Cent. Eur. J. Math. • 11(1) • 2013 • 177-187
DOI: 10.2478/s11533-012-0068-9
Central European Journal of Mathematics
On principal iteration semigroups
in the case of multiplier zero
Research Article
Dorota Krassowska1∗ , Marek C. Zdun2†
1 Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. Licealna 9, 65-417 Zielona Góra,
Poland
2 Institute of Mathemathics, Pedagogical University of Kraków, ul. Podchorążych 2, 30-084 Kraków, Poland
Received 15 November 2011; accepted 14 February 2012
Abstract: We collect and generalize various known definitions of principal iteration semigroups in the case of multiplier zero
and establish connections among them. The common characteristic property of each definition is conjugating of
an iteration semigroup to different normal forms. The conjugating functions are expressed by suitable formulas
and satisfy either Böttcher’s or Schröder’s functional equation.
MSC:
39B12, 26A18
Keywords: Schröder’s functional equation • Böttcher’s equation • Iteration semigroup • Conjugacy
© Versita Sp. z o.o.
1.
Introduction
Let I be an interval in R and let f be a mapping of I into itself. A family {f t : t ∈ R} of continuous self-mappings of an
interval I is said to be an iteration group of the function f, if f s ◦ f t = f s+t for all s, t ∈ R and f 1 = f. An iteration group
is called continuous if for every x ∈ I the function t 7→ f t (x) is continuous. Note that if f is invertible, then f 0 = id,
f t [I] = I and f t is strictly increasing for every t ∈ R. In fact, since f(f 0 (x)) = f(x) and f is invertible, f 0 (x) = x for x ∈ I.
Moreover f t ◦ f −t = f −t ◦ f t = f 0 = id for every t ∈ R. As a consequence, f t are invertible and f −t = (f t )−1 . Hence
Range f t = Dom f −t , thus f t is onto I. Since f t are invertible and continuous, they are strictly monotonic. We can write
f t = f t/2 ◦ f t/2 so f t is strictly increasing as a composition of two strictly monotonic functions with the same type of the
∗
†
E-mail: [email protected]
E-mail: [email protected]
177
Unauthenticated
Download Date | 7/12/17 7:18 PM
On principal iteration semigroups in the case of multiplier zero
monotonicity. A one parameter family {f t : t ≥ 0} of continuous functions f t : I → I such that f t ◦ f s = f t+s , t, s ≥ 0, and
f 1 = f is said to be an iteration semigroup of the function f. If for every x ∈ I the mapping t 7→ f t (x) is continuous then
an iteration semigroup is said to be continuous. Obviously, f n for n ∈ N is an n-th iterate of f. Similarly as for groups
 if f is invertible then f 0 = id. More information on iteration groups and semigroups can be found, for example, in the
books [3, 4, 6] and the papers [1, 7, 12, 13].
From now on let I be the interval [0, a), where 0 < a ≤ +∞, and let f fulfill the following hypothesis:
(H)
f : I → I is continuous and strictly increasing with 0 < f(x) < x for 0 < x < a.
Remark 1.1 (see [10, Theorems 5.1 and 11.1]).
If f : I → I satisfies (H) and is not surjective then every iteration semigroup {f t : t ≥ 0} of f is continuous and all f t are
strictly increasing.
Every iteration semigroup of f satisfying (H) can be uniquely extended to the “relative iteration group”, cf. [9]. Namely,
for a given iteration semigroup {f t : t ≥ 0} of f define
t
F =
(
ft
for t ≥ 0,
−t −1
for t < 0,
(f )
where Dom F t = I and Dom F −t = f t [I] for t > 0. It is easy to observe that {F t : t ∈ R} is a relative iteration group, i.e.
F t ◦ F s (x) = F t+s (x) for all x for which this formula makes sense. Moreover, if f is a homeomorphism, then {F t : t ∈ R}
is a relative iteration group. In the paper [9] the following result was proved, see also [11].
Proposition 1.2.
Let {f t : t ∈ R} be a relative iteration group of a continuous function f defined on the interval [0, a] and such that f[0,a)
satisfies (H). If f is not surjective, then the function h(t) = f t (a) for t ≥ 0 is continuous, injective and
f t (x) = h(t + h−1 (x))
for
(x, t) ∈ D,
x=
6 0,
where D = [0, a] × [0, ∞) ∪ {(x, t) : t < 0, 0 < x ≤ h(−t)}. If f is a surjection then {f t : t ∈ R} is not necessarily a
continuous iteration group; but if for some x0 ∈ (0, a) the function h(t) = f t (x0 ) is continuous, then f t (x) = h(t + h−1 (x))
for x ∈ (0, a), t ∈ R.
