In this work, we study the topological properties of Julia sets of the quadratic polynomial maps  .

Complex dynamics is the study of iteration of maps which map the complex plane into itself . In general , their dynamics are quite complicated and hard to explain ,their dynamics are quite complicated and hard to explain ,but for some classes of maps , many interesting results can be proved .

 

For example , one often studies the Julia sets of polynomial maps . 

 

The Julia set of the quadratic map of the form  was studied extensively .

 

From a dynamical systems point of view , all of the interesting behavior of a complex analytic map occurs on its Julia set , it is this set that contains the interesting topology .

 

The idea behind the Julia sets is to study whether the absolute value of a point in the complex plane converges towards infinity or not ,

 when it is iterated under a map . All the points that do not go toward infinity ,

when iterated , are in the Julia set .