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Roland
Roland
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Some years ago, I attended a course where Julia and Mandelbrot sets were part of. It was called "Chaotic Dynamical Systems", and was held once a week for 90 minutes for a total of 14 lectures. It was made for master students in math, although advanced bachelor students could also have participated. This doesn't completely fit the topics of your course, but it might be a solid base. Without understanding of complex numbers, this course wouldn't have worked, although complex analysis wasn't strictly required.

The contents of the course were:

  1. Introduction of discrete dynamical systems; the definition mentions topological spaces, but the script says "Usually, we have more than that". Orbit, invariant sets, conjugate systems, examples are usually on $\mathbb{R},\mathbb{R}^+$ and intervals. Main toy example is the logistic system/family.
  2. Fixed points, periodic points, stability of orbits, attractive points, repellent points, characterization of these points for functions which are continouosly differentiable on $\mathbb{R}$ or $\mathbb{C}$, Bifurcation for a family of dynamical systems, Sharkovsky's theorem (no proof), theorem of Li-Yorke (with proof)
  3. Definition of a chaotic system (Devaney, see references), symbolic dynamics
  4. Complex dynamic systems: Newton's method, Julia sets, Fatou set, Mandelbrot set. proofs partly omitted, partly shortened to a sketch

References for the course:

The desired topics were only covered in the last few weeks of the course, but I think that if you go lighter on the chaotic part, you might be able to give more emphasis to the sets you are interested in.

I think that a course on discrete dynamical systems is a good context for these kind of sets - unfortunately I can't quite tell how to fit in Cantor/Sierpinski/Merger - usually I'd include these in a course on measure theory.

Some years ago, I attended a course where Julia and Mandelbrot sets were part of. It was called "Chaotic Dynamical Systems", and was held once a week for 90 minutes. It was made for master students in math, although advanced bachelor students could also have participated. This doesn't completely fit the topics of your course, but it might be a solid base. Without understanding of complex numbers, this course wouldn't have worked, although complex analysis wasn't strictly required.

The contents of the course were:

  1. Introduction of discrete dynamical systems; the definition mentions topological spaces, but the script says "Usually, we have more than that". Orbit, invariant sets, conjugate systems, examples are usually on $\mathbb{R},\mathbb{R}^+$ and intervals. Main toy example is the logistic system/family.
  2. Fixed points, periodic points, stability of orbits, attractive points, repellent points, characterization of these points for functions which are continouosly differentiable on $\mathbb{R}$ or $\mathbb{C}$, Bifurcation for a family of dynamical systems, Sharkovsky's theorem (no proof), theorem of Li-Yorke (with proof)
  3. Definition of a chaotic system (Devaney, see references), symbolic dynamics
  4. Complex dynamic systems: Newton's method, Julia sets, Fatou set, Mandelbrot set. proofs partly omitted, partly shortened to a sketch

References for the course:

The desired topics were only covered in the last few weeks of the course, but I think that if you go lighter on the chaotic part, you might be able to give more emphasis to the sets you are interested in.

I think that a course on discrete dynamical systems is a good context for these kind of sets - unfortunately I can't quite tell how to fit in Cantor/Sierpinski/Merger - usually I'd include these in a course on measure theory.

Some years ago, I attended a course where Julia and Mandelbrot sets were part of. It was called "Chaotic Dynamical Systems", and was held once a week for 90 minutes for a total of 14 lectures. It was made for master students in math, although advanced bachelor students could also have participated. This doesn't completely fit the topics of your course, but it might be a solid base. Without understanding of complex numbers, this course wouldn't have worked, although complex analysis wasn't strictly required.

The contents of the course were:

  1. Introduction of discrete dynamical systems; the definition mentions topological spaces, but the script says "Usually, we have more than that". Orbit, invariant sets, conjugate systems, examples are usually on $\mathbb{R},\mathbb{R}^+$ and intervals. Main toy example is the logistic system/family.
  2. Fixed points, periodic points, stability of orbits, attractive points, repellent points, characterization of these points for functions which are continouosly differentiable on $\mathbb{R}$ or $\mathbb{C}$, Bifurcation for a family of dynamical systems, Sharkovsky's theorem (no proof), theorem of Li-Yorke (with proof)
  3. Definition of a chaotic system (Devaney, see references), symbolic dynamics
  4. Complex dynamic systems: Newton's method, Julia sets, Fatou set, Mandelbrot set. proofs partly omitted, partly shortened to a sketch

References for the course:

The desired topics were only covered in the last few weeks of the course, but I think that if you go lighter on the chaotic part, you might be able to give more emphasis to the sets you are interested in.

I think that a course on discrete dynamical systems is a good context for these kind of sets - unfortunately I can't quite tell how to fit in Cantor/Sierpinski/Merger - usually I'd include these in a course on measure theory.

Source Link
Roland
Roland
  • 1.8k
  • 14
  • 19

Some years ago, I attended a course where Julia and Mandelbrot sets were part of. It was called "Chaotic Dynamical Systems", and was held once a week for 90 minutes. It was made for master students in math, although advanced bachelor students could also have participated. This doesn't completely fit the topics of your course, but it might be a solid base. Without understanding of complex numbers, this course wouldn't have worked, although complex analysis wasn't strictly required.

The contents of the course were:

  1. Introduction of discrete dynamical systems; the definition mentions topological spaces, but the script says "Usually, we have more than that". Orbit, invariant sets, conjugate systems, examples are usually on $\mathbb{R},\mathbb{R}^+$ and intervals. Main toy example is the logistic system/family.
  2. Fixed points, periodic points, stability of orbits, attractive points, repellent points, characterization of these points for functions which are continouosly differentiable on $\mathbb{R}$ or $\mathbb{C}$, Bifurcation for a family of dynamical systems, Sharkovsky's theorem (no proof), theorem of Li-Yorke (with proof)
  3. Definition of a chaotic system (Devaney, see references), symbolic dynamics
  4. Complex dynamic systems: Newton's method, Julia sets, Fatou set, Mandelbrot set. proofs partly omitted, partly shortened to a sketch

References for the course:

The desired topics were only covered in the last few weeks of the course, but I think that if you go lighter on the chaotic part, you might be able to give more emphasis to the sets you are interested in.

I think that a course on discrete dynamical systems is a good context for these kind of sets - unfortunately I can't quite tell how to fit in Cantor/Sierpinski/Merger - usually I'd include these in a course on measure theory.