Book contents
- Frontmatter
- Contents
- List of Figures
- Preface
- 1 Topological Roots
- 2 Measure Theoretic Roots
- 3 Beginning Symbolic and Topological Dynamics
- 4 Beginning Measurable Dynamics
- 5 A First Example: The 2∞ Map
- 6 Kneading Maps
- 7 Some Number Theory
- 8 Circle Maps
- 9 Topological Entropy
- 10 Symmetric Tent Maps
- 11 Unimodal Maps and Rigid Rotations
- 12 β-Transformations, Unimodal Maps, and Circle Maps
- 13 Homeomorphic Restrictions in the Unimodal Setting
- 14 Complex Quadratic Dynamics
- Bibliography
- Index
13 - Homeomorphic Restrictions in the Unimodal Setting
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- List of Figures
- Preface
- 1 Topological Roots
- 2 Measure Theoretic Roots
- 3 Beginning Symbolic and Topological Dynamics
- 4 Beginning Measurable Dynamics
- 5 A First Example: The 2∞ Map
- 6 Kneading Maps
- 7 Some Number Theory
- 8 Circle Maps
- 9 Topological Entropy
- 10 Symmetric Tent Maps
- 11 Unimodal Maps and Rigid Rotations
- 12 β-Transformations, Unimodal Maps, and Circle Maps
- 13 Homeomorphic Restrictions in the Unimodal Setting
- 14 Complex Quadratic Dynamics
- Bibliography
- Index
Summary
Given a unimodal map ƒ: [0, 1] → [0, 1], we are interested in closed invariant subsets E ⊂ [0, 1] such that the restriction of ƒ to E (denoted f|E) is an onto homeomorphism (recall Definition 1.1.37). If such an E were a finite set, it would consist of a finite number of periodic orbits. Hence, we are interested in the case when E is not finite. Chapter 3 and Sections 6.1, 6.2, and 11.1 contain background material for this chapter.
Our interest in this dynamical behavior is motivated by the following example and question. In Chapter 5 we investigated the 2∞ map, g*, from the unimodal logistic family ga{x) - ax{1 – x). We found that ω(c, g*) was minimal and Cantor, and that g*|ω(c,g*) is an onto homeomorphism (recall Exercises 3.2.5, 5.1.9, and 5.4.7). There are many other maps in the logistic family such that:
ω(c, ga) is minimal.
ω(c, ga) is a Cantor set.
ga|ω(c,ga) is an onto homeomorphism.
In fact, any infinitely renormalizable map in this family has these three properties; moreover, the Lebesgue measure of such an ω(c,ga) is zero [78, 115]. One then asks, are there nonrenormalizable maps with these three properties? If some ga has an attracting periodic orbit, then w(c,ga) is precisely that orbit. Hence, we are interested in maps from the logistic family that are not renormalizable and which do not have an attracting periodic orbit. Any such map is topologically conjugate to a symmetric tent map (recall Theorem 3.4.27 and see [115, 39]).
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- Topics from One-Dimensional Dynamics , pp. 216 - 249Publisher: Cambridge University PressPrint publication year: 2004