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Register a new userPrevalence of Odometers in Cellular Automata
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We consider a left permutive cellular automaton Phi, with no memory and positive anticipation, defined on the space of all doubly infinite sequences with entries from a finite alphabet. For each such automaton that is not one-to-one, there is a dense set of points X (which is large in another sense too) such that the Phi-orbit closure of each x in X is topologically conjugate to an odometer (the ``+1 map on a projective limit of finite cyclic groups). We identify this odometer in several cases.
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We say that a finite asynchronous cellular automaton (or more generally, any sequential dynamical system) is pi-independent if its set of periodic points are independent of the order that the local functions are applied. In this case, the local functions permute the periodic points, and these permutations generate the dynamics group. We have previously shown that exactly 104 of the possible 256 cellular automaton rules are pi-independent. In this article, we classify the periodic states of these systems and describe their dynamics groups, which are quotients of Coxeter groups. The dynamics groups provide information about permissible dynamics as a function of update sequence and, as such, connect discrete dynamical systems, group theory, and algebraic combinatorics in a new and interesting way. We conclude with a discussion of numerous open problems and directions for future research.
We investigate the mean dimension of a cellular automaton (CA for short) with a compact non-discrete space of states. A formula for the mean dimension is established for (near) strongly permutative, permutative algebraic and unit one-dimensional automata. In higher dimensions, a CA permutative algebraic or having a spaceship has infinite mean dimension. However, building on Meyerovitchs example, we give an example of algebraic surjective cellular automaton with positive finite mean dimension.
We describe the infinite interval exchange transformations obtained as a composition of a finite interval exchange transformation and the von Neumann-Kakutani map, called the rotated odometers. We show that with respect to Lebesgue measure on the unit interval, every such transformation is measurably isomorphic to the first return map of a rational parallel flow on a translation surface of finite area with infinite genus and a finite number of ends. We describe the dynamics of rotated odometers by means of Bratteli-Vershik systems, and derive several of their topological and ergodic properties. In particular, we show that every rotated odometer has a unique minimal subsystem, and that there exist rotated odometers whose minimal subsystem does not factor onto the dyadic odometer.
Weighted shifts are an important concrete class of operators in linear dynamics. In particular, they are an essential tool in distinguishing variety dynamical properties. Recently, a systematic study of dynamical properties of composition operators on $L^p$ spaces has been initiated. This class of operators includes weighted shifts and also allows flexibility in construction of other concrete examples. In this article, we study one such concrete class of operators, namely composition operators induced by measures on odometers. In particular, we study measures on odometers which induce mixing and transitive linear operators on $L^p$ spaces.
A cellular automata (CA) configuration is constructed that exhibits emergent failover. The configuration is based on standard Game of Life rules. Gliders and glider-guns form the core messaging structure in the configuration. The blinker is represented as the basic computational unit, and it is shown how it can be recreated in case of a failure. Stateless failover using primary-backup mechanism is demonstrated. The details of the CA components used in the configuration and its working are described, and a simulation of the complete configuration is also presented.
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