Put
α(x) = h−1 (x),
φ(x) = exp (α(x) log s),
ψ(x) = exp (exp (α(x) log p)),
where s > 0, p > 1 and x ∈ I. It is easy to see that if {f t : t ≥ 0} is a continuous iteration semigroup of a function f
satisfying condition (H), then f t is given by the formulas
f t (x) = α −1 (t + α(x)),
(A)
f t (x) = φ−1 (st φ(x)),
t f t (x) = ψ −1 ψ p (x) ,
(S)
(B)
for t ≥ 0 and x ∈ (0, a). These functions satisfy Abel’s, Schröder’s, Böttcher’s functional equations, respectively:
α(f(x)) = α(x) + 1,
x ∈ (0, a),
(AE)
φ(f(x)) = sφ(x),
x ∈ [0, a),
(SE)
ψ(f(x)) = ψ p (x),
x ∈ [0, a].
(BE)
178
Unauthenticated
Download Date | 7/12/17 7:18 PM
D. Krassowska, M.C. Zdun
Formulas (A), (S) and (B) are equivalent.
If a function f is differentiable at zero and f(0) = 0, then f 0 (0) is said to be a multiplier. It is obvious that the
assumption (H) implies the inequalities 0 ≤ f 0 (0) ≤ 1. Depending on the value of the multiplier, we make a suitable
choice from one of the three possibilities mentioned above: if the multiplier is zero, we use the form (B), if the multiplier
is between 0 and 1, we use the form (S), and if the multiplier is 1, we use the form (A).
From the theory of functional equations, see for example [3–5], it is known that each of the above equations possesses
infinitely many homeomorphic solutions. They depend on an arbitrary function. This phenomenon raises the question
whether it is possible to determine a ‘best’ solution or a ‘best’ form of an iteration group. The problem was solved in
two ways  by introducing the idea of principal iteration groups and the idea of regular iteration groups. In the cases
f 0 (0) ∈ (0, 1) and f 0 (0) = 1 these groups have been thoroughly investigated and compared, see [3]. However for f 0 (0) = 0,
in the literature, see [2, 3, 5], one can meet various notions of principal iteration groups. The results are independent of
each other.
The aim of the present paper is to show connections and differences among these various concepts. Here we confine
ourselves to presenting the idea of notion of principal iteration groups for the case (S) which is useful also in the
case (B). Let us recall that if f satisfies (H) and there exists φ(x) = limn→∞ f n (x)/f n (x0 ) for x ∈ I and some x0 ∈ I \ {0},
then it is a solution of Schröder’s equation. The function φ is called the principal Schröder’s function. The principal
solution is determined up to a multiplicative constant. This solution is non-decreasing, though not necessarily strictly
increasing, see [3, p. 143].
Definition 1.3.
If φ is a continuous and strictly monotonic principal Schröder’s function in I, then {f t : t ∈ R}, with f t (x) = φ−1 (st φ(x)),
is said to be the principal iteration group of a function f with multiplier s, where 0 < s < 1.
2.
Various definitions of the principal semigroup in the case of multiplier zero
Throughout this section we make the following general assumption:
(Hp ) I = [0, a) ⊂ [0, 1], the function f satisfies (H) and for some p > 1 the following one-sided limit exists and is
(strictly) positive:
f(x)
(1)
b = lim+ p .
x→0
x
We start with
Lemma 2.1.
Under the above assumptions, the function
h = L ◦ f ◦ L−1,
(2)
where

− 1
log x
L(x) =

0
if x > 0,
(3)
if x = 0,
satisfies the hypothesis (H) in the interval J = L[I] = [0, L(a)). Moreover, h is differentiable at 0 and h0 (0) = 1/p.
Proof.
We have
(
−1/x
L (x) = e ,
0,
−1
x > 0,
x = 0,
hence
h(x) = −
1
,
log f(e−1/x )
x > 0.
