In others words, we study how transitivity and point-transitivity of a dynamical system ( ( X, d ), f ) and the dynamical system ( F ( X ), f ^ ), where f ^ is the Zadeh’s extension of f and F ( X ) is the hyperspace of all normal fuzzy sets of X. We study the relationship between the individual behavior and fuzzy collective behavior for transitivity and point-transitivity. To see the relationship between Γ-convergence and the sendograph metric, the interested reader can consult. It can be characterized by means of the notion of Γ-convergence (see for details). For example, it is used in fuzzy reference by fuzzy numbers defined on the unit interval (see ). The endograph metric has many applications in fuzzy theory. The endograph (respectively, sendograph) metric is defined by means of the Hausdorff distance between the endographs (respectively, sendographs) of two normal fuzzy sets. Given a metric space ( X, d ), the Skorokhod metric on F ( X ) was defined in. Joo and Kim introduced the Skorokhod metric in the field of fuzzy numbers which has been also studied in the context of F ( R n ) (see ). It plays an important role for the convergence of probability measures on D, namely the convergence in distribution of stochastic processes with jumps: indeed, many central limit results and invariance principles were obtained (see ). In, Billingsley showed that the Skorokhod topology is metrizable, actually it proves that D endowed with the Skorokhod topology is a separable complete metric space. The Skorokhod topology was introduced by Skorokhod in as an alternative to the topology of uniform convergence on the set D of right-continuous functions on having limits to the left at each t ∈ ( 0, 1 ]. It is worth noting that the space F 0 ( X ) is relevant in the theory of fuzzy numbers and it is the least studied in the theory of fuzzified discrete dynamical systems. Among other things, the following results are presented: (1) If ( X, d ) is a metric space, then the following conditions are equivalent: (a) ( X, f ) is weakly mixing, (b) ( ( F ( X ), d ∞ ), f ^ ) is transitive, (c) ( ( F ( X ), d 0 ), f ^ ) is transitive and (d) ( ( F ( X ), d S ) ), f ^ ) is transitive, (2) if f : ( X, d ) → ( X, d ) is a continuous function, then the following hold: (a) if ( ( F ( X ), d S ), f ^ ) is transitive, then ( ( F ( X ), d E ), f ^ ) is transitive, (b) if ( ( F ( X ), d S ), f ^ ) is transitive, then ( X, f ) is transitive and (3) if ( X, d ) be a complete metric space, then the following conditions are equivalent: (a) ( X × X, f × f ) is point-transitive and (b) ( ( F ( X ), d 0 ) is point-transitive. In this context, we consider the Zadeh’s extension f ^ of f to F ( X ), the family of all normal fuzzy sets on X, i.e., the hyperspace F ( X ) of all upper semicontinuous fuzzy sets on X with compact supports and non-empty levels and we endow F ( X ) with different metrics: the supremum metric d ∞, the Skorokhod metric d 0, the sendograph metric d S and the endograph metric d E. Given a metric space ( X, d ), we deal with a classical problem in the theory of hyperspaces: how some important dynamical properties (namely, weakly mixing, transitivity and point-transitivity) between a discrete dynamical system f : ( X, d ) → ( X, d ) and its natural extension to the hyperspace are related.
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