دانلود کتاب Topics from One-Dimensional Dynamics

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Contents


List of Figures


Preface


1 Topological Roots
1.1 Basics from Topology
1.2 Middle Third Cantor Set

2 Measure Theoretic Roots 12
2.1 Basics of Lebesgue Measure on 1R
2.2 A Nonmeasurable Set
2.3 Lebesgue Measure of Cantor Sets
2.3.1 The Middle Third Cantor Set
2.3.2 Other Cantor Sets
2.4 Sets of Lebesgue Measure Zero

3 Beginning Symbolic and Topological Dynamics
3.1 Periodic Behavior
3.2 Nonwandering and W—Limit Sets
3.3 Topological Conjugacy
3.4 Transitive Behavior
3.5 Recurrence
3.6 Shift Spaces

4 Beginning Measurable Dynamics
4.1 Preliminaries
4.2 Measurable Maps on I
4.3 Poincare Recurrence
4.4 Ergodicity
4.4.1 Integration of Measurable Functions
4.4.2 Averaging Measurable Functions Along Orbits
4.4.3 A Connection to Topological Dynamics

5 A First Example: The 2°°Map
5.1 Logistic Family
5.2 A Bit of Combinatorics
5.3 Construction of the Cantor Set w(c,g)
5.4 Cantor Set and Adding Machines
5.5 A Toeplitz Sequence

6 Kneading Maps
6.1 Hofbauer Towers and Kneading Maps
6.2 First Uses of Kneading Maps
6.3 Shadowing
6.4 Examples of Kneading Maps

7 Some Number Theory
7.1 Farey Tree
7.2 Continued Fractions
7.3 Continued Fractions and the Farey Tree

8 Circle Maps
8.1 Circle Homeomorphisms
8.2 Degree One Circle Maps
8.3 Irrational Rotations and Return Maps
8.4 Cantor Thread

9 Topological Entropy
9.1 Basic Properties of Topological Entropy
9.2 Entropy of Subshifts
9.3 Lapnumbers and Markov Extensions
9.4 Lapnumbers and Entropy
9.5 Semiconjugacy to a Piecewise Linear Map
9.6 The Monotonicity Problem

10 Symmetric Tent Maps
10.1 Preliminary Combinatorics
10.2 w—Limit Sets
10.3 Phase Portrait
10.4 Measure Results
10.5 Slow Recurrence and the CE Condition
10.6 Attractors
10.7 Combinatorics and Renormalization

11 Unimodal Maps and Rigid Rotations
11.1 Adding Machines in Unimodal Maps
11.2 Rigid Rotations in Unimodal Maps— I
11.3 Rigid Rotations in Unimodal Maps — II

12 fi—Transformations, Unimodal Maps, and Circle Maps
12.1 fi—Transformations and fi—Expansions
12.2 Flip—Half—of—the-Graph Trick
12.3 A Relation Between Unimodal Maps and Circle Maps
12.4 Comparing fi-Transformations and Tent Maps
12.5 Ledrappier\'s Example
12.6 Maps with Slope < 2

13 Homeomorphic Restrictions in the Unimodal Setting
13.1 First Observations
13.2 A 2°° Trapezoidal Map
13.3 The Adding Machine (Q,P)
13.4 The Case Q(k)--> + oo

14 Complex Quadratic Dynamics
14.1 Julia Sets and External Rays
14.2 The Mandelbrot Set
14.3 Itineraries and Hubbard Trees

Bibliography

Index