The codenominator is a function that extends the Fibonacci sequence to the index set of positive rational numbers, formula_1. Many known Fibonacci identities carry over to the codenominator. One can express Dyer's outer automorphism formula_2 of the extended modular group in terms of the codenominator. This automorphism can be viewed as an automorphism group of the trivalent tree. The real formula_2-covariant modular function Jimm on the real line formula_4 is defined via the codenominator. Jimm relates the Stern-Brocot tree to the Bird tree.
Jimm induces an involution of the moduli space of rank-2 pseudolattices and is related to the arithmetic of real quadratic irrationals.
Definition of the codenominator.
The codenominator function formula_5 is defined by the following system of functional equations:
formula_6
with the initial condition formula_7. The function formula_8 is called the conumerator. (The name `codenominator' comes from the fact that the usual denominator function formula_9
can be defined by the functional equations
formula_10
and the initial condition formula_11.)
The codenominator takes every positive integral value infinitely often.
Connection with the Fibonacci sequence.
For integer arguments, the codenominator agrees with the standard Fibonacci sequence, satisfying the recurrence:
formula_12
The codenominator extends this sequence to positive rational arguments. Moreover, for every rational formula_13, the sequence
formula_14 is the so-called Gibonacci sequence (also called the generalized Fibonacci sequence) defined by
formula_15,
formula_16
and the recursion
formula_17.
Properties of the codenominator.
The codenominator has the following properties:
1. Fibonacci recursion: The codenominator function satisfies the Fibonacci recurrence for rational arguments:
formula_18
2. Fibonacci invariance:
For any integer formula_19 and formula_20
formula_21
3. Symmetry: If formula_22, then
formula_23
4. Continued fractions: For a rational number formula_24 expressed as a simple continued fraction formula_25, the value of formula_26 can be computed recursively using Fibonacci numbers as:
formula_27
5. Involutivity: The numerator function formula_28 can be expressed in terms of the codenominator as formula_29, which implies
formula_30
6. Reversion:
formula_31
7. Splitting: Let formula_32 be integers. Then:
formula_33
where formula_34 is the least index such that formula_35
(if formula_36 then set formula_37).
8. Periodicity:
For any positive integer formula_38, the codenominator
formula_39 is periodic in each partial quotient formula_40 modulo formula_41 with period divisible with formula_42 , where formula_43 is the Pisano period.
9. Fibonacci identities:
Many known Fibonacci identities admit a codenominator version. For example,
if at least two among formula_44 are integral, then
formula_45
where formula_46 is the codiscriminant (also called the 'characteristic number').
This reduces to Tagiuri's identity when formula_47; which in turn is a generalization of the famous Catalan identity. Any Gibonacci identity can be interpreted as a codenominator identity. There is also a combinatorial interpretation of the codenominator.
The codiscriminant is a 2-periodic function.
Involution Jimm.
The Jimm (ج) function is defined on positive rational arguments via
formula_48
This function is involutive and admits a natural extension to non-zero rationals via formula_49 which is also involutive.
Let formula_50 be the simple continued fraction expansion of formula_51. Denote by formula_52 the sequence formula_53 of length formula_54. Then:
formula_55
with the rules:
formula_56
and
formula_57.
The function formula_58
admits an extension to the set of non-zero real numbers by taking limits (for positive real numbers one can use the same rules as above to compute it).
This extension (denoted again formula_58) is 2-1 valued on golden -or noble- numbers (i.e. the numbers in the -orbit of the golden ratio formula_60).
The extended function formula_58
and moreover satisfies the functional equations
formula_64 (except on the set of golden irrationals),
formula_66 (provided formula_67),
formula_69,
formula_71.
These four functional equations in fact characterize Jimm. Additionally, Jimm satisfies
formula_72
Let formula_73 be the jump of formula_74 at formula_75. Then
formula_76
Dyer's outer automorphism and Jimm.
The extended modular group admits the presentation
formula_77
where (viewing as a group of Möbius transformations) formula_78, formula_79 and formula_80.
The map formula_81 of generators
formula_82
defines an involutive automorphism formula_83 , called Dyer's outer automorphism.
It is known that Out()formula_84 is generated by formula_81.
The modular group 
formula_86
 is not invariant under formula_81. However, the subgroup
formula_88 is formula_81-invariant. Conjugacy classes of subgroups of formula_90 is in 1-1 correspondence with bipartite trivalent graphs, and formula_81 thus defines a duality of such graphs. This duality transforms zig-zag paths on a graph
formula_92 to straight paths on its formula_93-dual graph and vice versa.
Dyer's outer automorphism can be expressed in terms of the codenumerator, as follows:
Suppose formula_94 and formula_95. Then
formula_96
The covariance equations above implies that formula_97 is a representation of formula_81 as a map formula_99 , i.e.
formula_100 whenever formula_101 and
formula_102. Another way of saying this is that
formula_97 is a formula_2-covariant map.
