Skip to main content

Embeddings and Factor Maps

  • Chapter
Symbolic Dynamics

Part of the book series: Universitext ((UTX))

  • 1188 Accesses

Abstract

In this chapter we will examine some questions about embeddings and factor maps. An embedding is a continuous, invertible, shift commuting map from one subshift of finite type into another. A factor map is a continuous, shift commuting map from one subshift of finite type onto another. We will concentrate on two-sided subshifts of finite type and then see how these results carry over to one-sided subshifts of finite type.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. Adler L.W. Goodwyn and B. Weiss, Equivalence of Topological Markov Shifts, Israel Journal of Mathematics no. 27 (1977), 49–63.

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Adler and B. Marcus, Topological Entropy and Equivalence of Dynamical Systems, Memoirs of the American Mathematical Society no. 219 (1979).

    Google Scholar 

  3. R. Adler and B. Weiss, Entropy, a Complete Metric Invariant for Automorphisms of the Torus, Proceedings of the National Academy of Sciences, USA no. 57 (1967), 1573–1576.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Ashley, Bounded-to-1 Factors of an Aperiodic Shift of Finite Type Are 1-to-1 Almost Everywhere Factors Also, Ergodic Theory and Dynamical Systems 10 (1990) 615–625.

    MathSciNet  MATH  Google Scholar 

  5. J. Ashley, Resolving Factor Maps for Shifts of Finite Type with Equal Entropy, Ergodic Theory and Dynamical Systems 11 (1991), 219–240.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Boyle, Lower Entropy Factors of Sofic Systems Ergodic Theory and Dynamical Systems, 4 (1984), 541–557.

    Article  MathSciNet  Google Scholar 

  7. M. Boyle, Constraints on the Degree of Sofic Homomorphisms and the Induced Multiplication of Measures on Unstable Sets, Israel Journal of Mathematics 53 (1986), 52–68.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Boyle, Factoring Factor Maps, preprint.

    Google Scholar 

  9. M. Boyle, B. Marcus and P. Trow, Resolving Maps and the Dimension Group for Shifts of Finite Type, Memoirs of the American Mathematical Society no. 377 (1987).

    Google Scholar 

  10. E. Coven and M. Paul, Endomorphisms of Irreducible Subshifts of Finite Type, Mathematical Systems Theory 8 (1974), 167–175.

    Article  MathSciNet  Google Scholar 

  11. J. Friedman, On the Road Coloring Problem, Proceedings of the American Mathematical Society 110 (1990), 1133–1135.

    Article  MathSciNet  MATH  Google Scholar 

  12. G.A. Hedlund, Transformations Commuting with the Shift, Topological Dynamics (J. Auslander and W. Gottschalk, eds.), W.A. Benjamin, 1968.

    Google Scholar 

  13. G.A. Hedlund, Endomorphisms and Automorphisms of the Shift Dynamical System, Mathematical Systems Theory 3 no. 4 (1969), 320–375.

    Article  MathSciNet  MATH  Google Scholar 

  14. B. Kitchens, An Invariant for Continuous Factors of Markov Shifts, Proceedings of the American Mathematical Society 83 (1981), 825–828.

    Article  MathSciNet  MATH  Google Scholar 

  15. B. Kitchens, P. Trow and B. Marcus, Eventual Factor Maps and Compositions of Closing Maps, Ergodic Theory and Dynamical Systems 11 (1991), 857–913.

    Article  MathSciNet  Google Scholar 

  16. W. Krieger, On the Periodic Points of Topological Markov Chains, Mathematische Zeitschrift 169 (1979), 99–104.

    Article  MathSciNet  MATH  Google Scholar 

  17. W. Krieger, On Subsystems of Topological Markov Chains, Ergodic Theory and Dynamical Systems 2 (1982), 195–202.

    Article  MathSciNet  MATH  Google Scholar 

  18. B. McMillan, The Basic Theorems of Information Theory, Annals of Mathematical Statistics 24 (1953), 196–219.

    Article  MathSciNet  MATH  Google Scholar 

  19. B. Marcus, Factors and Extensions of Full Shifts, Monatshefte fü Mathematik 88 (1979), 239–247.

    Article  MATH  Google Scholar 

  20. M. Nasu, Uniformly Finite-to-one and Onto Extensions of Homomorphisms Between Strongly Connected Graphs, Discrete Mathematics no. 39 1982, 171–197.

    Article  MathSciNet  MATH  Google Scholar 

  21. G.L. O’Brien, The Road Coloring Problem, Israel Journal of Mathematics 39 (1981), 145–154.

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Trow, Resolving Maps which Commute with a Power of the Shift, Ergodic Theory and Dynamical Systems 6 (1986), 281–293.

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Trow, Degrees of Finite-to-one Factor Maps, Israel Journal of Mathematics 71 (1990), 229–238.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Kitchens, B.P. (1998). Embeddings and Factor Maps. In: Symbolic Dynamics. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58822-8_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-58822-8_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62738-8

  • Online ISBN: 978-3-642-58822-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics