##
**On iterated maps of the interval.**
*(English)*
Zbl 0664.58015

Dynamical systems, Proc. Spec. Year, College Park/Maryland, Lect. Notes Math. 1342, 465-563 (1988).

[For the entire collection see Zbl 0653.00011.]

The paper sets up an effective calculus for describing the qualitative behaviour of the successive iterates of a piecewise monotone mapping f of an interval into itself. Section 1 studies the sequence of lap numbers \((=\) numbers of subintervals of monotonicity) \(\ell (f)\), \(\ell (f^ 2)\),... and the growth number \(s=\lim_{k\to \infty}\ell (f^ k)^{1/k}.\) Sections 2 and 3 investigate the itinerary of a point and an invariant coordinate function. The kneading matrix and its determinant are introduced in section 4. An explicit method for computing the lap numbers in terms of the kneading matrix is given in section 5. Section 6 studies convergence properties of formal power series introduced so far. If \(s>1\) then f is topologically semiconjugate to a piecewise linear map with slope \(\pm s\), which is shown in section 7. Section 8 surveys a number of known methods and theorems concerning periodic points.

Section 9 and 10 state several versions of the main theorem of the authors which computes the Artin-Mazur zeta function (a formal power series) in terms of the kneading determinant. As an application it is shown that \(s>1\) iff there exists a periodic point whose period is not a power of two. Furthermore there are only finitely many distinct periods (all are powers of two) iff the sequence \(\ell (f^ k)\) is bounded by a power of k. Section 12 characterizes those power series which can actually occur as kneading determinant for some f with \(\ell (f)=2\). Also continuity properties of \(s=s(f)\) under smooth deformations of f are studied. A monotonicity theorem for the kneading invariant of a family of quadratic maps is proved in section 14. The last section constructs a number of illustrative examples.

Most of the results were obtained before 1983.

The paper sets up an effective calculus for describing the qualitative behaviour of the successive iterates of a piecewise monotone mapping f of an interval into itself. Section 1 studies the sequence of lap numbers \((=\) numbers of subintervals of monotonicity) \(\ell (f)\), \(\ell (f^ 2)\),... and the growth number \(s=\lim_{k\to \infty}\ell (f^ k)^{1/k}.\) Sections 2 and 3 investigate the itinerary of a point and an invariant coordinate function. The kneading matrix and its determinant are introduced in section 4. An explicit method for computing the lap numbers in terms of the kneading matrix is given in section 5. Section 6 studies convergence properties of formal power series introduced so far. If \(s>1\) then f is topologically semiconjugate to a piecewise linear map with slope \(\pm s\), which is shown in section 7. Section 8 surveys a number of known methods and theorems concerning periodic points.

Section 9 and 10 state several versions of the main theorem of the authors which computes the Artin-Mazur zeta function (a formal power series) in terms of the kneading determinant. As an application it is shown that \(s>1\) iff there exists a periodic point whose period is not a power of two. Furthermore there are only finitely many distinct periods (all are powers of two) iff the sequence \(\ell (f^ k)\) is bounded by a power of k. Section 12 characterizes those power series which can actually occur as kneading determinant for some f with \(\ell (f)=2\). Also continuity properties of \(s=s(f)\) under smooth deformations of f are studied. A monotonicity theorem for the kneading invariant of a family of quadratic maps is proved in section 14. The last section constructs a number of illustrative examples.

Most of the results were obtained before 1983.

Reviewer: G.Jetschke