Inverse semigroups associated to subshifts

The dynamics of a one-sided subshift $\mathsf{X}$ can be modeled by a set of partially defined bijections. From this data we define an inverse semigroup $\mathcal{S}_{\mathsf{X}}$ and show that it has many interesting properties. We prove that the Carlsen-Matsumoto C*-algebra $\mathcal{O}_\mathsf{X}$ associated to $\mathsf{X}$ is canonically isomorphic to Exel's tight C*-algebra of $\mathcal{S}_{\mathsf{X}}$. As one consequence, we obtain that $\mathcal{O}_\mathsf{X}$ can be written as a partial crossed product of a commutative C*-algebra by a countable group.Comment: 21 pages. Version 3 fixes more typos and adds more references. This version matches the published version in J. Algebr

Virulence as a Model for Interplanetary and Interstellar Colonisation - Parasitism or Mutualism

In the light of current scientific assessments of human-induced climate change, we investigate an experimental model to inform how resource-use strategies may influence interplanetary and interstellar colonisation by intelligent civilisations. In doing so, we seek to provide an additional aspect for refining the famed Fermi Paradox. The model described is necessarily simplistic, and the intent is to simply obtain some general insights to inform and inspire additional models. We model the relationship between an intelligent civilisation and its host planet as symbiotic, where the the relationship between the symbiont and the host species (the civilisation and the planets ecology, respectively) determines the fitness and ultimate survival of both organisms. We perform a series of Monte Carlo Realisation simulations, where civilisations pursue a variety of different relationships/strategies with their host planet, from mutualism to parasitism, and can consequently 'infect' other planets/hosts. We find that parasitic civilisations are generally less effective at survival than mutualist civilisations, provided that interstellar colonisation is inefficient (the maximum velocity of colonisation/infection is low). However, as the colonisation velocity is increased, the strategy of parasitism becomes more successful, until they dominate the 'population'. This is in accordance with predictions based on island biogeography and r/K selection theory. While heavily assumption dependent, we contend that this provides a fertile approach for further application of insights from theoretical ecology for extraterrestrial colonisation - while also potentially offering insights for understanding the human-Earth relationship and the potential for extraterrestrial human colonisation.Comment: 18 pages, 7 figures, published in the International Journal of Astrobiolog

Two-sided shift spaces over infinite alphabets

Ott, Tomforde, and Willis proposed a useful compactification for one-sided shifts over infinite alphabets. Building from their idea we develop a notion of two-sided shift spaces over infinite alphabets, with an eye towards generalizing a result of Kitchens. As with the one-sided shifts over infinite alphabets our shift spaces are compact Hausdorff spaces but, in contrast to the one-sided setting, our shift map is continuous everywhere. We show that many of the classical results from symbolic dynamics are still true for our two-sided shift spaces. In particular, while for one-sided shifts the problem about whether or not any $M$-step shift is conjugate to an edge shift space is open, for two-sided shifts we can give a positive answer for this question.Comment: 32 page