zbMATH — the first resource for mathematics

Hausdorff dimension of the set of nonergodic foliations of a quadratic differential. (English) Zbl 0780.30032
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. 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)]}.

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
Full Text: DOI
[1] L. Bers, Quasiconformal mappings and Teichmüller’s theorem , Analytic functions ed. R. Nevanlinna, et al., Princeton Univ. Press, Princeton, N.J., 1960, pp. 89-119. · Zbl 0100.28904
[2] A. Fathi, F. Laudenbach, and V. Poenaru, Travaux de Thurston sur les surfaces , Astérisque, vol. 66, Société Mathématique de France, Paris, 1979. · Zbl 0446.57018
[3] F. P. Gardiner, Teichmüller theory and quadratic differentials , Pure and Applied Mathematics, John Wiley & Sons Inc., New York, 1987. · Zbl 0629.30002
[4] F. Gardiner, Extremal length geometry of Teichmüller space , Complex Variables Theory Appl. 16 (1991), no. 2-3, 209-237. · Zbl 0702.32019
[5] M. Keane, Non-ergodic interval exchange transformations , Israel J. Math. 26 (1977), no. 2, 188-196. · Zbl 0351.28012
[6] S. Kerckhoff, H. Masur, and J. Smillie, Ergodicity of billiard flows and quadratic differentials , Ann. of Math. (2) 124 (1986), no. 2, 293-311. JSTOR: · Zbl 0637.58010
[7] H. B. Keynes and D. Newton, A “minimal”, non-uniquely ergodic interval exchange transformation , Math. Z. 148 (1976), no. 2, 101-105. · Zbl 0308.28014
[8] B. Maskit, Comparison of hyperbolic and extremal lengths , Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 381-386. · Zbl 0527.30036
[9] H. Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space , Duke Math. J. 43 (1976), no. 3, 623-635. · Zbl 0358.32017
[10] H. Masur, Interval exchange transformations and measured foliations , Ann. of Math. (2) 115 (1982), no. 1, 169-200. JSTOR: · Zbl 0497.28012
[11] H. Masur, The growth rate of trajectories of a quadratic differential , Ergodic Theory Dynam. Systems 10 (1990), no. 1, 151-176. · Zbl 0706.30035
[12] H. Masur, Hausdorff dimension of divergent Teichmüller geodesics , Trans. Amer. Math. Soc. 324 (1991), no. 1, 235-254. JSTOR: · Zbl 0733.32018
[13] H. Masur and J. Smillie, Hausdorff dimension of sets of nonergodic measured foliations , Ann. of Math. (2) 134 (1991), no. 3, 455-543. JSTOR: · Zbl 0774.58024
[14] D. Mumford, A remark on Mahler’s compactness theorem , Proc. Amer. Math. Soc. 28 (1971), 289-294. · Zbl 0215.23202
[15] E. A. Sataev, The number of invariant measures for flows on orientable surfaces , Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 4, 860-878.
[16] K. Strebel, Quadratic differentials , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. · Zbl 0547.30001
[17] W. A. Veech, Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem \(\mathrm mod\;2\) , Trans. Amer. Math. Soc. 140 (1969), 1-33. · Zbl 0201.05601
[18] W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards , Invent. Math. 97 (1989), no. 3, 553-583. · Zbl 0676.32006
[19] W. A. Veech, Moduli spaces of quadratic differentials , J. Analyse Math. 55 (1990), 117-171. · Zbl 0722.30032
[20] P. Walters, An introduction to ergodic theory , Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York, 1982. · Zbl 0475.28009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.