179
Unauthenticated
Download Date | 7/12/17 7:18 PM
On principal iteration semigroups in the case of multiplier zero
Obviously h satisfies (H). Define
R(y) = log f(y) − log b − p log y,
y ∈ I \ {0}.
Setting y = e−1/x we get
log f(e−1/x ) = R(e−1/x ) + log b −
Hence we have
h(x) −
p
.
x
x
−1
x
log b + R(e−1/x )
=
− = 2
x2.
−1/x
p
log f(e ) p
p − px log b − pxR(e−1/x )
By (1),
lim R(x) = lim+ log
x→0+
and, as a consequence,
lim
x→0+
p2
x→0
f(x)
= log 1 = 0
bx p
log b
log b + R(e−1/x )
.
=
− px log b − pxR(e−1/x )
p2
Hence there exists K > 0 such that |h(x) − x/p| ≤ K · x 2 for suitably small x and from this it follows that
h(x)
1
− = O(x 2 ),
x
p
x → 0,
which means that h is differentiable at zero and h0 (0) = 1/p.
Let us collect various definitions which have appeared in the literature. Following J. Ger and A. Smajdor [2] the function
n
ψ(x) = lim (f n (x))c/p ,
(4)
n→∞
where c is an arbitrary positive real constant and x ∈ I, if this limit exists, is called the principal solution of Böttcher’s
equation. Clearly this function satisfies (BE).
Remark 2.2.
If there exists an x0 > 0 such that ψ(x0 ) = 0, then ψ ≡ 0.
Definition GS (Ger, Smajdor [2]).
t
An iteration semigroup {f t : t ≥ 0} of f is said to be GS principal if f t (x) = ψ −1 (ψ(x))p
is a continuous strictly increasing function given by (4).
for t ≥ 0 and x ∈ I, where ψ
Let us note that f t , t ≥ 0, does not depend on the choice of c in (4). In what follows we put c = 1.
From Lemma 2.1 we have that the multiplier of h at 0 belongs to the interval (0, 1) and so its principal iteration semigroup
is given by the formula
ht (x) = φ−1 (p−t φ(x)),
x ∈ J, t ≥ 0,
(5)
where φ is a continuous and strictly increasing function satisfying the condition
hn (x)
n→∞ hn (y0 )
φ(x) = lim
(6)
for x ∈ J and an arbitrary given y0 ∈ J \ {0}, if this limit exists. (See [3, p. 200].) Note that changing y0 changes the
limit (6) up to a multiplicative constant. In particular, the form of (5) remains unchanged. This observation allows us to
introduce the following definition.
180
Unauthenticated
Download Date | 7/12/17 7:18 PM
D. Krassowska, M.C. Zdun
Definition s-K (Kuczma [3]).
An iteration semigroup {f t : t ≥ 0} of a function f is said to be (strong Kuczma’s) s-K principal if it is of the form
f t (x) = L−1 (ht (L(x))),
x ∈ I,
t ≥ 0,
(7)
where L and h are given by (3) and (2) respectively, and {ht : t ≥ 0} is expressed by (5), where φ is a continuous
strictly increasing function given by
φ(x) = lim pn hn (x),
x ∈ J,
(8)
n→∞
if such a function exists.
Definition K.
An iteration semigroup {f t : t ≥ 0} of a function f is said to be (Kuczma’s) K principal if it is of the form (7), where L
is given by (3) and {ht : t ≥ 0} is the principal iteration semigroup of h which is given by (5) with the function φ given
by (6).
Definition s-S (Szekeres [5]).
An iteration semigroup {f t : t ≥ 0} of a function f of the form
f t (x) = γ −1 (pt γ(x)),
x ∈ I \ {0},
t ≥ 0,
(9)
is said to be a (strong Szekeres’) s-S principal iteration semigroup of f if there exists a continuous and invertible
mapping γ given by the formula
−n
γ(x) = lim log (f n (x))−p .
(10)
n→∞
Obviously, γ satisfies the equation
γ(f(x)) = pγ(x),
x ∈ I.
Definition S (Szekeres [5]).
An iteration semigroup {f t : t ≥ 0} of a function f is said to be an (Szekeres’) S principal iteration semigroup of f if it
is given by (9), where γ is a continuous and invertible function defined by the formula
log f n (x)
n→∞ log f n (x0 )
γ(x) = lim
(11)
for x ∈ I and where x0 is any fixed but arbitrary point of I \ {0}.
