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Any infinite sequence of substitutions with the same matrix of the Pisot type defines a symbolic dynamical system which is minimal. We prove that, to any such sequence, we can associate a compact set (Rauzy fractal) by projection of the stepped line associated with an element of the symbolic system on the contracting space of the matrix. We show that this Rauzy fractal depends continuously on the sequence of substitutions, and investigate some of its properties; in some cases, this construction gives a geometric model for the symbolic dynamical system.
As a guiding example, the diffraction measure of a random local mixture of the two classic Fibonacci substitutions is determined and reanalysed via self-similar measures of Hutchinson type, defined by a finite family of contractions. Our revised appr
Given a finite set of quasi-periodic cocycles the random product of them is defined as the random composition according to some probability measure. We prove that the set of $C^r$, $0leq r leq infty$ (or analytic) $k+1$-tuples of quasi periodic coc
There has been a long-standing and at times fractious debate whether complex and large systems can be stable. In ecology, the so-called `diversity-stability debate arose because mathematical analyses of ecosystem stability were either specific to a p
We use moment method to understand the cycle structure of the composition of independent invariant permutations. We prove that under a good control on fixed points and cycles of length 2, the limiting joint distribution of the number of small cycles
The study of generic properties of quantum states has led to an abundance of insightful results. A meaningful set of states that can be efficiently prepared in experiments are ground states of gapped local Hamiltonians, which are well approximated by