Abstract

Sets that enjoy with recurrence properties near zero, in particular, sets that force recurrence near zero, sets that contain broken IP-set near zero, sets that force uniform recurrence near zero and sets that contain broken syndetic set near zero have been introduced and some relations among them are established on countable dense subsemigroup of the additive semigroup of positive real numbers. The article begins with a characterization of recurrent point near zero that affects the results exhibited here. It is also showed that how IP-set near zero can be realized dynamically.

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