On the path properties of a lacunary power series

Jensen, Gerd; Pommerenke, Christian; Ramírez, Jorge

doi:10.1090/S0002-9939-2014-12077-6

On the path properties of a lacunary power series
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by Gerd Jensen, Christian Pommerenke and Jorge M. Ramírez PDF

Proc. Amer. Math. Soc. 142 (2014), 1591-1606 Request permission

Abstract:

A power series $f(z)$ which converges in $\mathbb {D}=\{|z|<1\}$ maps the radii $[0,\zeta )$ onto paths $\Gamma (\zeta )$, $\zeta \in \mathbb {T}=\partial \mathbb {D}$. These are studied under several aspects in the case of the special lacunary series $f(z)=z+z^2+z^4+z^8+\ldots$. First, the $\Gamma (\zeta )$ are considered as random functions on the probability space $(\mathbb {T},\mathscr {B},\mathrm {mes}/2\pi )$, where $\mathscr {B}$ is the $\sigma$-algebra of Borel sets and $\mathrm {mes}$ the Lebesgue measure. Then analytical properties of the $\Gamma (\zeta )$ are discussed which hold on subsets $A$ of $\mathbb {T}$ with Hausdorff dimension 1 in spite of $\mathrm {mes}{A}=0$. Furthermore, estimates of the derivative of $f$ and of the arc length of sections of the $\Gamma (\zeta )$ are given. Finally, these results are used to derive connections between the distribution of critical points of $f$ and the overall behaviour of the paths.

References

J. M. Anderson, J. Clunie, and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12–37. MR 361090
J. M. Anderson and Loren D. Pitt, On recurrence properties of certain lacunary series. I. General results, J. Reine Angew. Math. 377 (1987), 65–82. MR 887400
I. Berkes, An almost sure invariance principle for lacunary trigonometric series, Acta Math. Acad. Sci. Hungar. 26 (1975), 209–220. MR 426085, DOI 10.1007/BF01895964
Patrick Billingsley, Probability and measure, 3rd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR 1324786
Patrick Billingsley, Convergence of probability measures, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR 1700749, DOI 10.1002/9780470316962
K. G. Binmore, Analytic functions with Hadamard gaps, Bull. London Math. Soc. 1 (1969), 211–217. MR 244473, DOI 10.1112/blms/1.2.211
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
P. Erdös and I. S. Gál, On the law of the iterated logarithm. I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 65–76, 77–84. MR 0069309
K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
D. Gnuschke-Hauschild and Ch. Pommerenke, On dominance in Hadamard gap series, Complex Variables Theory Appl. 9 (1987), no. 2-3, 189–197. MR 923219, DOI 10.1080/17476938708814262
D. Gnuschke and Ch. Pommerenke, On the absolute convergence of power series with Hadamard gaps, Bull. London Math. Soc. 15 (1983), no. 5, 507–512. MR 705531, DOI 10.1112/blms/15.5.507
D. Gnuschke and Ch. Pommerenke, On the growth of functions with Hadamard gaps, J. London Math. Soc. (2) 30 (1984), no. 3, 441–450. MR 810953, DOI 10.1112/jlms/s2-30.3.441
John Hawkes, Probabilistic behaviour of some lacunary series, Z. Wahrsch. Verw. Gebiete 53 (1980), no. 1, 21–33. MR 576895, DOI 10.1007/BF00531609
Philip Hartman and Aurel Wintner, On the law of the iterated logarithm, Amer. J. Math. 63 (1941), 169–176. MR 3497, DOI 10.2307/2371287
Jean-Pierre Kahane, A century of interplay between Taylor series, Fourier series and Brownian motion, Bull. London Math. Soc. 29 (1997), no. 3, 257–279. MR 1435557, DOI 10.1112/S0024609396002913
N. G. Makarov, Smooth measures and the law of the iterated logarithm, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 2, 439–446 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 2, 455–463. MR 998306, DOI 10.1070/IM1990v034n02ABEH000664
N. G. Makarov, Probability methods in the theory of conformal mappings, Algebra i Analiz 1 (1989), no. 1, 3–59 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 1, 1–56. MR 1015333
Peter Mörters and Yuval Peres, Brownian motion, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 30, Cambridge University Press, Cambridge, 2010. With an appendix by Oded Schramm and Wendelin Werner. MR 2604525, DOI 10.1017/CBO9780511750489
Takafumi Murai, The value-distribution of lacunary series and a conjecture of Paley, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 1, vii, 135–156 (English, with French summary). MR 613032
Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706, DOI 10.1007/978-3-662-02770-7
Walter Philipp and William Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc. 2 (1975), no. 161,, 161, iv+140. MR 433597, DOI 10.1090/memo/0161
S. Rohde, Hausdorffmaß und Randverhalten analytischer Funktionen, Thesis, Technische Universität Berlin, 1989.
Volker Strassen, Almost sure behavior of sums of independent random variables and martingales, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 315–343. MR 0214118
R. Salem and A. Zygmund, On lacunary trigonometric series, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 333–338. MR 22263, DOI 10.1073/pnas.33.11.333
Mary Weiss, The law of the iterated logarithm for lacunary trigonometric series, Trans. Amer. Math. Soc. 91 (1959), 444–469. MR 108681, DOI 10.1090/S0002-9947-1959-0108681-8
A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587
A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587