142 research outputs found
Stable components in the parameter plane of transcendental functions of finite type
We study the parameter planes of certain one-dimensional, dynamically-defined
slices of holomorphic families of entire and meromorphic transcendental maps of
finite type. Our planes are defined by constraining the orbits of all but one
of the singular values, and leaving free one asymptotic value. We study the
structure of the regions of parameters, which we call {\em shell components},
for which the free asymptotic value tends to an attracting cycle of
non-constant multiplier. The exponential and the tangent families are examples
that have been studied in detail, and the hyperbolic components in those
parameter planes are shell components. Our results apply to slices of both
entire and meromorphic maps. We prove that shell components are simply
connected, have a locally connected boundary and have no center, i.e., no
parameter value for which the cycle is superattracting. Instead, there is a
unique parameter in the boundary, the {\em virtual center}, which plays the
same role. For entire slices, the virtual center is always at infinity, while
for meromorphic ones it maybe finite or infinite. In the dynamical plane we
prove, among other results, that the basins of attraction which contain only
one asymptotic value and no critical points are simply connected. Our dynamical
plane results apply without the restriction of finite type.Comment: 41 pages, 13 figure
Enumerating Palindromes and Primitives in Rank Two Free Groups
Let be a rank two free group. A word in is {\sl
primitive} if it, along with another group element, generates the group. It is
a {\sl palindrome} (with respect to and ) if it reads the same forwards
and backwards. It is known that in a rank two free group any primitive element
is conjugate either to a palindrome or to the product of two palindromes, but
known iteration schemes for all primitive words give only a representative for
the conjugacy class. Here we derive a new iteration scheme that gives either
the unique palindrome in the conjugacy class or expresses the word as a unique
product of two unique palindromes. We denote these words by where
is rational number expressed in lowest terms. We prove that is
a palindrome if is even and the unique product of two unique palindromes
if is odd. We prove that the pairs generate the group
when . This improves the previously known result that held only for
and both even. The derivation of the enumeration scheme also gives a
new proof of the known results about primitives.Comment: Final revisions, to appear J Algebr
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