An iterated function system (IFS) on a manifold M is a tuple of smooth maps f1, ...,fs : M → M. One of the reasons for studying IFS's is that they (more precisely, associated step skew products over Bernoulli shift) provide a nice model example of partially hyperbolic skew products. If some interesting robust property is found for the IFS's, it is often possible to find this property for a locally generic set of diffeomorphisms (see, e.g., [1]).

Generic IFS's on the interval were studied by V. Kleptsyn and D. Volk ([2]). Among other things, they proved that the associated skew product has nitely many attractors and nitely many physical measures. However, it is unknown whether the supports of these physical measures coincide with the attractors. 

Unlike IFS's on the interval, IFS's on the circle can be minimal (i.e., the (f1, . . . , fs)-orbit of each point is dense). It turns out that this is the only di erence from the interval case. Namely, for an open and dense set of IFS's on the circle such that f1,...,fs are orientation-preserving diffeomorphisms an alternative holds: 

• either the IFS is minimal
• or there is an absorbing domain a nontrivial nite union I of open intervals such 

that fi(I) ⊂ I for each i = 1,...,s. In this case the results from [2] can be applied. 

References 

[1] Yu.S. Ilyashenko, Thick attractors of boundary preserving di eomorphisms, Indagationes Mathematicae, 2011 

[2] V. Kleptsyn, D. Volk, Physical measures for nonlinear random walks on interval, Mosc. Math. J., 2014