Hausdorff dimension of the set of nonergodic foliations of a quadratic differential.

*(English)*Zbl 0780.30032Let \(q=\{q_ z(z)\}\) 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. S. Kerckhoff, the author and 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 A. N. Zemlyakov and A. B. Katok [Mat. Zametki 18, No. 2, 291-300 (1975; Zbl 0315.58014)]}.

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. S. Kerckhoff, the author and 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 A. N. Zemlyakov and A. B. Katok [Mat. Zametki 18, No. 2, 291-300 (1975; Zbl 0315.58014)]}.

Reviewer: Masakazu Shiba (J-Hrose)

##### MSC:

30F30 | Differentials on Riemann surfaces |

28A78 | Hausdorff and packing measures |

37A99 | Ergodic theory |

37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |

57M50 | General geometric structures on low-dimensional manifolds |

57R30 | Foliations in differential topology; geometric theory |

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##### References:

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