Reem Yassawi (Open University)
Given two commuting transformations 蟽, 桅 : X 鈫� X acting on a compact metric聽space, what are the measures on X which are invariant under the action of 蟽 and 桅? This general question includes the open problem posed by Furstenberg, which is to find the measures on the unit interval which are invariant under both x 1鈫� 2x mod 1 and x 鈫� 3x mod 1. Let p be a prime number and let Fp be the field of cardinality p. A linear cellular聽automaton 桅 : FZ 鈫� FZ is an Fp-linear map that commutes with the (left) shift map 蟽 : FZ 鈫� FZ. A famous linear cellular automaton is Ledrappier鈥檚, defined by p p 桅(x) = x + 蟽(x), where, in contrast to a symbolic version of Furstenberg鈥檚 question, addition is performed 鈥渂itwise鈥� and without carry. For a linear cellular automaton 桅, examples of measures which are (桅, 蟽)- invariant are the uniform measure on FZ, and measures supported on a finite set. In work by Einsiedler from the early 2000鈥檚, if we recast linear cellular automata in the setting of Markov subgroups, we find a new family of nontrivial (蟽, 桅)-invariant measures. In recent joint work with Eric Rowland, we find another family of of nontrivial (蟽, 桅)-invariant measures, using constant length substitutions, and their characterisation by Christol. I will describe how we obtain these measures, and compare them to Einsiedler鈥檚 construction.