| Some results regarding to the Bifurcation set for the family Double Standard Maps on the circle |
| Diego Lugo |
| Universidad Pontificia Católica de Valparaíso (Chile) |
| Resumen: |
| Let \[f\] a continuous map on the circle \[\mathbb{S}^1\] and consider for an open set \[H\subset\mathbb{S}^1\] the set \[S_H=\{\theta\in\mathbb{S}^1:\;f^n(\theta)\notin H\;\forall n\geq0\}\] called the Surviving set for\[H\] . Focusing on the set of all the open intervals of the circle, the Bifurcation set \[\mathcal{B}_f\] $\mathcal{B}_f$ is defined to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of their position. |
| There are interesting results for the set \[\mathcal{B}_f\] when the map \[f\] is transitive regarding with its geometric form and topological properties. We pretend to extend this results to the non-transitive world. As an approach, we point out on the family of Double Standard maps of the circle given by the formula \[f_{\alpha,\beta}(\theta)=2\theta+\alpha+\frac{\beta}{\pi}\sin(2\pi\theta)(\mathrm{mod}\,1).\] |
| The principal reasons to use this family are the semiconjugacy of each member or the family with the Doubling map and its simple non-transitive form given by the presence of a single attracting or indifferent periodic orbit. Besides the set of parameters \[\alpha,\beta\] for which an attractting or neutral periodic orbit of fixed period exists for \[f_{\alpha,\beta}\] is known as an analogue of the Arnold Tongues for its family of circle diffeomorphisms (The Arnold maps or The Arnold Family). |