![]() ![]() ![]() Our aim is to provide such a general result also for the Moore property. One of the most general versions of the Myhill property was considered by Li in where he proves that every expansive action of an amenable group on a compact metrizable space having the weak specification property satisfies the Myhill property. It did not take long for the Garden of Eden theorem to be considered for many other dynamical systems (see e.g. This result was generalized to all amenable groups in and later also for certain subshifts of amenable groups (see ). A dynamical system for which this equivalence holds is said to satisfy the Garden of Eden theorem. Is pre-injective, respectively conversely every such pre-injective map is surjective (we shall define pre-injectivity later). , Moore in, respectively Myhill in, proved that every continuous surjective G-equivariant map One of those is the Garden of Eden theorem. These are the objects of study of symbolic dynamics, whose many results and notions have inspired and have been generalized to more general dynamical systems. Such a topological dynamical system is called a (topological) shift and any of its (topologically) closed G-invariant subsets are called subshifts. , with the product topology, is a compact topological space on which G naturally acts by homeomorphisms. If it has additionally the weak specification property, the set of such points is dense. We show that every expansive action of such a group having a certain Markov topological property, again implied by the pseudo-orbit tracing property, has a weakly periodic point. Second, we generalize the recent result of Cohen that any subshift of finite type of a finitely generated group having at least two ends has weakly periodic points. We hint how to easily generalize this result even for uncountable amenable groups and general compact, not necessarily metrizable, spaces. This together with an earlier result of Li (where the strong topological Markov property is not needed) of the Myhill property, which we also re-prove here, establishes the Garden of Eden theorem for all expansive actions of countable amenable groups on compact metrizable spaces satisfying the weak specification and strong topological Markov properties. First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, that is, every surjective endomorphism of such dynamical system is pre-injective. L.We present several applications of the weak specification property and certain topological Markov properties, recently introduced by Barbieri, García-Ramos, and Li, and implied by the pseudo-orbit tracing property, for general expansive group actions on compact spaces. L.: Fractal Geometry–Mathematical Foundations and Application, John Wiley and Sons, Chichester, 1990 C.: The Hausdorff dimension of a chaotic set of shift map in symbolic space. Kan, I.: A chaotic function possessing a scrambled set with positive Lebesgue measure. Smítal, J.: A chaotic function with some extremal properties. C.: Chaos caused by strong–mixing measure–preserving transformations. Walters, P.: An Introduction to Ergodic Theory, Spring–Verlag, New York, 1982 J.: Chaos and SS chaos are not equivalent. ![]() Schweizer, B., Smítal, J.: Measure of chaos and a spectral decomposition of dynamical system on the interval. D.: Devaney’s chaos or 2–scattering implies Li–Yorke’s chaos. G.: Chaos caused by a topologically mixing maps, In dynamical systems and related topics, World Scientific Press, Singapore, 550–572, 1992ĭevaney, R.: An introduction to chaotic dynamical systems, Addison–Wesley, Redwood city, 1989 ![]()
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