Abstract

We examine the topological dynamics of the automorphism groups of $\omega$-categorical sparse graphs resulting from Hrushovski constructions. Specifically, we consider the fixed points on type spaces property, which a structure $M$ has if, for all $n\in \mathbb{N}$, every $\mathrm{Aut}<!--FUNCTION\; APPLICATION-->(M)$-subflow of the space $S_n(M)$ of $n$-types has a fixed point. Extending a result of Evans, Hubička and Nešetřil, we show that there exists an $\omega$-categorical structure $M$, resulting from a Hrushovski construction, such that no $\omega$-categorical expansion of $M$ has the fixed points on type spaces property.

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