Maximal commuting families of functions
Dave Rusin
A recent thread in this group, and a colloquium talk I just attended,
both skirted around this issue, so I would like to ask about what is known.
A family F of analytic functions f : C --> C is _commuting_ if
for all f and g in F we have f o g = g o f. (Note that we insist the
functions be entire so that the composites are always defined, although
we could also consider e.g. maps on the Riemann sphere and thus include
families of rational functions.)
Here are some examples:
#1. The set of all translations f(z) = z + c .
#2. The set of all scalings f(z) = c z .
#3. For a fixed f0, the set of all iterates f = (f0)^n (n = 1, 2, 3, ...)
#4. The powers of x . (Allowing maps on the Riemann sphere, this can
include all the negative powers of x too.)
#5. The Chebyshev polynomials phi_n (since phi_n o phi_m = phi_{n+m} ).
I might mention as well that for any fixed f0 we can consider the
commutator of f0 in some other family of functions (the sets of
polynomials, or rational functions, or entire functions say) : this
is the set of all functions g with g o f0 = f0 o g . There is no
reason a priori to think that all elements of the commutator commute
with each other, although in many cases that is true. For example,
the commutator of the Chebyshev polynomial phi_2 in the set of
all polynomials is precisely the family #5 above (I think), and I heard
today that if f0(z) = a exp( b z ) + c with non-zero values of
a and b, then the only entire functions which commute with f0
are the iterates of f0, giving family #3. (In particular, such an f0
has no "square-root" under composition -- another oft-asked question here.)
Note that a linear function h(z) = a z + b with nonzero a is
invertible and entire, so we can conjugate any commuting family F of
functions by h to get another commuting family
{ h o f o h^(-1) ; f \in F }
Using this device we can conjugate any family of functions with a common
fixed point into a family of functions all of which fix the origin; the
family #2 is then just one representative of a larger conjugacy class.
It looks to me like every degree-2 polynomial of the form
f(x) = x + Q(x) where Q is a quadratic with a discriminant of 1
can be embedded into a conjugate of #4, and likewise f can be embedded
into a conjugate of #5 iff the discriminant of Q is 9. For
other quadratics f, there is no cubic polynomial which commutes with f,
and I would be willing to believe that the commutator is exactly
the family shown in #3 (and in particular, all commutators commute
with each other). Is that true?
Here are some other questions which arise.
Q1 : Are families #1 and #2 the only uncountable families of commuting
functions (up to conjugacy)?
Q2 : Are there MAXIMAL families of commuting functions which are _finite_?
(In view of the constrution in #3, this requires that f be periodic
on the image of some power of f, for each f in the family.)
Q3 : A family of commuting functions is ordered by divisibility (we say
f | g if g = f o h for some h in the family). Then families #3 and #5
are naturally isomorphic to the natural numbers ordered by " < ", while
family #4 is naturally isomorphic to the natural numbers ordered by "|".
What other order types are possible?
Q4 : What other invariants can be used to distinguish commuting families?
(E.g. I understand that a commuting family is conjectured to share a
common Julia set, so that we could distinguish conjugacy classes of
these families by properties of their Julia sets as well.)
I would be interested in learning what is known about (maximal) families of
commuting functions, especially those of specific types (e.g. rational
functions, functions of finite order, etc.)
dave
Nico Benschop
>
> A recent thread in this group, and a colloquium talk I just attended,
> both skirted around this issue, so I would like to ask about what is
> known.
>
> A family F of analytic functions f : C --> C is _commuting_ if
> for all f and g in F we have f o g = g o f. (Note that we
> insist the functions be entire so that the composites are always
> defined, although we could also consider e.g. maps on the Riemann
> sphere and thus include families of rational functions.)
Just a suggestion of common context for these questions: since function
composition is associative, and 'family' suggests closure, I take it
that the relevant context is that of semigroup theory, and in fact
commutative semigroups. That's a huge area indeed, especially if
infinite (continuous function) semigroups are included.
And since each function on a finite domain/codomain generates a unique
idempotent (Frobenius, I believe), that resriction leads to Boolean
algebra and Lattices... -- NB
Nico Benschop
>
> A recent thread in this group, and a colloquium talk I just attended,
> both skirted around this issue, so I would like to ask about what is
> known.
>
> A family F of analytic functions f : C --> C is _commuting_ if
> for all f and g in F we have f o g = g o f. (Note that we
> insist the functions be entire so that the composites are always
> defined, although we could also consider e.g. maps on the Riemann
> sphere and thus include families of rational functions.)
Just a suggestion of common context for these questions: since function
composition is associative, and 'family' suggests closure, I take it
that the relevant context is that of semigroup theory, and in fact
commutative semigroups. That's a huge area indeed, especially if
infinite (continuous function) semigroups are included.
And since each function on a finite domain/codomain generates a unique
idempotent (Frobenius, I believe), that restriction leads to Boolean