Definition R.
An iteration semigroup {f t : t ≥ 0} of a function f satisfying (Hp ) for some p > 1 is said to be regular if for every t > 0
t
the one-sided limit lim+ f t (x)/x p exists and is finite.
x→0
Lemma 2.3.
If {f t : t ≥ 0} is a regular iteration semigroup of f satisfying (Hp ) for some p > 1 then, for every t ≥ 0,
lim+
x→0
f t (x)
t
= b(p −1)/(p−1) .
x pt
(12)
181
Unauthenticated
Download Date | 7/12/17 7:18 PM
On principal iteration semigroups in the case of multiplier zero
Proof.
t
Define δ(t) = lim+ f t (x)/x p , t ≥ 0. By (Hp ), f s (x) > 0 for x > 0, s ≥ 0. By the equation f t (f s (x)) = f t+s (x),
x→0
t, s > 0, we have
t
f t+s (x)
f t (f s (x)) (f s (x))p
·
=
,
t
s
t
(f s (x))p
(x p )p
x pt+s
Letting x → 0+ gives
x > 0.
t
δ(t) · (δ(s))p = δ(t + s),
t, s ≥ 0.
(13)
Note that the function δ is non-vanishing on the interval [0, ∞). To see this, note first that it follows from the definition
of this function given above that it must be nonnegative since f t ≥ 0 and x ≥ 0. If δ(t0 ) = 0 for a t0 > 0 then, by (13), we
infer that δ(t) = 0 for t ≥ t0 . On the other hand, putting t = s = t0 /2 in (13) gives that δ(t0 /2) = 0 and further, by the
induction, δ(t0 /2n ) = 0, whence δ(t) = 0 for all t > t0 /2n , n ∈ N, but this contradicts the assumption that δ(1) = b > 0.
Now set η(t) = log δ(t), t ≥ 0. Then (13) can be expressed in the form
η(t) + pt η(s) = η(t + s),
t, s ≥ 0.
(14)
Putting in turn s = 1 and t = 1 in (14) we obtain
η(t) + pt η(1) = η(t + 1),
so that
t ≥ 0,
and
η(1) + pη(s) = η(s + 1),
η(1) + pη(t) = η(t) + pt η(1),
Thus
η(t) =
whence
s ≥ 0,
t ≥ 0.
pt − 1
pt − 1
η(1) =
log b ,
p−1
p−1
t
δ(t) = b(p −1)/(p−1) ,
t ≥ 0.
Remark 2.4.
If we extend a semigroup {f t : t ≥ 0} to a relative iteration group {f t : t ∈ R}, then formula (12) holds for every t ∈ R.
Proof.
In fact, if t > 0,
t
−t
f −t (f t (x))
x
(x p )p
=
=
.
−t
−t
(f t (x))p
(f t (x))p
(f t (x))p−t
Letting x → 0+ we get
lim+
x→0
f −t (x) (pt −1)/(p−1) −1 p−t
−t
= b
= b(p −1)/(p−1) .
−t
p
x
Corollary 2.5.
If {f t : t ≥ 0} is a regular iteration semigroup of f then its extension to a relative iteration group {f t : t ∈ R} is regular
in the sense of Szekeres, that is, (12) holds for every t ∈ R; see definitions in [3, 5].
182
Unauthenticated
Download Date | 7/12/17 7:18 PM
D. Krassowska, M.C. Zdun
3.
Comparison of the principal iteration semigroups
In this section we state and prove connections between various definitions given in the previous section. We start with
the scheme below showing equivalences and implications:
RKS
s-K
KS ks
—-
+3 GS ks
3+ s-S
KS
—-
K ks
+3 S
We now turn to justifying these relations. We first prove the following.
Theorem 3.1.
A semigroup {f t : t ≥ 0} is GS-principal if and only if it is s-K principal.
t
Assume that {f t : t ≥ 0} is a GS-principal iteration semigroup of f satisfying (Hp ), so that f t (x) = ψ −1 (ψ(x))p
with ψ satisfying (4). Clearly ψ is a solution of Böttcher’s equation (BE). Putting ω = ψ −1 ◦ L−1 , on J \ {0} we get
Proof.
ω ◦ (L ◦ f ◦ L−1 ) =
1
1
1
◦ L−1 ◦ (L ◦ f ◦ L−1 ) =
=
= ωp .
