Hausdorff dimension of the set of nonergodic foliations of a quadratic differential
Let \(q=\{q_ z(z)\}\) be a (normalized integrable) quadratic differential on a compact Riemann surface \(X\). That is, \(q_ z(z)dz^ 2\) is invariant under change of local parameters of \(X\) and satisfies \(\int_ X| q_ z(z)dz^ 2|=1\). The metric defined by \(| q^{1/2} dz|\) has singularities at the poles and zeroes of \(q\). Except at these points the metric is Euclidean. A saddle connection of \(q\) means a geodesic segment which joins two singularities of \(q\) and has no singularities in its interior. The vertical trajectories of \(q\) are those curves along which \(q_ z(z)dz^ 2<0\). For any real \(\theta\), \(|\theta|\leq \pi/2\), denote by \(F_ \theta\) the measured foliation whose leaves are the vertical trajectories of the quadratic differential \(e^{2i\theta} q\). A measured foliation \(F\) is called minimal if every closed set which is a union of leaves either is empty or its complement is. Replacing the terms ``closed'' by ``measurable'' and ``empty'' by ``of measure zero'' in the definition of minimality, we have the notion of ergodicity. Let \(\text{NE}(q)\) be the set of \(\theta\) such that \(F_ \theta\) is not ergodic. The author proves that the Hausdorff dimension of \(\text{NE}(q)\) does not exceed \({1\over 2}\). Actually, he proves more: Let \(\text{NUE}(q)\) be the set of \(\theta\) such that \(F_ \theta\) is not uniquely ergodic, that is, \(F_ \theta\) has more than one -- even if we ignore scalar multiplications -- transverse invariant measure. The set \(\text{NUE}(q)\) has Lebesgue measure zero [cf. \textit{S. Kerckhoff}, the author and \textit{J. Smillie}, Ann. Math., II. Ser. 124, No. 2, 293-311 (1986; Zbl 0637.58010)]. The main theorem now reads: The Hausdorff dimension of \(\text{NUE}(q)\) is at most \({1\over 2}\). To prove the main theorem the author first improves an earlier result of his [cf. Ann. Math., II. Ser. 115, No. 1, 169-200 (1982; Zbl 0497.28012)] and shows that if \(f_ t: X\to X_ t\) is the Teichmüller map defined by \(t\) and \(q\), then \(X_ t\) eventually leaves every compact set in the moduli space of \(X\) as \(t\to\infty\). In other words, the terminal quadratic differential \(g_ t(e^{2i\theta}q)\) induced by \(f_ t\) leaves every compact subset of the moduli space. Since the set \(\text{NM}(q)\) of \(\theta\) such that \(F_ \theta\) is minimal is known to be countable, it suffices to show that the Hausdorff dimension of \(\text{NUE}(q)\backslash\text{NM}(q)\) is less than or equal to \({1\over 2}\). The above theorem implies that, for any \(a>0\), \(\text{NUE}(q)\backslash\text{NM}(q)\subset\bigcup_ T\text{Div}(a,T)\), where \(\text{Div}(a,T):=\{\theta\in[-\pi/2,\pi/2]\): \(F_ \theta\) is minimal, and for any \(t\geq T\), there exists a saddle connection whose \(g_ t(e^{2i\theta} q)\)-length is at most \(a\)\}. For any saddle connection \(\beta\) of \(q\) there exists an angle \(\theta_ \beta\) such that \(\beta\) is a vertical trajectory of \(e^{2i\theta} q\). Let \(I(\beta,a)\) be the interval \(\{|\theta-\theta_ \beta|<2a^ 2/|\beta|^ 2_ q\}\), \(|\beta|_ q\) being the \(q\)-length of \(\beta\). The proof of the main theorem is now accomplished by picking out a subset of ``good'' saddle connections \(\beta\) and studying the covering of \(\text{Div}(a,T)\) by the union of the corresponding intervals \(I(\beta,a)\). The author prepares a number of technical lemmas and propositions as well as a series of estimates before proving his main theorem. Some of them are complicated; for example, Proposition 2.3 requires a proof eight pages long. Readers will appreciate the author's ideas and sketchy proofs which precede the formal proof. \{Reviewer's remark: The paper [ZK],cited in the introduction but not found in the bibliography, is ``Topological transitivity of billiards in polygons'' by \textit{A. N. Zemlyakov} and \textit{A. B. Katok} [Mat. Zametki 18, No. 2, 291-300 (1975; Zbl 0315.58014)]\}.
scientific article; zbMATH DE number 78545
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