Cellular automata on spaces of probability measures

In this paper, we propose Cellular Automata on probability Measures (CAMs), a novel framework which extends the classical notion of cellular automata (CA). In this setting, the state of each cell is described by a probability measure over the set of possible states, enabling the modelling of systems with inherent uncertainty and spatially-varying dynamics. We show that the CAM framework collapses to classical CA when Dirac probability measures are used and develop a significant theoretical foundation for the generalization. A significant aspect of this theory is our proof that a version of the classical Curtis–Hedlund–Lyndon theorem holds for CAMs. We also find some families of topological conjugacies between CAMs and prove that every CAM has at least one fixed point. A particularly interesting example of our framework is CAMs with fractal probability measures. We establish results on the convergence properties of both fractal and generic CAMs under certain hypotheses. Finally, we present the foundation for solving inverse problems for CAMs with fractal measures. This study lays the groundwork for future exploration of CAMs, offering a flexible and robust framework for modelling uncertainty in CA and opening new directions for both theoretical analysis and practical applications.