ψ
(ψ ◦ f) ◦ L−1
(ψ ◦ L−1 )p
Hence, for h given by (2), ω ◦ h = ωp on J \ {0}. Clearly ω is continuous, strictly decreasing and positive on this interval.
Define
Ω = log ω.
(15)
By (4) we have
−n
Ω(x) = − log ψ ◦ L−1 (x) = − log lim (f n ◦ L−1 (x))p = lim −p−n log f n ◦ L−1 (x)
n→∞
n→∞
−1
= lim p−n (− log f n ◦ L−1 (x))−1
= lim p−n (L ◦ f n ◦ L−1 (x))−1 ,
n→∞
n→∞
Since hn = L ◦ f n ◦ L−1 , we get
x ∈ J \ {0}.
Ω(x) = lim p−n (hn (x))−1.
n→∞
Define φ = 1/Ω on J \ {0} and φ(0) = 0. Hence φ(x) = limn→∞ pn hn (x) for x ∈ J \ {0}. It is easy to see that φ is
continuous and strictly increasing on J, and it satisfies Schröder’s equation for h with s = 1/p. Let {ht : t ≥ 0} be given
by (5). We have to show that {f t : t ≥ 0} is of the form (7). By (15), Ω = − log ψ ◦ L−1 , so
ψ = e−Ω◦L = L−1 ◦
1
= L−1 ◦ φ ◦ L.
Ω◦L
Hence ψ −1 = L−1 ◦ φ−1 ◦ L. Thus
t
f t (x) = ψ −1 (ψ(x))p
t
= L−1 ◦ φ−1 ◦ L (ψ(x))p
t
= L−1 ◦ φ−1 − log(ψ(x))p
= L−1 ◦ φ−1 (p−t L ◦ ψ(x)) = L−1 ◦ φ−1 (p−t φ ◦ L(x)) = L−1 ◦ ht ◦ L(x),
−1 (16)
what means that {f t : t ≥ 0} is an s-K principal semigroup.
To prove the converse implication let {f t : t ≥ 0} be an s-K principal semigroup, i.e. {f t : t ≥ 0} is of the form (7),
where ht is given by (5) for a function φ continuous, strictly increasing and given by (8). Let
ψ = L−1 ◦ φ ◦ L.
(17)
t
Hence ψ −1 = L−1 ◦ φ−1 ◦ L. Again equalities (16) are satisfied and, by (7), we have f t (x) = ψ −1 (ψ(x))p . It is obvious
that ψ is continuous, positive, strictly increasing and, by (3) and (17), of the form (4) with c = 1.
183
Unauthenticated
Download Date | 7/12/17 7:18 PM
On principal iteration semigroups in the case of multiplier zero
Theorem 3.2.
A semigroup {f t : t ≥ 0} is s-S principal if and only if it is GS-principal.
Proof. Let {f t : t ≥ 0} be s-S principal, i.e. be a semigroup given by f t (x) = γ −1 (pt γ(x)), where continuous γ is as
in (10). It is obvious that γ is strictly decreasing. Put
ψ(x) = e−γ(x) ,
x ∈ I \ {0}.
(18)
Clearly ψ is continuous, strictly increasing and positive. Moreover, by (10) and (4),
−n
−n
ψ(x) = exp lim − log (f n (x))−p
= lim (f n (x))p .
n→∞
n→∞
It remains to prove that f t is of the form (B). Indeed, by (18), γ(x) = − log ψ(x), γ −1 = ψ −1 (e−x ) and, as a consequence,
t
f t (x) = γ −1 (pt γ(x)) = ψ −1 ep
log ψ(x)
t
= ψ −1 (ψ(x))p .
Hence {f t : t ≥ 0} is GS-principal. To prove the converse implication it is enough to repeat the same reasoning with
γ(x) = − log ψ(x).
Corollary 3.3.
Definitions GS, s-S and s-K are equivalent.
Theorem 3.4.
Every iteration semigroup which is s-K principal is also K principal.
Let the semigroup {f t : t ≥ 0} be s-K principal. Then there exists φ(x) = limn→∞ pn hn (x) for x ∈ J and this
function is continuous and strictly increasing. Given any fixed x0 ∈ J \ {0}, the limit limn→∞ pn hn (x0 ) clearly exists and is
positive. Hence for any x ∈ J, the limit limn→∞ hn (x)/hn (x0 ) also exists. Showing that this limit is equal to φ = φ/φ(x0 )
completes the proof.
