Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Transversality Family of Expanding Rational Semigroups   This paper deals with both complex dynamical systems and conformal iterated
function systems. We study finitely generated expanding semigroups of rational
maps with overlaps on the Riemann sphere. We show that if a $d$-parameter
family of such semigroups satisfies the transversality condition, then for
almost every parameter value the Hausdorff dimension of the Julia set is the
minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff
dimension of the exceptional set of parameters is estimated. We also show that
if the zero of the pressure function is greater than 2, then typically the
2-dimensional Lebesgue measure of the Julia set is positive. Some sufficient
conditions for a family to satisfy the transversality conditions are given. We
give non-trivial examples of families of semigroups of non-linear polynomials
with transversality condition for which the Hausdorff dimension of the Julia
set is typically equal to the zero of the pressure function and is less than 2.
We also show that a family of small perturbations of Sierpi\'nski gasket system
satisfies that for a typical parameter value, the Hausdorff dimension of the
Julia set (limit set) is equal to the zero of the pressure function, which is
equal to the similarity dimension. Combining the arguments on the
transversality condition, thermodynamical formalisms and potential theory, we
show that for each complex number $a$ with $|a|\neq 0,1$, the family of small
perturbations of the semigroup generated by ${z^{2}, az^2} $ satisfies that for
a typical parameter value, the 2-dimensional Lebesgue measure of the Julia set
is positive.

23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
28
The categories of this paper are: alg-geom
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn