one can associate a commutative (possibly
nonassociative) algebra. The correspondence between quadratic maps
and algebras is one-to-one. Classical Julia sets actually
originate from studying the iteration of the complex squaring map.
However, the algebra of complex numbers is just one of many
nonisomorphic commutative algebras existing in
. Concerning the algebraic properties like
idempotents and nilpotents we searched for the algebra which is
most similar to the algebra of complex numbers. The associated
discrete dynamical system in this particular algebra exhibits
similar behavior on the boundary of basin of attraction of the
origin as the complex-squaring map on the classical Julia set. We
used a one-to-one projection to the unit circle to prove the
chaotic properties of the invariant set.
Key words: discrete dynamical system, quadratic maps, algebra, chaos
