Robustly Transitive Translations of Boole-like transformations
DOI:
https://doi.org/10.37135/unach.ns.001.01.01Keywords:
Periodic Orbits, Robustly Transitive, Transitivity, Boole Transformation, TranslationsAbstract
The study of the renowned Boole transformation and its different parametrizations have been made from the context of the ergodic theory for infinite measurements. Very little is known about the study of these transformations from the perspective of topological dynamic systems (periodic points, invariant sets, transitivity, non-errant set, etc.), and much less is known about the stability of the dynamic phenomena found in this type of transformation. In this work we show a geometric model of Boole transformation, which results to be transitive, that is, it has a dense orbit in ℝ. Also, we show that a type of translation (one type of parameterization) of the geometric model of the Boole transformation has a transitive invariant Cantor set, whose periodic orbits are dense in such a set. For this, we use the classical method to obtain dynamically defined Cantor sets. Finally, adapting methods for unbounded transformations with a discontinuity, we proved that, if B is a Boole transformation, then for each e > 0 the translation Be is robustly transitive.
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