数理解析研究所講究録
1042 巻 1998 年 92-98

92

Dyn.aof skew tent maps$\mathrm{m}\mathrm{i}\mathrm{C}\mathrm{s}$Keiko ICHIMURA (市村圭子) and Michiho ITO (伊藤みちほ)
Department of Information Sciences, Ochanomizu University

1Introduction
We consider skew tent maps defined by$f_{a,b}(_{X)}$$(a, b)$

$=$

$\in$

$\{$

$ax+1$ ,$(x\leq 0)$$-bx+1$ ,$(x\geq 0)$$D:=\{(a, b) :a>0, b>1, a+b\geq ab\}$ .

By the dynamical behavior of, the parameter domain $D$ is divided into subdomains
defined by some algebraic curves:. In each
thereare subdomains
wherehas unique attracting periodic orbit of period
andwherehas $2k$ or chaotic intervals, which is analyzed by the methodof renormalization.
this paper we give the relation between
product for kneading sequence and renormalization as Theorem 1. We also givean explicit proof of Theorem A in [MV91], which states monotonicity of kneading
sequence of this family. Moreover we correct Corollary of Theorem 2 in [MV92]
as Proposition 4.
We denote kneading sequence ofby $K(a, b)$ and topological entropy ofby $h(a, b)$ . We refer basic definitions and notations from [Ich] [CE80].
$f_{a,b}$$D_{k}$$D=\Sigma_{k=2}^{\infty}D_{k}$$D_{k}^{A}$$D_{k}^{B}$$D_{k}$$f_{a,b}$

$k$

$k$

$f_{a,b}$$([\mathrm{l}\mathrm{c}\mathrm{h}][\mathrm{I}\mathrm{t}\mathrm{o}])$$\ln$$\star-$$f_{a,b}$2$f_{a,b}$Renormalization and -product$\star$We denoteby in this section. For getting maximal level of renormalization, we assume sequenceis prime. Let
be the length of sequence$(J)$and int
interior of an interval . For the definitions of renormalization and-product, see [Ito] and [CE80] respectively.
Definitionis called prime if does not have any finite sequence
ofandand any finite or infinite sequencesuch that.
$f_{a,b}$$f$$\underline{B}$$|\underline{A}|$$\underline{A}$$J$$\star$$\underline{S}$$\underline{S}$$\underline{A}(\neq 0)$$L’ \mathrm{s}$$R’ \mathrm{s}$$\underline{B}(\neq C)$$\underline{S}=\underline{A}\star\underline{B}$93
andis finite sequence of
whereTheorem 1
is prime if and only if there exist invariant closed intervals
and$fJ_{i}=J_{i+1}$ $(i=0, \cdots , |\underline{A}|-1),$
andsuch that.
int $(J_{i})\cap int(J_{i}’)=\emptyset(i\neq i’)$ . can not have any refinement of$K(a, b)=\underline{A}\star\underline{B}$$L’ \mathrm{s}$$\underline{A}(\neq\emptyset)$$\underline{B}(\neq C)$$R’ \mathrm{s}$$fJ_{|\underline{A}|}=J_{0}$$J_{|\underline{A}|}\ni 0,$$\{J_{i}\}_{i=0,\cdots,|\underline{A}}|$$f$$\{J_{i}\}$oris finite sequence ofwhereProof As.sume.
andis prime. Set $x_{n}=f^{n}(1)(n\geq 0),$and, . Then, we have
Let be convex hull of $\{Xi+k(p+1) : k=0,1, \cdots\}$ for $i=0,$$fJ_{i}=J_{i+1}$ $(i=0, \cdots , p-1)$ andis monotone on each except of $i=p$ . Remark.
on each has same slopes. It follows thatthat.
We consider the following two cases.
does not contain both and R.
The first case:contains a turning point as an
is finite in this case. It follows thatfor all $i(0\leq i\leq p)$ ,end point of it. Hence, $fJ_{p}=J_{0}$ . As is monotone on
is monotone and surjective on . Hence, its slope is $-1$ .
restricted onsuch thatare disjoint or there would exist someThen
continuity of . The latter can not occur
are disjoint. Notice that the first. Therefore
because of the assumption.
andcase corresponds to boundary curve ofcontains both and R.
The second case:has unique turning point inside . We set
In this case $fJ_{p}=J_{0}$ and$\alpha(>0),$ $\beta(<0)$ . We divide
andinto two subinterval
two slopes ofis surjective on , we have that
corresponding to slope and . As$\sup\{|\alpha|, |\beta|\}>1$ .
$K(a, b)=\underline{A}\star\underline{B}$$\underline{A}=A_{0^{A}}1\ldots Ap-1$$p=|\underline{A}|$$\underline{B}(\neq C)$$\cdots$$J_{i}$$f$$f^{p+1}$$R’ \mathrm{s}$$L’ \mathrm{s}$$\underline{A}(\neq\emptyset)$$p$$J_{i}$$f^{p+1}|_{j_{i}}\sim f^{p+1}.|_{J_{i}},$ $(i$$J_{i}$$\neq i’)$$\mathrm{L}$$\underline{B}$$0$$J_{p}$$\underline{B}$$f$$f^{p+1}$$J_{i}$$J_{i}$$J_{i}$$i,$ $i’$$\{J_{i}\}’ \mathrm{s}$$J_{i}=J_{i’}\mathrm{h}\mathrm{o}\mathrm{m}$$f$$\{J_{i}\}$$\mathrm{o}\mathrm{f}\underline{A}$$D_{k}^{B}$$D_{k}^{A}$$\mathrm{L}$$\underline{B}$$f^{p+1}|_{J_{i}}$$f^{p+1}|_{J_{i}}$$I_{\beta_{i}}$$J_{i}$$c_{i}$$J_{i}$$\alpha$$\beta$$f^{p+1}|_{J_{i}}$Figure 1: The graph of $f^{p+1}$ on$\sqrt i’$$I_{\alpha_{i}}$$J_{i}$

