There are two-dimensional expanding shift radix systems (SRS) which have some periodic orbits. The aim of the present paper is to describe such unusual points as well as possible. We give all regions that contain parameters the corresponding SRS of which generate obvious cycles like (1),(1),(1, 1),(1, 0),(1, 0). We prove that if r = (r0, r1) 2 R2 neither belongs to the aforementioned regions nor to the finite region 1 ? r0 ? 4/3, r0 ? r1 < r01, then r only has the trivial bounded orbit 0, which is a natural generalization of the established finiteness property for SRS with non-periodic orbits. The further reduction should be quite involving, because for all 1 ? r0 < 4/3 there exists at least one interval I such that for the point (r0, r1) this is not true whenever r1 2 I.
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