Area and Hausdorff dimension of Julia sets of entire functions

We show the Julia set of A sin(z) has positive area and the action of A sin(z) on its Julia set is not ergodic; the Julia set of A exp(z) has Hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero.

[1]  Michał Misiurewicz,et al.  On iterates of ez , 1981, Ergodic Theory and Dynamical Systems.

[2]  A. Avez,et al.  Ergodic theory of dynamical systems , 1966 .

[3]  D. Sullivan.  Conformal dynamical systems , 1983 .

[4]  P. J. Rippon,et al.  Iteration of exponential functions , 1984 .

[5]  J. M. Thomas,et al.  Conformal Invariants. , 1926, Proceedings of the National Academy of Sciences of the United States of America.

[6]  A. Douady,et al.  Systèmes dynamiques holomorphes , 1983 .

[7]  I. N. Baker.  The domains of normality of an entire function , 1975 .

[8]  P. Fatou,et al.  Sur l'itération des fonctions transcendantes Entières , 1926 .

[9]  D. Sullivan,et al.  On the measurable dynamics of z → ez , 1985, Ergodic Theory and Dynamical Systems.

[10]  L. Keen,et al.  A finiteness theorem for a dynamical class of entire functions , 1986, Ergodic Theory and Dynamical Systems.

[11]  L. Ahlfors.  Conformal Invariants: Topics in Geometric Function Theory , 1973 .

[12]  J. Kahane,et al.  Ensembles parfaits et séries trigonométriques , 1963 .

[13]  Robert L. Devaney,et al.  Dynamics of exp (z) , 1984, Ergodic Theory and Dynamical Systems.

[14]  F. John.  On quasi‐isometric mappings, II , 1968 .

[15]  I. N. Baker.  Some entire functions with multiply-connected wandering domains , 1985, Ergodic Theory and Dynamical Systems.

[16]  F. Gehring.  Injectivity of local quasi-isometries , 1982 .