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On the Garden of Eden theorem for B-free subshifts

Gerhard Keller, Mariusz Lemanczyk, Christoph Richard, Daniel Sell

Abstract

We prove that on B-free subshifts, with B satisfying the Erdös condition, all cellular automata are determined by monotone sliding block codes. In particular, this implies the validity of the Garden of Eden theorem for such systems.

On the Garden of Eden theorem for B-free subshifts

Abstract

We prove that on B-free subshifts, with B satisfying the Erdös condition, all cellular automata are determined by monotone sliding block codes. In particular, this implies the validity of the Garden of Eden theorem for such systems.

Paper Structure

This paper contains 23 sections, 26 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Sets of multiples and B-free sets
  3. Endomorphisms of subshifts
  4. Results of the paper
  5. Plan of the paper
  6. B-free dynamical systems
  7. Some generalities on subshifts
  8. B-free subshifts and their hereditary closures
  9. Tautness in B-free subshifts
  10. Proximal B-free subshifts and Behrend subshifts
  11. B-admissible subshifts
  12. B-admissible subshifts and the Moore property
  13. Monotonicity in B-admissible subshifts
  14. Proof of the Moore property
  15. Hereditary subshifts and the Myhill property
  16. ...and 8 more sections

Key Result

Theorem 1.1

Assume that $\mathscr{B}$ is an Erdös set. Then the corresponding $\mathscr{B}$-free subshift satisfies the GoE theorem. If a CA is onto, then it equals a power of the shift. Moreover, every CA is given by a monotone code, modulo a power of the shift.

Theorems & Definitions (48)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Theorem 2.2: Ku-Le-We,Dy-Ka-Ku-Le
  • Theorem 2.3: Ke,Ku-Le(jr)
  • Theorem 2.4: Dy-Ka-Ku-Le
  • Theorem 2.5: Ke
  • Proposition 2.6: Ku-Le(jr)
  • Corollary 2.7
  • proof
  • ...and 38 more