In particular, formula_97 sends -orbits to -orbits, thereby inducing an involution of the moduli space of rank-2 pseudo lattices, , where is the projective line over the real numbers.
Given formula_106, the involution formula_97 sends the geodesic on the hyperbolic upper half plane formula_108 through formula_109
to the geodesic through formula_110, thereby inducing an involution of geodesics on the modular curve
\formula_108. It preserves the set of closed geodesics because formula_58 sends real quadratic irrationals to real quadratic irrationals (with the exception of golden numbers, see below) respecting the Galois conjugation on them.
Jimm as a tree automorphism.
Djokovic and Miller constructed formula_113 as a group of automorphisms of the infinite trivalent tree. In this context, formula_93 appears as an automorphism of the infinite trivalent tree.
formula_113 is one of the 7 groups acting with finite vertex stabilizers on the infinite trivalent tree.
Jimm and the Stern-Brocot tree.
Applying Jimm to each node of the Stern-Brocot tree permutes all rationals in a row and otherwise preserves each row, yielding a new tree of rationals called "Bird's tree", which was first described by Bird.
Reading the denominators of rationals on Bird's tree from top to bottom and following each row from left to right gives Hinze's sequence:
formula_116
The sequence of conumerators is:
formula_117
Properties of the plot of Jimm and the golden ratio.
By involutivity, the plot of formula_58 is symmetric with respect to the diagonal formula_119, and by covariance with formula_120, the plot is symmetric with respect to the diagonal formula_121. The fact that the derivative of formula_58 is 0 a.e. can be observed from the plot.
The plot of Jimm hides many copies of the golden ratio formula_123 in it.
For example
More generally, for any rational
formula_124, the limit
formula_125 is of the form
formula_126 with formula_127 and formula_128. The limit formula_129 is its Galois conjugate
formula_130.
Conversely, one has formula_131.
Jimm on real quadratic irrational numbers.
Jimm sends real quadratic irrationals to real quadratic irrationals, except the golden irrationals, which it sends to rationals in a 2–1 manner. It commutes with the Galois conjugation on the set of non-golden quadratic irrationals, i.e. if
formula_132, then
formula_133, with
formula_134 and
formula_135 positive non-squares.
For example:
formula_136
2-variable form of functional equations.
The functional equations can be written in the two-variable form as:
 formula_137
 formula_139
 formula_141
 formula_143
As a consequence of these, one has:
formula_144
Therefore formula_74 sends the pair formula_146 of complementary Beatty sequences
to the pair formula_147 of complementary Beatty sequences; where formula_109 are non-golden irrationals with formula_149.
If formula_150 is a real quadratic irrational, which is not a golden number, then as a consequence of the two-variable version of functional equations of formula_97 one has
1. formula_152
2. formula_153
3. formula_154
4. formula_155
where formula_156 denotes the norm and formula_157 denotes the trace of formula_158.
On the other hand, formula_58 may send two members of one real quadratic number field to members of two different real quadratic number fields; i.e. it does not respect individual class groups.
Jimm on Markov irrationals.
Jimm sends the Markov irrationals to 'simpler' quadratic irrationals, see table below.
Jimm and dynamics.
Jimm conjugates
the Gauss map formula_160 (see Gauss–Kuzmin–Wirsing operator) to the so-called Fibonacci map formula_161
, i.e.
formula_162.
The expression of Jimm in terms of continued fractions shows that, if a real number formula_24 obeys the Gauss-Kuzmin distribution, then
the asymptotic density of 1's among the partial quotients of formula_164 is one, i.e. formula_164 does not obey the Gauss-Kuzmin statistics. For example
21/3=formula_166
formula_58(21/3)=formula_168
This argument also shows that formula_97 sends the set of real numbers obeying the Gauss-Kuzmin statistics, which is of full measure, to a set of null measure.
Jimm on higher algebraic numbers.
It is widely believed that if formula_24 is an algebraic number of degree formula_171, then it obeys the Gauss-Kuzmin statistics. By the above remark, this implies that formula_164 violates the Gauss-Kuzmin statistics. Hence, according to the same belief, formula_164 must be transcendental. This is the basis of the conjecture that Jimm sends algebraic numbers of degree formula_174
to transcendental numbers. A stronger version of the conjecture states that any two algebraically related formula_175, formula_176 are in the same -orbit, if formula_109 are both algebraic of degree formula_174.
Functional equations and equivariant modular forms.
Given a representation formula_179, a meromorphic function formula_180 on formula_181 is called a formula_182-covariant function if
formula_183
(sometimes formula_180 is also called a formula_182-equivariant function). It is known that there exists meromorphic covariant functions formula_186 on the upper half plane formula_187, i.e. functions satisfying
formula_188. The existence of meromorphic functions satisfying a version of the functional equations for formula_97 is
also known.
Some codenumerator values.
Below is a table of some codenominator values formula_190, where 41 is an arbitrarily chosen number.