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 12 - EQUAL ENTROPY FACTORS
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
The Lower Entropy Factor Theorem of §10.3 completely solves the problem of when one irreducible shift of finite type factors onto another of strictly lower entropy. In contrast, the corresponding problem for equal entropy does not yet have a satisfactory solution. This chapter contains some necessary conditions for an equal-entropy factor code, and also some sufficient conditions. We will also completely characterize when one shift of finite type “eventually” factors onto another of the same entropy.
In §12.1 we state a necessary and sufficient condition for one irreducible shift of finite type to be a right-closing factor of another. While the proof of this is too complicated for inclusion here, we do prove an “eventual” version: for irreducible shifts of finite type X and Y with equal entropy we determine when Ym is a right-closing factor of Xm for all sufficiently large m. In §12.2 we extend this to determine when Xm factors onto Ym for all sufficiently large m (where the factor code need not be right-closing). This is analogous to Theorem 7.5.15, which showed that two irreducible edge shifts are eventually conjugate if and only if their associated matrices are shift equivalent.
At the end of §10.3 we proved a generalization of the Finite-State Coding Theorem from Chapter 5, where the sofic shifts had different entropies. This amounted to the construction of right-closing finite equivalences. In §12.3 we show that given two irreducible edge shifts with equal entropy log λ, the existence of a right-closing finite equivalence forces an arithmetic condition on the entries of the corresponding Perron eigenvectors.
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- An Introduction to Symbolic Dynamics and Coding , pp. 402 - 430Publisher: Cambridge University PressPrint publication year: 1995