is said to be a topological conjugacy between
and
(
and
are then called topologically conjugate), whereas an arbitrary function
fulfilling (1.1) is called a conjugacy between them (so the conjugacy needs not to be continuous, surjective, or injective).
A continuous function is said to be a horseshoe map (see [1]) if there exist an integer
and a sequence
of reals such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ2_HTML.gif)
and for every ,
is a homeomorphism of the interval
(which is called a lap of
) onto
. We say that horseshoe maps
and
having that the same number of laps are of the same type if
and
are of the same type of monotonicity on their leftmost laps.
It is known (see [1, 2]) that two horseshoe maps of the same type and without homtervals (i.e., intervals on which all their iterates are monotone) are topologically conjugate. So are also transitive horseshoe maps having two laps each, and, in this case, topological conjugacy is only one (see [3]). Moreover, if is transitive and
is an arbitrary function, then every increasing, continuous, and surjective solution
of (1.1) is homeomorphic (see [4]). We will show (Example 3.7) that if we omit the assumption of the transitivity of
, then such a solution needs not to be injective even if
is continuous and transitive and
is continuous and piecewise monotone.
However, the main purpose of this paper is to find some regularity conditions on and
ensuring the uniqueness of conjugacy between them as well as implying that conjugacy is topological. The following fixed point principle plays a crucial role in the proof of our main result (Theorem 3.1).
Theorem 1.1 (see [5, Theorem , page
] and also [6, Theorem
, page 12]).
Let be a complete metric space and
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ3_HTML.gif)
for a nondecreasing function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ4_HTML.gif)
then has a unique fixed point.
2. Preliminaries
We begin by recalling the basic definitions and introducing some notation.
Throughout the paper stands for the integer part function.
Let be a metric space. A function
is called strictly contractive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ5_HTML.gif)
Given a nondecreasing function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ6_HTML.gif)
for and a selfmap
of
, we say that
is
-contractive if it satisfies condition (1.3).
Given horseshoe maps having laps
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ7_HTML.gif)
respectively, for every put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ8_HTML.gif)
A horseshoe map having
laps
is said to be piecewise expansive (resp., piecewise
-expansive) if for every
,
is strictly contractive (resp.,
-contractive for a
.
We start with the following.
Proposition 2.1.
Let and
be horseshoe maps having
laps
and
, respectively. Assume also that
is a monotone and surjective solution of (1.1).
If is increasing, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ9_HTML.gif)
and for
.
If is decreasing, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ10_HTML.gif)
and for
.
Proof.
Let us first note that the fact that is monotone and surjective together with (1.1) gives
for
, and consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ11_HTML.gif)
Suppose that for an
. Then, using the fact that
is monotone, we see that
consists of a single point, and therefore so does
. But from (1.1) and the surjectivity of
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ12_HTML.gif)
a contradiction. We have thus shown that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ13_HTML.gif)
Assume that is increasing. Then, by (2.9), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ14_HTML.gif)
which together with (2.7) gives (2.5). Hence we immediately see that for any ,
.
The rest of the proof runs as before.
Proposition 2.2.
Assume that
(H) and
are horseshoe maps of the same type and having
laps
and
, respectively,
and is piecewise expansive. If
is a continuous and nonconstant solution of (1.1), then
is surjective. If, moreover,
is even and
is injective, then
is strictly increasing.
Proof.
Let with
be such that
. Then, by (1.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ15_HTML.gif)
This and the fact that for every ,
is strictly contractive shows that
and
are not in the same interval
. Therefore,
or
, and consequently, by (2.11),
or
.
Assume that . Since
,
, (2.11) now gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ16_HTML.gif)
Therefore, . Similarly,
implies
. We have thus shown that
is a surjection.
Now, assume that is even (which obviously yields
). Suppose also, contrary to our claim, that
is decreasing, and let us consider the case when
. Then
and
, and (1.1) now gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ17_HTML.gif)
which contradicts the fact that and
are of the same type. Similar considerations apply to the case when
.
Lemma 2.3.
Let be odd, and let assumption (H) hold. If
is a solution of (1.1) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ18_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ19_HTML.gif)
If, moreover, is piecewise expansive, then (2.6) holds true.
Conversely, every function satisfying (2.15) is a solution of (1.1) fulfilling (2.14).
Proof.
It is obvious that if fulfils (1.1) and (2.14), then (2.15) holds.
