Iterates of maps on an interval.

*(English)*Zbl 0582.58001
Lecture Notes in Mathematics. 999. Berlin-Heidelberg-New York-Tokyo: Springer-Verlag. VII, 205 p. DM 28.00; $ 11.60 (1983).

The book is an introduction and simultaneously a review of selected topics in the advanced theory of iterates of maps of an interval, and it contains several new important results and some new proofs of known results as well. A good preliminary impression about these topics is given by the titles of chapters: 1. Introduction, 2. Piecewise monotone functions, 3. Well-behaved piecewise monotone functions, 4. Property R and negative Schwarzian derivatives, 5. The iterates of functions in \({\mathcal S}\), 6. Reductions, 7. Getting rid of homtervals, 8. Kneading sequences, 9. An ”almost all” version of Theorem 5.1, 10. Occurrence of the different types of behaviour, References, Indexes.

The fundamental classes of functions discussed in the book are: functions piecewise monotone and \({\mathcal S}:=\{f: [0,1]\to [0,1]|\) f is continuous and there exists \(a\in (0,1)\) such that f is strictly increasing in [0,a] and strictly decreasing in [a,1]\(\}\). The main questions investigated by the author are of the type: what kind of behaviour can be exhibited by the iterates of a function in \({\mathcal S}?\) In Sec. 1 (Introduction) there are given some fundamental examples based on a short discussion of mappings \(f_{\mu}\) and \(g_ r\) defined by formulae \(f_{\mu}(x)=\mu x(1-x)\) \((0<\mu \leq 4)\) \(g_ r(x)=rx \exp (1- rx)\) \((r>1)\) and showing that there are several possibilities of qualitative properties of that mappings, depending on particular values of \(\mu\) and r. In this section we have also a short review of problems investigated in the sequel together with references and remarks concerning papers of other authors.

In Sec. 2 the author considers piecewise monotone functions and introduces the notion of a trapped periodic point, besides the usual discussion and classification of periodic points as being either stable, one-sided stable or unstable. A periodic point x with a period n is said to be trapped if there exist \(y<x<z\) and \(\delta >0\) such that the 2n-th iteration \(f^{2n}\) of f (being the function under consideration) is increasing on the interval \([y-\delta,z+\delta]\) and \(f^{2n}(y)\leq y\), \(f^{2n}(z)\geq z\). If the periodic point is trapped then all points of its orbit are also; then the orbit [x] is said to be trapped as well. The main result of the section implies that if \(f\in {\mathcal S}\) then has at most one periodic orbit \([x]:=\{x,f(x),...,f^{n-1}(x)\}\) such that (i) [x] is either stable or one-sided stable, (ii) [x] is not trapped, (iii) x is not a fixed point of f in [0,a) (where a is the turning point of f). Moreover if such an orbit [x] exists then for some \(\delta >0\) all the points in \((a-\delta,a)\cup (a,a+\delta)\) are attracted to [x].

The aim of Sec. 3 is to find sufficient conditions under which for a given function f: [a,b]\(\to [a,b]\) being piecewise monotone, the set \(A(f):=\{x\in [a,b]:\) lim \(f^{nm}(x)=1\) (m\(\to \infty)\) exists for some \(n\geq 1\}\), contains a dense open set in [a,b], i.e. under which a ”typical” point in [a,b] gets attracted to a periodic orbit. Interesting answers to this problem are given by lemmas and theorems presented in that section.

Sec. 4 deals with the subclass of piecewise monotone functions having the property R in the sense P. Collet and J.-P. Eckmann [Iterated maps on the interval as dynamical systems (1980; Zbl 0458.58002)]. All their trapped orbits are unstable. Among other results of Sec. 4 there is the following one: if \(f\in {\mathcal S}\) and f has the continuous second derivative f” in (0,a)\(\cup (a,1)\), f’(x)\(\neq 0\) for all \(x\in (0,a)\cup (a,1)\) and \(f(z)>z\) for all \(z\in (0,a)\), then f has at most one stable or one-sided stable periodic orbit; if that orbit exists then the set of points which get attracted to it contains a dense open subset of [0,1], if it does not exist then A(f) is countable.

The result which is claimed as the main one in the book is presented in Sec. 5 (Theorem 5.2). It classifies the types of behaviour that can be exhibited by the iterates of functions in \({\mathcal S}\). As a special version of this classification the author obtained results of J. Guckenheimer [Commun. Math. Phys. 70, 133-160 (1979; Zbl 0429.58012)] and M. Misiurewicz [Publ. Math., Inst. Hautes Etud. Sci. 53, 17-51 (1981; Zbl 0477.58020)]. In Sec. 8 the author discusses kneading sequences \(\{f_ k(n):\) \(n\geq 0\}\) for \(f\in {\mathcal S}:\) \(k_ f(n)=-1\) if \(f^ n(a)<a\), \(k_ f(n)=0\) if \(f^ n(a)=a\) and \(k_ f(n)=1\) if \(f^ n(a)>0\), where a is the turning point of f. The connections between properties of f and \(\{k_ f(n)\}\) are investigated. In Sec. 9 in particular a theorem is proved saying that if \(f\in {\mathcal S}\) and it satisfies suitable additional conditions (in particular: the condition R and the regularity of the class \(C^ 2)\) then from the inequality \(f(z)>z\) for all \(z\in (0,a)\) it follows that: if f has a stable periodic orbit [x] then the set of points which are attracted to [x] has Lebesgue measure one. Sec. 10 shows that each of the three types of properties of \(f\in {\mathcal S}\) introduced in the classification presented in Sec. 5 actually occurs. It is shown that we can construct one-parameter families of functions (such as \(f_{\mu}(x):=\mu x(1-x)\) \(2<\mu \leq 4\), \(f_{\mu}(x)=\sin (\mu x)\) \(\pi /2<\mu \leq \pi\), \(f_{\mu}(x)=\mu x \exp (1-\mu x)\), \(\mu >1)\) containing infinitely many functions of each type.