Proof.
The converse implication is not valid. To prove this statement we present
Example 3.5.
Let p > 1. Put
Z x
h(x) =
1
p
1−
0
1
log t
dt
for x ∈ (0, a] and h(0) = 0, where 0 < a < 1 is such that h(a) < a. We have h0 (0) = 1/p. The function h is of class C 1 on
[0, a] and it is convex. Therefore, by Lundberg’s and Kuczma’s theorems [3, Theorems 6.6 and 6.8] for every x, x0 ∈ (0, a)
there exists the limit
hn (x)
φ(x) = lim n
,
(19)
n→∞ h (x0 )
and φ is strictly increasing. Define
f t = L−1 ◦ φ−1 (p−t φ ◦ L),
t ≥ 0.
Obviously {f t : t ≥ 0} is a K principal iteration semigroup. We show that {f t : t ≥ 0} is not s-K principal. In [3, p. 139]
it is shown that the sequence
1
d(pn hn (x))/dx
184
Unauthenticated
Download Date | 7/12/17 7:18 PM
D. Krassowska, M.C. Zdun
converges to zero for every x ∈ (0, a). Since d(pn hn (x))/dx > 0, limn→∞ d(pn hn (x))/dx = ∞ in (0, a). Now we use the
following result [8, Theorem 1]: If a convex function g of class C 1 satisfies (H) and 0 < g0 (0) < 1, then the limit
gn (x)
n→∞ dgn (x)/dx
γ(x) = lim
exists for x ∈ [0, a), γ is continuous, differentiable at zero and γ 0 (0) = 1. Obviously γ(x) > 0 for x ∈ (0, δ), where δ > 0.
Applying the above result for the function h, which is convex, we obtain
pn hn (x)
d n n
· lim
(p h (x)) = γ(x) · ∞ = ∞,
n→∞ d(pn hn (x))/dx n→∞ dx
lim pn hn (x) = lim
n→∞
but this means that {f t : t ≥ 0} is not s-K principal.
Theorem 3.6.
Every iteration group which is s-S principal is also S principal.
Assume that there exists a continuous and invertible mapping γ given by (10) and f t are given by (9). Then,
for arbitrary fixed x0 ∈ I \ {0} the limit
−pn
lim
=C
n→∞ log f n (x0 )
Proof.
exists and is finite. If we then set
γ(x) = lim
n→∞
log f n (x)
log f n (x)
−pn
= lim
·
= C γ(x),
n
n
n→∞
log f (x0 )
−p
log f n (x0 )
for x ∈ I \ {0}, then condition (11) is satisfied (up to a multiplicative constant). Since f t (x) = γ −1 (pt γ(x)) = γ −1 (pt γ(x)),
f t is given by (9) with γ as in (11).
We now show
Theorem 3.7.
An iteration semigroup is K principal if and only if it is S principal.
Let an iteration semigroup {f t : t ≥ 0} be K principal. By (7), (5) and (6) f t = L−1 ◦ ht ◦ L, where ht (x) =
φ (p φ(x)) and φ is as in (19), for all x ∈ J and some x0 ∈ J \ {0}. Since hn = L ◦ f n ◦ L−1 we have, by (3),
Proof.
−1
−t
L ◦ f n ◦ L−1 (y)
log f n ◦ L−1 (y0 )
=
lim
n→∞ L ◦ f n ◦ L−1 (y0 )
n→∞ log f n ◦ L−1 (y)
φ(y) = lim
(20)
for every y ∈ J \ {0} and some y0 ∈ J \ {0}. Fix an x0 ∈ I \ {0}, set y0 = L(x0 ) and define
γ(x) =
Hence, by (20),
γ(x) =
1
,
φ(L(x))
x ∈ I \ {0}.
1
log f n ◦ L−1 ◦ L(x)
log f n (x)
= lim
=
lim
.
φ(L(x)) n→∞ log f n ◦ L−1 ◦ L(x0 ) n→∞ log f n (x0 )
On the other hand, φ−1 (x) = L ◦ γ −1 (1/x), so f t (x) = L−1 ◦ ht ◦ L(x) = L−1 ◦ φ−1 (p−t φ ◦ L(x)) = L−1 ◦ L ◦ γ −1 (pt /φ ◦ L(x)) =
γ −1 (pt γ(x)). Thus {f t : t ≥ 0} is an S principal semigroup.
