Topological entropy of unimodal maps. Monotonicity for quadratic polynomials. (English) Zbl 0923.58018
Branner, Bodil (ed.) et al., Real and complex dynamical systems. Proceedings of the NATO Advanced Study Institute held in Hillerød, Denmark, June 20-July 2, 1993. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 464, 65-87 (1995).
The author considers the family \(f_c(x)=x^2+c\) acting on \(\overline\mathbb{R}\), where \(c\in\mathbb{R}\), and shows the following main theorem:
(a) the topological entropy \(c\to h_{\text{top}}(f_c)\) as a map from \(\mathbb{R} \to[0,\log 2]\) is a (weakly) decreasing and continuous function.
(b) \(h_{ \text{top}}(f_c)>0\) iff \(c<c_F\) (Feigenbaum point).
(c) \(h_{\text{top}} (f_x)=h_{\text{top}}(f_{c'})>0\) and only if \(c\) and \(c'\) are both tuned from the same \(c_0\) with \(h_{\text{top}}(f_{c_0})>0\).
The tuning process is defined in section 4 and may be considered as the inverse operation to renormalization.
For the entire collection see [Zbl 0818.00014].
(a) the topological entropy \(c\to h_{\text{top}}(f_c)\) as a map from \(\mathbb{R} \to[0,\log 2]\) is a (weakly) decreasing and continuous function.
(b) \(h_{ \text{top}}(f_c)>0\) iff \(c<c_F\) (Feigenbaum point).
(c) \(h_{\text{top}} (f_x)=h_{\text{top}}(f_{c'})>0\) and only if \(c\) and \(c'\) are both tuned from the same \(c_0\) with \(h_{\text{top}}(f_{c_0})>0\).
The tuning process is defined in section 4 and may be considered as the inverse operation to renormalization.
For the entire collection see [Zbl 0818.00014].
Reviewer: M.Denker (Göttingen)
MSC:
| 37E99 | Low-dimensional dynamical systems |
| 37A99 | Ergodic theory |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 26A18 | Iteration of real functions in one variable |
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 54C70 | Entropy in general topology |