The fundamental classes of functions discussed in the book are: functions piecewise monotone and \({\mathcal S}:=\{f: [0,1]\to [0,1]|\) f is continuous and there exists \(a\in (0,1)\) such that f is strictly increasing in [0,a] and strictly decreasing in [a,1]\(\}\). The main questions investigated by the author are of the type: what kind of behaviour can be exhibited by the iterates of a function in \({\mathcal S}?\) In Sec. 1 (Introduction) there are given some fundamental examples based on a short discussion of mappings \(f_{\mu}\) and \(g_ r\) defined by formulae \(f_{\mu}(x)=\mu x(1-x)\) \((0<\mu \leq 4)\) \(g_ r(x)=rx \exp (1- rx)\) \((r>1)\) and showing that there are several possibilities of qualitative properties of that mappings, depending on particular values of \(\mu\) and r. In this section we have also a short review of problems investigated in the sequel together with references and remarks concerning papers of other authors.

In Sec. 2 the author considers piecewise monotone functions and introduces the notion of a trapped periodic point, besides the usual discussion and classification of periodic points as being either stable, one-sided stable or unstable. A periodic point x with a period n is said to be trapped if there exist \(y<x<z\) and \(\delta >0\) such that the 2n-th iteration \(f^{2n}\) of f (being the function under consideration) is increasing on the interval \([y-\delta,z+\delta]\) and \(f^{2n}(y)\leq y\), \(f^{2n}(z)\geq z\). If the periodic point is trapped then all points of its orbit are also; then the orbit [x] is said to be trapped as well. The main result of the section implies that if \(f\in {\mathcal S}\) then has at most one periodic orbit \([x]:=\{x,f(x),...,f^{n-1}(x)\}\) such that (i) [x] is either stable or one-sided stable, (ii) [x] is not trapped, (iii) x is not a fixed point of f in [0,a) (where a is the turning point of f). Moreover if such an orbit [x] exists then for some \(\delta >0\) all the points in \((a-\delta,a)\cup (a,a+\delta)\) are attracted to [x].

The aim of Sec. 3 is to find sufficient conditions under which for a given function f: [a,b]\(\to [a,b]\) being piecewise monotone, the set \(A(f):=\{x\in [a,b]:\) lim \(f^{nm}(x)=1\) (m\(\to \infty)\) exists for some \(n\geq 1\}\), contains a dense open set in [a,b], i.e. under which a ”typical” point in [a,b] gets attracted to a periodic orbit. Interesting answers to this problem are given by lemmas and theorems presented in that section.

Sec. 4 deals with the subclass of piecewise monotone functions having the property R in the sense P. Collet and J.-P. Eckmann [Iterated maps on the interval as dynamical systems (1980; Zbl 0458.58002)]. All their trapped orbits are unstable. Among other results of Sec. 4 there is the following one: if \(f\in {\mathcal S}\) and f has the continuous second derivative f” in (0,a)\(\cup (a,1)\), f’(x)\(\neq 0\) for all \(x\in (0,a)\cup (a,1)\) and \(f(z)>z\) for all \(z\in (0,a)\), then f has at most one stable or one-sided stable periodic orbit; if that orbit exists then the set of points which get attracted to it contains a dense open subset of [0,1], if it does not exist then A(f) is countable.

The result which is claimed as the main one in the book is presented in Sec. 5 (Theorem 5.2). It classifies the types of behaviour that can be exhibited by the iterates of functions in \({\mathcal S}\). As a special version of this classification the author obtained results of J. Guckenheimer [Commun. Math. Phys. 70, 133-160 (1979; Zbl 0429.58012)] and M. Misiurewicz [Publ. Math., Inst. Hautes Etud. Sci. 53, 17-51 (1981; Zbl 0477.58020)]. In Sec. 8 the author discusses kneading sequences \(\{f_ k(n):\) \(n\geq 0\}\) for \(f\in {\mathcal S}:\) \(k_ f(n)=-1\) if \(f^ n(a)<a\), \(k_ f(n)=0\) if \(f^ n(a)=a\) and \(k_ f(n)=1\) if \(f^ n(a)>0\), where a is the turning point of f. The connections between properties of f and \(\{k_ f(n)\}\) are investigated. In Sec. 9 in particular a theorem is proved saying that if \(f\in {\mathcal S}\) and it satisfies suitable additional conditions (in particular: the condition R and the regularity of the class \(C^ 2)\) then from the inequality \(f(z)>z\) for all \(z\in (0,a)\) it follows that: if f has a stable periodic orbit [x] then the set of points which are attracted to [x] has Lebesgue measure one. Sec. 10 shows that each of the three types of properties of \(f\in {\mathcal S}\) introduced in the classification presented in Sec. 5 actually occurs. It is shown that we can construct one-parameter families of functions (such as \(f_{\mu}(x):=\mu x(1-x)\) \(2<\mu \leq 4\), \(f_{\mu}(x)=\sin (\mu x)\) \(\pi /2<\mu \leq \pi\), \(f_{\mu}(x)=\mu x \exp (1-\mu x)\), \(\mu >1)\) containing infinitely many functions of each type.

Reviewer: A.Pelczar

##### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

54H20 | Topological dynamics (MSC2010) |

26A18 | Iteration of real functions in one variable |

37G99 | Local and nonlocal bifurcation theory for dynamical systems |