Book contents
- Frontmatter
- Contents
- PREFACE
- CHAPTER 1 SHIFT SPACES
- CHAPTER 2 SHIFTS OF FINITE TYPE
- CHAPTER 3 SOFIC SHIFTS
- CHAPTER 4 ENTROPY
- CHAPTER 5 FINITE-STATE CODES
- CHAPTER 6 SHIFTS AS DYNAMICAL SYSTEMS
- CHAPTER 7 CONJUGACY
- CHAPTER 8 FINITE-TO-ONE CODES AND FINITE EQUIVALENCE
- CHAPTER 9 DEGREES OF CODES AND ALMOST CONJUGACY
- CHAPTER 10 EMBEDDINGS AND FACTOR CODES
- CHAPTER 11 REALIZATION
- CHAPTER 12 EQUAL ENTROPY FACTORS
- CHAPTER 13 GUIDE TO ADVANCED TOPICS
- BIBLIOGRAPHY
- NOTATION INDEX
- INDEX
CHAPTER 10 - EMBEDDINGS AND FACTOR CODES
Published online by Cambridge University Press: 30 November 2009
- Frontmatter
- Contents
- PREFACE
- CHAPTER 1 SHIFT SPACES
- CHAPTER 2 SHIFTS OF FINITE TYPE
- CHAPTER 3 SOFIC SHIFTS
- CHAPTER 4 ENTROPY
- CHAPTER 5 FINITE-STATE CODES
- CHAPTER 6 SHIFTS AS DYNAMICAL SYSTEMS
- CHAPTER 7 CONJUGACY
- CHAPTER 8 FINITE-TO-ONE CODES AND FINITE EQUIVALENCE
- CHAPTER 9 DEGREES OF CODES AND ALMOST CONJUGACY
- CHAPTER 10 EMBEDDINGS AND FACTOR CODES
- CHAPTER 11 REALIZATION
- CHAPTER 12 EQUAL ENTROPY FACTORS
- CHAPTER 13 GUIDE TO ADVANCED TOPICS
- BIBLIOGRAPHY
- NOTATION INDEX
- INDEX
Summary
When can we embed one shift of finite type into another? When can we factor one shift of finite type onto another? The main results in this chapter, the Embedding Theorem and the Lower Entropy Factor Theorem, tell us the answers when the shifts have different entropies. For each theorem there is a simple necessary condition on periodic points, and this condition turns out to be sufficient as well. In addition, these periodic point conditions can be verified with relative ease.
We state and prove the Embedding Theorem in §10.1. The necessity of the periodic point condition here is easy. The sufficiency makes use of the fundamental idea of a marker set to construct sliding block codes. In §10.2 we prove the Masking Lemma, which shows how to represent embeddings in a very concrete form; we will use this in Chapter 11 to prove a striking application of symbolic dynamics to linear algebra. §10.3 contains the statement and proof of the Lower Entropy Factor Theorem, which is in a sense “dual” to the Embedding Theorem. The proof employs a marker construction similar to that of the Embedding Theorem. One consequence is an unequal entropy version of the Finite Equivalence Theorem.
The Embedding Theorem
Suppose that X and Y are irreducible shifts of finite type. When is there an embedding from X into Y? If the embedding is also onto, then X and Y are conjugate.
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- An Introduction to Symbolic Dynamics and Coding , pp. 337 - 366Publisher: Cambridge University PressPrint publication year: 1995