34 research outputs found
Entropy of Transcendental entire functions
We prove that all entire transcendental entire functions have infinite
topological entropy.Comment: 13 page
A separation theorem for entire transcendental maps
We study the distribution of periodic points for a wide class of maps, namely
entire transcendental functions of finite order and with bounded set of
singular values, or compositions thereof. Fix and assume that all
dynamic rays which are invariant under land. An interior -periodic
point is a fixed point of which is not the landing point of any periodic
ray invariant under . Points belonging to attracting, Siegel or Cremer
cycles are examples of interior periodic points. For functions as above we show
that rays which are invariant under , together with their landing points,
separate the plane into finitely many regions, each containing exactly one
interior periodic point or one parabolic immediate basin invariant under
. This result generalizes the Goldberg-Milnor Separation Theorem for
polynomials, and has several corollaries. It follows, for example, that two
periodic Fatou components can always be separated by a pair of periodic rays
landing together; that there cannot be Cremer points on the boundary of Siegel
discs; that "hidden components" of a bounded Siegel disc have to be either
wandering domains or preperiodic to the Siegel disc itself; or that there are
only finitely many non-repelling cycles of any given period, regardless of the
number of singular values
Dynamics of transcendental H\'enon maps-II
Transcendental H\'enon maps are the natural extensions of the well
investigated complex polynomial H\'enon maps to the much larger class of
holomorphic automorphisms. We prove here that transcendental H\'enon maps
always have non-trivial dynamical behavior, namely that they always admit both
periodic and escaping orbits, and that their Julia sets are non-empty and
perfect
Singular values and non-repelling cycles for entire transcendental maps
Let be a map with bounded set of singular values for which periodic dynamic rays exist and land. We prove that each non-repelling cycle is associated to a singular orbit which cannot accumulate on any other non-repelling cycle. When has finitely many singular values this implies a refinement of the Fatou-Shishikura inequality