Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 787-810

ON MINKOWSKI DIMENSION OF JORDAN CURVES

Nhat Le Thanh Hoang and Michel Zinsmeister

Ho Chi Minh City University of Food Industry, Department of General Sciences
Division of Mathematics, Ho Chi Minh City, Vietnam; nhatlth 'at' cntp.edu.vn

Université d'Orléans, UFR Sciences, Bâtiment de Mathématiques
Rue de Chartres, B.P. 6759, 45067 Orléans cedex 2, France; zins 'at' univ-orleans.fr

Abstract. This paper has its origin in a question raised by McMullen [McM08]: Under what general circumstances does a smooth family of conformal maps φ_t : D → \overline C with φ_t = id satisfy

(a) d²/dt² H.dim(φ_t(∂D)) |_t=0 = lim_r→1 1/4π|log(1-r)|∫_|z|=r |\dot φ'₀(z)|²|dz|?

McMullen has shown that (a) is true for some families (φ_t) arising from some dynamical systems. In order to answer this question, we consider a general analytic 1-parameter family (φ_t), t ∈ U, a neighborhood of 0, conformal maps with φ₀ = id and φ_t(0) = 0, ∀ t ∈ U defined as φ_t(z) = ∫₀^z e^tb(u)du, b ∈ B, where B is the Bloch space. By using a probability argument, we first describe a relatively large class of functions in B for which (φ_t)_t∈U satisfies (a), where Hausdorff dimension is replaced by Minkowski dimension. This class is defined in terms of the square function of the associated dyadic martingale of Re(b). The second principal result of this paper is a counter-example which is reminiscent of Kahane and Piranian's construction of non-Smirnov domain. We have constructed a singular Bloch function b such that if we consider the associated family (φ_t) as above, then φ_t(∂D) is rectifiable for t < 0. Using the properties of this Bloch function b, we prove that there exists c > 0 such that M.dim(φ_t(∂D)) ≥ 1 + ct² (t > 0 small), thus contradicting (a), where the Hausdorff dimension replaced by the Minkowski dimension.

2010 Mathematics Subject Classification: Primary 30C62.

Key words: Bloch function, dyadic martingale, Hausdorff dimension, Kahane measure, lacunary series, Minkowski dimension.

Reference to this article: N. Le Thanh Hoang and M. Zinsmeister: On Minkowski dimension of Jordan curves. Ann. Acad. Sci. Fenn. Math. 39 (2014), 787-810.

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doi:10.5186/aasfm.2014.3931