Abstract
For a two parameter family of two-dimensional piecewise linear maps and for every natural number \begin{document} $n$ \end{document} , we prove not only the existence of intervals of parameters for which the respective maps are n times renormalizable but also we show the existence of intervals of parameters where the coexistence of at least \begin{document} $2^n$ \end{document} strange attractors takes place. This family of maps contains the two-dimensional extension of the classical one-dimensional family of tent maps.
Highlights
Several papers in the last century have been devoted to analytically prove the existence of strange attractors
The strange attractors found in the previous references are one-dimensional attractors in the sense that they only have one positive Lyapounov exponent
It remains as an open problem to show that the interval of parameters In in which the existence of at least 2n strange attractors is proved can be constructed inside the set of parameters where, according to the first statement of Theorem B, the map Λt is a n times renormalizable map
Summary
Several papers in the last century (see [2],[7],[8]) have been devoted to analytically prove the existence of strange attractors. It remains as an open problem to show that the interval of parameters In in which the existence of at least 2n strange attractors is proved can be constructed inside the set of parameters where, according to the first statement of Theorem B, the map Λt is a n times renormalizable map. We think this stronger result is true, we remark that the most important dynamical property, i.e. the coexistence of any arbitrarily large number of persistent (in an interval of parameters) different strange attractors is demonstrated along this paper.
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