Escaping subsets of cosine functions with the given Hausdorff dimension

Abstract

Exponential functions are the most typical transcendental entire functions. Their escaping sets are the important research object of transcendental entire function dynamic system, which has been deeply studied. It is well known that if the function is slightly disturbed, the properties of its dynamic system may vary greatly. We can't easily study different functions in the same way. Contrasting exponential function, we pay our main attention to the cosine functions in this paper. We construct some escaping subsets of cosine functions by Devaney-Krych codes such that the Hausdorff dimensions of the subsets are equal to the given number which is in the interval $(1,2)$.