185
Unauthenticated
Download Date | 7/12/17 7:18 PM
On principal iteration semigroups in the case of multiplier zero
Conversely, let {f t : t ≥ 0} be S principal. Every iterate f t , t ≥ 0, is given by (9), where γ is given by (11) with
arbitrarily chosen x0 ∈ I \ {0}. Put now φ(x) = 1/(γ(L−1 (x))), x ∈ J \ {0}. By (11) we get
log f n ◦ L−1 (x0 )
−(log f n ◦ L−1 (x))−1
L ◦ f n ◦ L−1 (x)
hn (x)
=
lim
=
lim
=
lim
.
n→∞ log f n ◦ L−1 (x)
n→∞ −(log f n ◦ L−1 (x0 ))−1
n→∞ L ◦ f n ◦ L−1 (x0 )
n→∞ hn (x0 )
φ(x) = lim
On the other hand, since γ −1 (x) = L−1 ◦ φ(1/x), we have f t (x) = γ −1 (pt γ(x)) = L−1 ◦ φ(p−t γ −1 (x)) = L−1 ◦ φ(p−t φ ◦ L(x)).
Thus {f t : t ≥ 0} is a K principal semigroup.
It remains to show the equivalence between definitions of regular and GS principal semigroups. In the paper [2] a weaker
version of the following theorem is proved.
Theorem 3.8.
An iteration semigroup {f t : t ≥ 0} of a function f satisfying (Hp ) is GS principal if and only if it is regular.
The equivalence R ⇔ GS is proved in [2] under the assumption (Hp ) for relative iteration groups and regular groups
defined so that the relation (12) is satisfied. However, by Lemma 2.3 we get our assertion.
We end the paper with the following remark which is easy to check.
Remark 3.9.
In all theorems of this paper, the words iteration semigroup can be replaced by relative iteration group.
Let us note that in the literature one can meet another designations for what we call a relative iteration group. For
example, in Kuczma’s book [3] the name iteration group with respect to 0 is used.
Acknowledgements
The authors would like to thank the reviewer for his suggestions which allowed to improve the text and enhance the
readability of the manuscript.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
Blanton G., Baker J.A., Iteration groups generated by C n functions, Arch. Math. (Brno), 1982, 18(3), 121–127
Ger J., Smajdor A., Regular iteration in the case of multiplier zero, Aequationes Math., 1972, 7, 127–131
Kuczma M., Functional Equations in a Single Variable, Monogr. Mat., 46, PWN, Warszawa, 1968
Kuczma M., Choczewski B., Ger R., Iterative Functional Equations, Encyclopedia Math. Appl., 32, Cambridge University Press, Cambridge, 1990
Szekeres G., Regular iteration of real and complex functions, Acta Math., 1958, 100, 203–258
Targoński Gy., Topics in Iteration Theory, Studia Math.: Skript, 6, Vandenhoeck & Ruprecht, Göttingen, 1981
Weitkämper J., Embeddings in iteration group and semigroups with nontrivial units, Stochastica, 1983, 7(3), 175–195
Zdun M.C., On the regular solutions of a linear functional equation, Ann. Polon. Math., 1974, 30, 89–96
Zdun M.C., Some remarks on iteration semigroups, Uniw. Śląski w Katowicach Prace Naukowe–Prace Mat., 7, 1977,
65–69
Zdun M.C., Continuous and Differentiable Iteration Semigroups, Prace Nauk. Uniw. Śląsk. Katowic., 308, Uniwersytet
Śląski, Katowice, 1979
186
Unauthenticated
Download Date | 7/12/17 7:18 PM
D. Krassowska, M.C. Zdun
[11] Zdun M.C., On differentiable iteration groups, Publ. Math. Debrecen, 1979, 26(1-2), 105–114
[12] Zdun M.C., On the structure of iteration group of homeomorphisms having fixed points, Aequationes Math., 1998,
55(3), 199–216
[13] Zdun M.C., Zhang W., Koenigs embedding flow problem with global C 1 smoothness, J. Math. Anal. Appl., 2011,
374(2), 633–643
187
Unauthenticated
Download Date | 7/12/17 7:18 PM