having two turning points.

94
If $|\beta|<1$ , then turning point is attracted to a fixe.d point on . It follows that
. This contradicts assumption of this case.
orIf $|\beta|=1$ , we reduce to the first case.
, hassuch that, there exists
If $|\beta|>1$ and int
such that $J_{i}=J_{i’}$ . In thetwo turning.points (see Figure 1) or there exist
for some $m(m\neq p)$ . This contradicts thatequals
latter case we haveincludes turning point in it. Hence we obtainbecauseis monotone on.
. Notice that the second case corresponds toint
in theConversely, if there exist disjoint invariant closed intervals
If is not prime,
withorem, we have
. Hence, is prime.
has refinement of$I_{\beta}$$R^{\infty}$$\underline{B}=L^{\infty}$$(J_{i})\cap int(J_{i’})\neq\emptyset$$f^{p+1}|_{J_{i}}$$J_{i’}$$i,$ $i’$$J_{m}$$J_{p}$$J_{m}$$f$$J_{p}$$(J_{i})\cap int(J_{i}’)=\emptyset$$D_{k}^{B}$$\{J_{i}\}_{i=0,\cdots,|\underline{A}|}$$K(a, b)=\underline{A}\star\underline{B}$$\underline{A}=A_{J_{0}}A_{j_{1}}\cdots AJp-1^{\cdot}$$\underline{B}$$\square f$$\{J_{i}\}$$\underline{B}$Now we have the relation of our renormalization (i.e.,
is a skew tent map of $D$ ) and -product.