Assume that is piecewise expansive. If
, then from (1.1) and (2.14) it follows that
is a fixed point of
, and the strict contractivity of
gives
.
Next, assume that , which clearly forces
,
, and
. Since by (2.14) we obtain
and
, (1.1) implies
and
. If it were true that
, we would conclude from the strict contractivity of
and
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ20_HTML.gif)
a contradiction.
We have thus shown that . In the same manner we can see that
.
Now, fix an . Suppose that
is even and note that from (1.1), the equalities
, and the fact that
is odd, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ21_HTML.gif)
Since, by (2.14), , the injectivity of
gives now
. As the same conclusion can be drawn for odd
, the proof of (2.6) is complete.
The rest of the proof is immediate.
Analysis similar to that in the proof of Lemma 2.3 (due to condition (2.18) it also applies to the case when is even) gives the following.
Lemma 2.4.
Let assumption (H) hold. If is a solution of (1.1) for which
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ22_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ23_HTML.gif)
If, moreover, is piecewise expansive, then (2.5) holds true.
Conversely, every function satisfying (2.19) is a solution of (1.1) fulfilling (2.18).
3. Main Results
We can now formulate our main results.
Theorem 3.1.
If assumption (H) holds and is piecewise
-expansive, then there exists a unique function
satisfying (1.1) and condition (2.18). This function is continuous, surjective and increasing. If, moreover,
is piecewise expansive, then
is also strictly increasing.
Proof.
Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ24_HTML.gif)
It is easily seen that all these spaces with the metric are complete.
Fix a , and set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ25_HTML.gif)
We will show that the above formula correctly defines a selfmap of . In order to do this let us first fix an
and observe that we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ26_HTML.gif)
In the same manner we can see that for
. Thus,
and, by Lemma 2.4,
is a solution of (1.1) fulfilling (2.18) if and only if it is a fixed point of
. Moreover, it is easily seen that
and
. Now, fix
, and note that by the facts that for every
,
is
-contractive and
is increasing, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ27_HTML.gif)
Thus, the function is
-contractive, and Theorem 1.1 now shows that
has a unique fixed point
. Similarly,
has a unique fixed point
, and
has a unique fixed point
. Therefore,
and, in consequence,
is a continuous, surjective and increasing solution of (1.1) fulfilling (2.18).
Assume, additionally, that is piecewise expansive. We will first show that
is constant in no neighbourhood of
or
.
To do this, suppose that is constant on a neighbourhood of
, and denote by
, where
, the maximal interval of constancy of
. Since
is surjective and satisfies (2.18),
is a proper subset of
. Therefore,
and from (1.1) it follows that
is constant on
.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ28_HTML.gif)
But from the strict contractivity of we also get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ29_HTML.gif)
a contradiction.
Now, assume that . Then
, so
is constant on a neighbourhood of
. Denote by
, where
, the maximal interval of constancy of
. Since
is surjective and satisfies (2.18),
is a proper subset of
. Therefore,
and from (1.1) it follows that
is constant on
.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ30_HTML.gif)
which contradicts the strict contractivity of .
Finally, assume that . Then
, so both
and
are intervals of constancy of
, and therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ31_HTML.gif)
This together with the strict contractivity of and
gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ32_HTML.gif)
a contradiction.
Analysis similar to the above shows that is constant in no neighbourhood of
.
Now, suppose that is constant on a neighbourhood of
for an
. Then from (1.1) it follows that
is constant on a neighbourhood of
or
, which is impossible.
We have thus shown that if is not injective, then any interval of its constancy is contained in
for an
. Let
be an interval of constancy of
having the maximal length. By (1.1),
is constant on
. But since
for an
, from the strict contractivity of
it follows that the interval
is of greater length than
, a contradiction.
Analysis similar to that in the proof of Theorem 3.1 with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ33_HTML.gif)
and application of Lemma 2.3 instead of Lemma 2.4 gives the following.
Theorem 3.2.
If is odd, assumption (H) holds, and
is piecewise
-expansive, then there exists a unique function
satisfying (1.1) and condition (2.14). This function is continuous, surjective, and decreasing. If, moreover,
is piecewise expansive, then
is also strictly decreasing.
Let us next note that an immediate consequence of Theorems 3.1 and 3.2 and Proposition 2.1 is what follows.
