Shifting and Lifting of Cellular Automata
- Cite this paper as:
- Acerbi L., Dennunzio A., Formenti E. (2007) Shifting and Lifting of Cellular Automata. In: Cooper S.B., Löwe B., Sorbi A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg
We consider the family of all the Cellular Automata (CA) sharing the same local rule but have different memory. This family contains also all the CA with memory m ≤ 0 (one-sided CA) which can act both on Aℤ and on Aℕ. We study several set theoretical and topological properties for these classes. In particular we investigate if the properties of a given CA are preserved when we consider the CA obtained by changing the memory of the original one (shifting operation). Furthermore we focus our attention to the one-sided CA acting on Aℤ starting from the one-sided CA acting on Aℕ and having the same local rule (lifting operation). As a particular consequence of these investigations, we prove that the long-standing conjecture [Surjectivity \(\Rightarrow\) Density of the Periodic Orbits (DPO)] is equivalent to the conjecture [Topological Mixing \(\Rightarrow\) DPO].
Keywordsdiscrete time dynamical systems cellular automata topological dynamics deterministic chaos
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