$(|\underline{A}|+1)$-renormalization$\star$IfCorollary 1$|\underline{A}|+1$$|\underline{B}|\neq 2$

in above theorem, then$f$is renormalizable of level.

because a turning point. In the first case, we have
Proof Let be. In the second case, we have $|\beta|>1$ andis 2-periodic point ofof on$f^{p+1}J_{i}=J_{i}$ .
. Therefore is $(p+1)$ -renormalizable
It follows(resp. $f^{p+1}(c_{i})<c_{i}$ ).
) if(ci)] (resp.
[$p$$f$$|\underline{B}|=2$$|\underline{A}|$$f^{p+1}$$J_{p}$$(\alpha, \beta)\in D$$f$$\mathrm{o}\mathrm{n}\square$$\mathrm{q},$$[f^{p+1}(\alpha),$ $\alpha]$$f^{p1}+$$c_{i}<f^{p+1}(\mathrm{q})$-periodic g-admissible
It is well known that for a smooth unimodal map. This fact issequence implies the existence of or $2n$ -periodic point
proved by Schwarzian derivative. But we have the following analogous fact forskew tent maps.
$g,$

$n$

$([\mathrm{D}\mathrm{e}\mathrm{V}89])$

$n$

and
is finite sequence of
whereCorollary 2 If
is prime, then has periodic points of period $|A|+1$ . Moreover if
and, then also has periodic points of period $2(|A|+1)$ .
$K(a, b)=\underline{A}\star\underline{B}$Remark$C,$ $L^{\infty},$ $R^{\infty}$

3$R’ \mathrm{s}$$f$$\underline{B}(\neq C)$$|\underline{B}|=2$$L’ \mathrm{s}$$\underline{A}(\neq\emptyset)$$f$

For showing Corollary 1 and 2, we need only theinstead of primarity of .

$\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{m}\mathrm{p}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}\underline{B}}\neq$$\underline{B}$Monotonicity of kneading sequences

In this section we will mention the monotonicity property of kneading sequencein the domain$\tilde{D}=\{(a, b)\in D;a\geq 1\}$ .

95
Let us define the order for parameter pairs as follows, according to M. Misiurewicz and E. Visinescu [MV91] :$(a, b)\succ(a’, b’)\Leftrightarrow a’\geq a,$$b’\geq b$, and at least one of these inequalities is strict.

Kneading sequences are monotone increasing with respect to this order.
Monotonicity Theorem (Theorem A in [MV91])
with $(a’, b’)\succ(a, b)$ , it holds that $K(a’, b’)>K(a, b)$ .

For $(a’, b’),$in$(a, b)$$\tilde{D}$This theorem is already proved in [MV91]. M. Misiurewicz and E. Visinescu
showed the claim by using the estimation of topological entropy. But we shallin [MV91], and renormalization
reprove it by using only thier results formethod, not via the topological entropy. For that purpose, we prepare Proposition
1, Proposition 2 and Proposition 3 (for the detailed proofs, see [Ich]).
$D^{\star}$

As to -product, we have the following.
$\star$

Proposition 1 Let. Then for all

and$n\geq \mathrm{I},$$\underline{A}\succ\underline{B}$3.1$\underline{A}$$\underline{B}$be symbolic sequences of $L,$

$R^{\star n}\star\underline{A}\succ R^{\star n}\star\underline{B}$Monotonicity in$R$, and$C$with.

$D^{\star}$be the domain $\{(a, b)\in D;a+b<ab^{2}, a>1\}$ .
LetM. Misiurewicz and E. Visinescu proved in [MV91] that $K(a’, b/)>K(a, b)$ foris characterlized by$(a’, b’),$ $(a, b)\in D^{\star}$ such that $(a’, b’)\succ(a, b)$ . This domain
the following.
$D^{\star}$$D^{\star}$

Fact 1 (Lemma 2.1 in [MV91])$(a, b)\in D^{\star}\Leftrightarrow K(a, b)\succ RLR^{\infty}$.

First, monotone increasing property of kneading sequence is proved inFact 2 (Proposition 4.3 in. [MV91]) lf $(a, b)$ and $(a’, b’)$ are in$(a, b)\prec(a’, b’)$ , then $K(a, b)\prec K(a’, b’)$ .

