Ordering by auto-expansions. (English) Zbl 0774.58011
For \(\lambda\in(1,2)\) the autoexpansion of \(\lambda\) is the sequence \((c_ i)^ \infty_{i=0}\), \(c_ i=-1,0,1\), defined inductively by \(c_ 0:=1\), \(x_ 0:=1\), and for \(n\geq 1\), if \(x_{n-1}<\lambda\), \(c_ n:=1\); if \(x_{n-1}>\lambda\), \(c_ n:=-1\); if \(x_{n-1}=\lambda\), \(c_ n:=0\); and \(x_ n:=x_{n-1}+c_ n/\lambda^ n\).
Defining an order on sequences by \((a_ i)^ \infty_{i=0}<(b_ i)^ \infty_{i=0}\) if \(a_ m<b_ m\), where \(a_ i=b_ i\), \(i=0,\dots,m- 1\), the author shows that the autoexpansion of \(\lambda\) is strictly monotonic with respect to \(\lambda\), i.e. if \((a_ i)^ \infty_{i=0}\), \((b_ i)^ \infty_{i=0}\) are the autoexpansions of \(\lambda\), \(\mu\) respectively, then \(\lambda<\mu\) implies \((a_ i)^ \infty_{i=0}<(b_ i)^ \infty_{i=0}\). This result was first claimed by B. Derrida, A. Gervois and Y. Pomeau [Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A, 29, No. 3, 305-356 (1978; Zbl 0416.28012); J. Phys. A 12, No. 3, 269-296 (1979; Zbl 0416.28011)], but their proof was incomplete. The author furnishes a complete proof in this paper.
The work has application to the kneading sequences of families of unimodal maps of the interval of the form \(T:x\to\lambda f(x)\).
Defining an order on sequences by \((a_ i)^ \infty_{i=0}<(b_ i)^ \infty_{i=0}\) if \(a_ m<b_ m\), where \(a_ i=b_ i\), \(i=0,\dots,m- 1\), the author shows that the autoexpansion of \(\lambda\) is strictly monotonic with respect to \(\lambda\), i.e. if \((a_ i)^ \infty_{i=0}\), \((b_ i)^ \infty_{i=0}\) are the autoexpansions of \(\lambda\), \(\mu\) respectively, then \(\lambda<\mu\) implies \((a_ i)^ \infty_{i=0}<(b_ i)^ \infty_{i=0}\). This result was first claimed by B. Derrida, A. Gervois and Y. Pomeau [Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A, 29, No. 3, 305-356 (1978; Zbl 0416.28012); J. Phys. A 12, No. 3, 269-296 (1979; Zbl 0416.28011)], but their proof was incomplete. The author furnishes a complete proof in this paper.
The work has application to the kneading sequences of families of unimodal maps of the interval of the form \(T:x\to\lambda f(x)\).
MSC:
| 37E99 | Low-dimensional dynamical systems |
| 28D99 | Measure-theoretic ergodic theory |
| 28D05 | Measure-preserving transformations |
References:
| [1] | Collet, P.; Eckmann, J. P., Iterated Maps on the Interval as Dynamical Systems (1980), Birkhaüser: Birkhaüser Boston · Zbl 0458.58002 |
| [2] | Derrida, B.; Gervois, A.; Pomeau, Y., Ann. Inst. H. Poincaré, 29, 305 (1978) · Zbl 0416.28012 |
| [3] | Derrida, B.; Gervois, A.; Pomeau, Y., J. Phys. A, 12, No. 3, 269 (1979) · Zbl 0416.28011 |
| [4] | Feigenbaum, M., J. Statist. Phys., 19, 25-52 (1978) · Zbl 0509.58037 |
| [5] | Louck, J. D.; Metropolis, N., (Symbolic Dynamics of Trapezoidal Maps (1986), Reidel: Reidel Dordrecht) · Zbl 0668.58040 |
| [6] | May, R., Nature, 261, 459 (1976) |
| [7] | Metropolis, N.; Stein, M. L.; Stein, P. R., J. Combin. Theory Ser. A, 15, 25 (1973) · Zbl 0259.26003 |
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