Some project suggestions and relevant literature

The "Chaos game":

Michael Barnsley, Fractals Everywhere (1993)  [Chapter X]

John H Elton, An ergodic theorem for iterated maps. Erg. Theory Dyn. Systems 7 (1987),  481-488. (link)

Pablo G. Barrientos, Fatemeh H. Ghane, Dominique Malicet and Aliasghar Sarizadeh, On the chaos game of iterated function systems. Topological Methods in Nonlinear Analysis 49 (2017),  105-132. (link)

Random circle maps and synchronisation:

Dominique Malicet, Random Walks on Homeo(S^1). Commun. Math. Phys.  356 (2017), 1083-1116. (link)

Julian Newman, Necessary and sufficient conditions for stable synchronization in random dynamical systems. Erg. Theory Dyn. Systems 38 (2018), 1857-1875. (link)

Yves Le Jan, Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants. Ann. Inst. Henri Poincaré Probab. Stat. 23 (1987), 111-120. (link)

Some general theory:

Ludwig Arnold, Random Dynamical Systems (1998)
Yuri Kifer, Ergodic Theory of Random Transformations (1986)
Hans Crauel and Franco Flandoli, Attractors for random dynamical systems, Prob. Theory Rel. Fields 100 (1994), 365-393. (link)
Marcelo Viana, Lectures on Lyapunov Exponents (2014) (link)