3.2

Renormalization and -product (forProposition 2
lent mutually.
(i) $(a, b)\in D_{0}$ .

$\star$Let $(a, b)$ be in$D$$\tilde{D}$$D^{\star}$$D^{\star}$.

with). The following three conditions are equiva-

96(ii) There exists a unique number $m\geq 1$ and a prime sequence.
is longer than 2 such that$\underline{B}$whose length$K(a, b)=R^{\star m}\star\underline{B}$(iii) There exists some number $m\geq 1$ such that
( , ab).

$\varphi^{m}(a, b)\in D^{\star}$, where $\varphi(a, b)=$$b^{2}$

Furthermore, there exist closed subintervals of$I_{a,b},$$\{I_{i}\}_{i=0},$$\cdots,$such$2^{m}-1$that their interiors are disjoint mutually, $f_{a,b}I_{i}=I_{i+1}$ for $0\leq i\leq 2^{m}-2$.
and $f_{a},bI2m-1=I_{0},$ $I2m-1\ni 0$ , and$f_{a,b}^{2^{m}}|_{I}i\sim f\varphi m(a,b)$

Proposition 3 Let $(a, b),$and$\varphi^{m}(a, b)\in D^{\star}$

3.3$(a’, b’)\in\tilde{D}\backslash D^{\star}$$\varphi^{n}(a’, b/)\in D^{\star}$such that $(a, b)\prec(a’, b’)$ .

If, then $m\geq n$ .

Proof of Monotonicity Theorem

Assume that $(a, b)\prec(a’, b’)$ .
(i) If both $(a, b)$ and $(a’, b’)$ belong toFact 2.

$D^{\star}$, then the proof is already given by. Then $(a’, b’)$ is in(ii) Assume that either $(a, b)$ or $(a’, b’)$ belongs tobecause $(a, b)\prec(a’, b’)$ . By virtue of Fact 1, $K(\varphi^{n}(a, b))\preceq RLR^{\infty}\prec$”
is$K(\varphi^{n}(a’, b/))$ . We have that $K(a, b)\prec K(a’, b/)$ as an order relation
total.
$D^{\star}$$D^{\star}$$”\prec$. Then, by Proposition(iii) Assume that $(a, b)$ and $(a’, b’)$ both belong to3, their kneading sequences are written as, for some $n\leq m$ ,$\tilde{D}\backslash D^{\star}$$K(a, b)=R^{\star m}\star K(\varphi^{m}(a, b))$and$K(a’, b’)=R^{\star n}\star K(\varphi^{n}(a, b))$.

since is an increasing
If $m=n$ , then we have thatfunction. Because $K(\varphi^{n}(a, b))\prec K(\varphi^{n}(a’, b/))$ and from Proposition 1, we
have that $K(a, b)\prec K(a’, b’)$ .
$\varphi^{n}(a, b)\prec\varphi^{n}(a’, b/)$

If $n<m$ , then we have thatof Fact 1, it follows that$\varphi^{n}(a, b)\neq D^{\star}$

and$\varphi$$\varphi^{n}(a’, b/)\in D^{\star}$$K(\varphi^{n}(a, b))\preceq RLR^{\infty}\prec K(\varphi^{n}(ab/,’))$

By Proposition 1, we have that $K(a, b)\prec K(a’, b/)$ .

. By virtue.
$\square$97

4Renormalization and topological entropy
Now we correct two statements of [MV92].
First: kneading sequence.

f.or boundary curve of$A_{m}(=D_{m+}^{A})1$

and $B_{m}(=$$D_{m+1}^{B})$In Theorem 1 of the paper [MV92], they say ;$(\lambda, \beta)(=(a, b))\in Am\Leftrightarrow K(\lambda, \beta)=(RLm)^{\infty}$$(\lambda,\beta)\in B_{m}\Leftrightarrow K(\lambda,\beta)=RL^{m-1}\star\underline{B}$with

where $M$ is set of kneading sequence for tent map,$\underline{B}\in M$$f_{\lambda,\lambda}(1<\lambda\leq 2)$.

