On some rational piecewise linear rotations
We study the dynamics of the piecewise planar rotations F¿(z)=¿(z-H(z)), with z¿C , H(z)=1 if Im(z)=0, H(z)=-1 if Im(z)<0, and ¿=eia¿C , being a a rational multiple of p. Our main results establish the dynamics in the so called regular set, which is the complementary of the closure of the set for...
| Autores: | , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/395443 |
| Acceso en línea: | https://hdl.handle.net/2117/395443 https://dx.doi.org/10.1080/10236198.2023.2260898 |
| Access Level: | acceso abierto |
| Palabra clave: | Differentiable dynamical systems Difference equations Periodic points Pointwise periodic maps Piecewise linear maps Fractal tessellations Sistemes dinàmics diferenciables Equacions en diferències Classificació AMS::37 Dynamical systems and ergodic theory::37B Topological dynamics Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory Classificació AMS::39 Difference and functional equations::39A Difference equations Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals |
| Sumario: | We study the dynamics of the piecewise planar rotations F¿(z)=¿(z-H(z)), with z¿C , H(z)=1 if Im(z)=0, H(z)=-1 if Im(z)<0, and ¿=eia¿C , being a a rational multiple of p. Our main results establish the dynamics in the so called regular set, which is the complementary of the closure of the set formed by the preimages of the discontinuity line. We prove that any connected component of this set is open, bounded and periodic under the action of F¿ , with a period l, that depends on the connected component. Furthermore, Fl¿ restricted to each component acts as a rotation with a period which also depends on the connected component. As a consequence, any point in the regular set is periodic. Among other results, we also prove that for any connected component of the regular set, its boundary is a convex polygon with certain maximum number of sides |
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