Chaotic and Topological Properties of Continued Fractions
Weibin Liu, Bing Li
TL;DR
This paper explores the complex chaotic and topological characteristics of continued fractions, specifically analyzing the Gauss map's scrambled sets and the orbit density properties of points.
Contribution
It establishes the existence of a full Hausdorff dimension scrambled set for the Gauss map and examines the topological nature of points with dense or non-dense orbits.
Findings
Existence of a scrambled set with full Hausdorff dimension for the Gauss map
Topological analysis of points with dense orbits
Topological analysis of points with non-dense orbits
Abstract
We prove that there exists a scrambled set for the Gauss map with full Hausdorff dimension. Meanwhile, we also investigate the topological properties of the sets of points with dense or non-dense orbits.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Advanced Mathematical Theories and Applications · Analytic and geometric function theory