. In our opinion this curve
have common boundary curve:and. We find our reason in theand fromshould be discussed separately fromfact that the kneading sequence on this curve is $RL^{m}RL^{m-}1C$ , not admitted by.
and onone on$A_{m}$$\lambda^{m}\mu=1$$B_{m}$$B_{m}$$A_{m}$$B_{m}$$A_{m}$

Second: topological entropy of $B_{1}(=D_{0})$ is not constant.
Corollary in [MV92], they say;
$\ln$

let

$(\lambda,\beta),$$(\lambda’,\beta’)\in\{(\lambda,\beta)\in D;\lambda\leq 1\}$$(\lambda, \beta),$Let$(\lambda, \beta)<(\lambda’, \beta’)$$(\lambda,\beta),$$B_{m}$,.

$(\lambda’, \beta/)\in Am\cup Bm\Rightarrow h(\lambda,\beta)=h(\lambda/, \beta’)$Namely, topological entropy onshow the followings:

Proposition 4such thatis constant for all $m(\geq 1)$ . But we can$(\lambda’, \beta’)\in\{(\lambda, \beta)\in D;\lambda\leq 1\}$$h(\lambda, \beta)<h(\lambda’,\beta’)$. If$(\lambda, \beta)<(\lambda’,\beta’)$,.

naturallyfor
Proof From [MT88] we obtain that topological entropy offollows from one of its renormalized map of subdomain $a\geq 1$ where themonotonicity holds.
A counter example to this statement is given in [Ich].
$f_{a,b}$$B_{1}$$\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{l}\mathrm{y}\square$

The author would like very much to thank Professor K.
Acknowledgments.
Nishizawa of Josai University and Professor M. Fujimura of National Defense
Academy for their valuable discussions and suggestions.

98

References[ITN79a] S. Ito, S. Tanaka, and H. Nakada. On unimodal linear transformations
and chaos I. Tokyo J. Math., $2(2):221^{-239}$ , 1979.
[ITN79b] S. Ito, S. Tanaka, and H. Nakada. On unimodal linear transformations
and chaos Il. Tokyo J. Math., $2(2):241^{-259}$ , 1979.

[NY95]

H. E. Nusse and J. A. Yorke. Border-collision bifurcations for piecewisesmooth one-dimensional maps. International Journal of Bifurcation
and Chaos, $5(1):189^{-}207$ , 1995.

[CE80]

P. Collet and J. P. Eckmann. Iterated maps on the interval as dynamicalsystems. Progress in Physics. Birkhauser, Boston, 1980.

[Dev89]R. L. Devaney. An
Wesley,$\grave{1}989$$i..ntroducti_{\mathit{0}}n$to chaotic dynamical systems. Addison-.

[MV91]J. C. Misiurewicz and E. Visinescu. Kneading sequences of skew tent
maps. Ann. Inst. Henri Poincare, 27:125-140, 1991.

[MV92]J. C. Marcuard and E. Visinescu. Monotonicity properties of someskew tent maps. Ann. Inst. Henri Poincare, 28:1-29, 1992.

[MT88]

J. Milnor and W. Thurston. On iterated maps on the interval. Springer
LNM 1342 (1988),465-563.

[IN97a]K. Ichimura and K. Nishizawa. Bifurcations for skew tent maps I (stairtype). RIMS Kokyuroku, 986:41-48. Kyoto Univ., 1997.

[IN97b]M. Ito and K. Nishizawa. Bifurcations for skew tent maps II (renormalization). RIMS Kokyuroku, 986:49-56. Kyoto Univ., 1997.

[lch]

K. Ichimura. Dynamics and kneading sequences of skew tent maps.
Science Bulletin of Josai university, Special Issue 4, to appear.

[Ito]M. lto. Renormalization and topological entropy of skew tent maps.
Science Bulletin of Josai university, Special Issue 4, to appear.