1. Discrete systems Discrete systems with state dependent parameters
Difference inclusions
Viability and invariance kernels under discrete systems
Viability and invariance characterization
The Mandelbrot function
Julia sets and Fatou dust
Fractals

2. Discrete systems with state dependent parameters
[Aubin, Saint-Pierre, 2004]

3. Difference inclusions

4. Viability kernels under discrete systems

5. Viability kernels under discrete systems

6. Invariance kernels under a discrete system

7. Invariance kernels under a discrete system

8. Discrete viability and invariance kernel

9. The Mandelbrot function
[Aubin, Saint-Pierre, 2004]

10. Viability characterization of the Mandelbrot function

11. Julia sets and Fatou dust
[Saint-Pierre, 2005]

12. Fractals (obtained with the viability algorithm)
[Saint-Pierre, 2005]