Corollary 3.3.
If assumption (H) holds, is piecewise
-expansive, and
is odd, then (1.1) has exactly two monotone and surjective solutions. One of them is increasing, while the other is decreasing.
The following fact follows immediately from Propositions 2.1 and 2.2.
Remark 3.4.
If assumption (H) holds, is piecewise expansive and
is even, then (1.1) has no homeomorphic solution satisfying condition (2.14).
On the other hand, we also have the following.
Theorem 3.5.
If assumption (H) holds and and
are piecewise expansive, then there exists a unique function
satisfying (1.1) and condition (2.18). This function is an increasing homeomorphism. If, moreover,
is odd, then there is also exactly one map
fulfilling (1.1) and condition (2.14). This map is a decreasing homeomorphism.
Proof.
Put for
. If
, then we also set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ34_HTML.gif)
while for we put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ35_HTML.gif)
By Theorem 3.1 (for every ,
is
-contractive with
for
, so its assumptions are satisfied) there exist increasing homeomorphisms
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ36_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ37_HTML.gif)
Moreover, by Proposition 2.1,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ38_HTML.gif)
Let us observe that in the case when is odd from Theorem 3.2 and Proposition 2.1 it follows that we can also take
which is a decreasing homeomorphism such that (3.14) holds true and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ39_HTML.gif)
Putting and using (3.13) and (3.14) we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ40_HTML.gif)
Furthermore, from (3.15) (resp., (3.16) it follows that condition (2.18) (resp., (2.14) holds true.
We have thus shown that (1.1) has a homeomorphic, increasing solution fulfilling (2.18) and if, moreover, is odd, then it also has a homeomorphic, decreasing solution satisfying (2.14).
To prove the uniqueness, assume that is a solution of (1.1) fulfilling condition (2.18) (resp., (2.14). Then, by (3.14), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ41_HTML.gif)
which means that satisfies (3.13). Moreover, from (3.15) (resp., (3.16) it follows that
for
. Hence, by Theorem 3.1,
and so
.
As an immediate consequence of Theorem 3.5, Proposition 2.1, and Remark 3.4, we get the following.
Corollary 3.6.
Let assumption (H) hold, and let and
be piecewise expansive. If
is even, then (1.1) has exactly one homeomorphic solution. This solution is strictly increasing. If
is odd, then (1.1) has exactly two homeomorphic solutions. One of them is strictly increasing, while the other is strictly decreasing.
Finally, we give two examples. The first of them shows that if one replaces the assumption that is piecewise expansive by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ42_HTML.gif)
then the last assertion of Theorem 3.1 is no longer true.
Example 3.7.
Let ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ43_HTML.gif)
and let be the standard tent map defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ44_HTML.gif)
It is well known (see, e.g., [7] which is also a good survey on transitive maps) that is transitive. It is also easily seen that
fulfills the assumption of Theorem 3.1 with
for
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ45_HTML.gif)
By Theorem 3.1 there exists a unique (continuous, surjective, and increasing) function satisfying equation (1.1) and condition (2.18). Put
, and note that
. Furthermore, from (1.1) it follows that
. Therefore,
has to be one of the intervals:
. Consequently,
is not injective, and thus (1.1) has no homeomorphic solution.
Example 3.8.
Let ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ46_HTML.gif)
and let be the standard tent map. It is known (see, e.g., [7–9]) that the function
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ47_HTML.gif)
is a topological conjugacy between and
. From Theorem 3.1 it follows that
is the unique conjugacy between these maps such that
and
. In particular, by Proposition 2.1,
is the only continuous, increasing, and surjective conjugacy between
and
.
Corollary 3.9.
The function given by (3.24) is the only one satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F230414/MediaObjects/13663_2008_Article_1125_Equ48_HTML.gif)
and such that for
and
for
.
Example 3.10.
If and
is a horseshoe map, then for any positive integer
the function
is a continuous and surjective solution of (1.1). It is obvious that this solution does not satisfy neither (2.14) nor (2.18). We thus see that in this case (1.1) can even have infinitely many solutions.
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Ciepliński, K., Zdun, M.C. On Uniqueness of Conjugacy of Continuous and Piecewise Monotone Functions. Fixed Point Theory Appl 2009, 230414 (2009). https://doi.org/10.1155/2009/230414
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DOI: https://doi.org/10.1155/2009/230414