Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>+</mo> <mi>&#964;</mi> <mi>y</mi> </mrow> </semantics> </math> </inline-formula> for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>&#8712;</mo> <mi mathvariant="double-struck">R</mi> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>&#964;</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> but <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#964;</mi> <mo>&#8800;</mo> <mo>&#177;</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, are the natural number system in which to encode geometric properties of the Minkowski space <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">R</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msup> </semantics> </math> </inline-formula>. We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets.

Abstract licence: CC BY

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