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Permutable entire functions and multiply connected wandering domains
Let f and g be permutable transcendental entire functions. We use a recent analysis of the dynamical behaviour in multiply connected wandering domains to make progress on the long standing conjecture that the Julia sets of f and g are equal; in particular, we show that J(f)=J(g) provided that neither f nor g has a simply connected wandering domain in the fast escaping set
On The Mackey Formula for Connected Centre Groups
Let be a connected reductive algebraic group over
and let be a
Frobenius endomorphism endowing with an -rational
structure. Bonnaf\'e--Michel have shown that the Mackey formula for
Deligne--Lusztig induction and restriction holds for the pair
except in the case where and has a quasi-simple component
of type , , or . Using their techniques we show
that if and is connected then the Mackey formula holds
unless has a quasi-simple component of type . This
establishes the Mackey formula, for instance, in the case where
is of type . Using this, together with work of
Bonnaf\'e--Michel, we can conclude that the Mackey formula holds on the space
of unipotently supported class functions if is connected.Comment: 7 pages; v2., minor changes, added Lemma 3.4 for clarit
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