Negative bases and automata

Automata, Logic and Semantics International audience We study expansions in non-integer negative base -beta introduced by Ito and Sadahiro. Using countable automata associated with (-beta)-expansions, we characterize the case where the (-beta)-shift is a system of finite type. We prove that, if beta is a Pisot number, then the (-beta)-shift is a sofic system. In that case, addition (and more generally normalization on any alphabet) is realizable by a finite transducer. We then give an on-line algorithm for the conversion from positive base beta to negative base -beta. When beta is a Pisot number, the conversion can be realized by a finite on-line transducer.

Arithmetic construction of sofic partitions of hyperbolic toral automorphisms

For each irreducible hyperbolic automorphism $A$ of the $n$-torus we construct a sofic system $(\Sigma,\sigma)$ and a bounded-to-one continuous semiconjugacy from $(\Sigma,\sigma)$ to $({\Bbb T}^n,A)$. This construction is natural in the sense that it depends only on the characteristic polynomial of $A$ and, furthermore, it has an arithmetic interpretation.

On the sofic limit sets of cellular automata

It is not known in general whether any mixing sofic system is the limit set of some one-dimensional cellular automaton. We address two aspects of this question. We prove first that any mixing almost of finite type (AFT) sofic system with a receptive fixed point is the limit set of a cellular automaton, under which it is attained in finite time. The AFT condition is not necessary: we also give examples of non-AFT sofic systems having the same properties. Finally, we show that near Markov sofic systems (a subclass of AFT sofic systems) cannot be obtained as limit sets of cellular automata at infinity.

A sofic system which is not spectrally of finite type

We exhibit a transitive sofic system for which the core matrix has negative trace, and hence cannot share the nonzero spectrum of any subshift of finite type cover. We also show that every transitive sofic system has an integral core matrix.