Fractals in Mathematics

Abstract In this paper we shall survey two topics concerning the Fractals: nowhere differentiable functions and self-similar sets in Euclidean space such as Cantor set, Koch curve, and Peano curves.

[1]  J. Gerver THE DIFFERENTIABILITY OF THE RIEMANN FUNCTION AT CERTAIN RATIONAL MULTIPLES OF pi. , 1969, Proceedings of the National Academy of Sciences of the United States of America.

[2]  G. Hardy Weierstrass’s non-differentiable function , 1916 .

[3]  V. Jarník Über die Differenzierbarkeit stetiger Funktionen , 1933 .

[4]  Masayoshi Hata,et al.  The Takagi function and its generalization , 1984 .

[5]  G. Faber ber stetige Funktionen: Zweite Abhandlung , 1910 .

[6]  The Theory Of Functions Of A Real Variable And The Theory Of Fouriers Series Vol-ii , 1926 .

[7]  G. Peano Sur une courbe, qui remplit toute une aire plane , 1890 .

[8]  D. Hilbert Ueber die stetige Abbildung einer Line auf ein Flächenstück , 1891 .

[9]  T. Takagi,et al.  A SIMPLE EXAMPLE OF THE CONTINUOUS FUNCTION WITHOUT DERIVATIVE , 1901 .

[10]  Pertti Mattila,et al.  On the structure of self-similar fractals , 1982 .

[11]  Z. Łomnicki,et al.  Sur la théorie de la mesure dans les espaces combinatoires et son application au calcul des probabilités. I. Variables indépendantes , 1934 .

[12]  On the functional equation $\frac{1}{p}\{f(\frac{x}{p})+\cdots+f(\frac{x+p-1}{p})\}=\lambda f(\mu x)$ , 1985 .

[13]  K. Falconer The geometry of fractal sets: Contents , 1985 .

[14]  Masayoshi Hata,et al.  Weierstrass's function and chaos , 1983 .

[15]  S. Kline On Curves of Fractional Dimensions , 1945 .

[16]  THE DIFFERENTIABILITY OF THE RIEMANN FUNCTION AT CERTAIN RATIONAL MULTIPLES OF pi. , 1969, Proceedings of the National Academy of Sciences of the United States of America.

[17]  James A. Yorke,et al.  The Lyapunov dimension of a nowhere differentiable attracting torus , 1984, Ergodic Theory and Dynamical Systems.

[18]  E. R. Love,et al.  Sur une classe de fonctionnelles linéaires , 1937 .

[19]  R. Salem On some singular monotonic functions which are strictly increasing , 1943 .

[20]  M. Hata On some properties of set-dynamical systems , 1985 .

[21]  J. Yorke,et al.  Period Three Implies Chaos , 1975 .

[22]  R. F. Williams Composition of contractions , 1971 .

[23]  H. D. Ursell,et al.  Sets of Fractional Dimensions (V) : On Dimensional Numbers of Some continuous Curves , 1937 .

[24]  Masayoshi Hata,et al.  On the structure of self-similar sets , 1985 .

[25]  P. Lax The differentiability of Pólya's function , 1973 .

[26]  K. Falconer The geometry of fractal sets , 1985 .

[27]  D. Hilbert Über die stetige Abbildung einer Linie auf ein Flächenstück , 1935 .

[28]  G. Faber Über stetige Funktionen , 1908 .