Ovaloids in parameter space
DOI:
https://doi.org/10.35819/remat2021v7i1id4191Keywords:
Global Geometry, Gaussian Curvature, Parameter Space, Morse TheoryAbstract
The topological dynamics induced by geometric inversions in the complex plane has already been addressed and the results were presented by Vieira et al. (2017). Subsequently, it was proved that, for a set of three inversions, the parameter space of the Markov measure supported by the attractor of the system is an open subset in R3 clad by compact level surfaces defined by metric entropy: isentropic surfaces (VIEIRA et al., 2018). The purpose of this article is to use Morse Theory to describe the global geometry of those surfaces. As functions of the levels, we proved that the area and diameter tend to zero and that the Gaussian curvature is unbounded, when the levels approach the critical level (maximum entropy). In particular, we demonstrated that there is a maximal open range of levels for which the surfaces are ovaloid.
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