## Abstract

We solve a functional equation connected to the algebraic characterization of generalized information functions. To prove the symmetry of the solution, we study a related system of functional equations, which involves two homographies. These transformations generate the modular group, and this fact plays a crucial role in solving the system. The method suggests a more general relation between conditional probabilities and arithmetic.

## Motivation and results

In this paper, we study the measurable solutions \(u:[0,1]\rightarrow {\mathbb {R}}\) of the functional equation

for all \(x,y \in [0,1)\) such that \(x+y\in [0,1]\). The parameter \(\alpha \) can take any positive real value.

This equation appears in the context of algebraic characterizations of information functions. Given a random variable *X* whose range is a finite set \(E_X\), a measure of its “information content” is supposed to be a function \(f[X]: \Delta (E_X) \rightarrow {\mathbb {R}}\), where \(\Delta (E_X)\) denotes the set of probabilities on \(E_X\),

The most important example of such a function is the Shannon-Gibbs entropy

where \(0\log 0\) equals 0 by convention.

Shannon entropy satisfies a remarkable property, called the *chain rule*, that we now describe. Let *X* (resp. *Y*) be a variable with range \(E_X\) (resp. \(E_Y\)); both \(E_X\) and \(E_Y\) are supposed to be finite sets. The couple (*X*, *Y*) takes values in a subset \(E_{XY}\) of \(E_X\times E_Y\), and any probability *p* on \(E_{XY}\) induce by marginalization laws \(X_*p\) on \(E_X\) and \(Y_*p\) on \(E_Y\). For instance,

The chain rule corresponds to the identities

where \(p|_{X=x}\) denotes the conditional probability \(y\mapsto p(y,x)/X_*p(x)\). These identities reflect the third axiom used by Shannon to characterize an information measure *H*: “if a choice be broken down into two successive choices, the original *H* should be the weighted sum of the individual values of *H*” [7].

There is a deformed version of Shannon entropy, called generalized entropy of degree \(\alpha \) [1, Ch. 6]. For any \(\alpha \in (0,\infty )\setminus \{1\}\), it is defined as

This function was introduced by Havrda and Charvát [4]. Constantino Tsallis popularized its use in physics, as the fundamental quantity of non-extensive statistical mechanics [8], so \(S_\alpha \) is also called Tsallis \(\alpha \)-entropy. It satisfies a deformed version of the chain rule:

Suppose now that, given \(\alpha >0\), we want to find the most general functions *f*[*X*]—for a given collection of finite random variables *X*—such that

- A.
\(f[X](\delta )=0\) whenever \(\delta \) is any Dirac measure—a measure concentrated on a singleton—, which means that variables with deterministic outputs do not give (new) information when measured;

- B.
the generalized \(\alpha \)-chain rule holds, i.e. for any variables

*X*and*Y*with finite range^{Footnote 1}$$\begin{aligned} f[(X,Y)](p)&= f[X](X_*p) + \sum _{x\in E_X} (X_*p(x))^\alpha f[Y](Y_*(p|_{X=x})), \end{aligned}$$(1.9)$$\begin{aligned} f[(X,Y)](p)&= f[Y](Y_*p) + \sum _{y\in E_Y} (Y_*p(y))^\alpha f[X](Y_*(p|_{Y=y})). \end{aligned}$$(1.10)

The simplest non-trivial case corresponds to \(E_X=E_Y=\{0,1\}\) and \(E_{XY}=\{(0,0),(1,0),(0,1)\}\); a probability *p* on \(E_{XY}\) is a triple \(p(0,0)=a\), \(p(1,0)=b\), \(p(0,1)=c\), such that \(X_*p=(a+c,b)\) and \(Y_*p=(a+b,c)\). The equality between the right-hand sides of (1.9) and (1.10) reads

for any triple \((a,b,c)\in [0,1]^2\) such that \(a+b+c=1\). Setting \(a=0\) and using assumption A, we conclude that \(f[X](c,1-c)=f[Y](1-c,c)=: u(c)\) for any \(c\in [0,1]\). Therefore, (1.11) can be written in terms of this unique unknown *u*; if moreover we set \(c=y\), \(b=x\) and consequently \(a=1-x-y\), we get the functional equation (1.1), with the stated boundary conditions.

The main result of this article is the following.

### Theorem 1.1

Let \(\alpha \) be a positive real number. Suppose \(u:[0,1]\rightarrow {\mathbb {R}}\) is a measurable function that satisfies (1.1) for every \(x,y \in [0,1)\) such that \(x+y\in [0,1]\). Then, there exists \(\lambda \in {\mathbb {R}}\) such that \(u(x)=\lambda s_\alpha (x)\), where

and

when \(\alpha \ne 1\).

By convention, \(0\log _2 0 := \lim _{x\rightarrow 0} x \log _2 x = 0\). For \(\alpha =1\), Theorem 1.1 is essentially Lemma 2 in [5]. Our proof depends on two independent results.

### Theorem 1.2

(Regularity) Any measurable solution of (1.1) is infinitely differentiable on the interval (0, 1).

### Theorem 1.3

(Symmetry) Any solution of (1.1) satisfies \(u(x) = u(1-x)\) for all \(x\in {\mathbb {Q}}\cap [0,1]\).

The first is proved analytically, by means of standard techniques in the field of functional equations (cf. [1, 5, 9]), and the second by a novel geometrical argument, relating the equation to the action of the modular group on the projective line.

Theorems 1.2 and 1.3 above imply that any measurable solution *u* of (1.1) must be symmetric, i.e. \(u(x) = u(1-x)\) for all \(x\in [0,1]\), and therefore

whenever \(x,y\in [0,1)\) and \(x+y \in [0,1]\). When \(\alpha =1\), this equation is called “the fundamental equation of information theory”; it first appeared in the work of Tverberg [9], who deduced it from a characterization of an “information function” that not only supposed a version of the chain rule, but also the invariance of the function under permutations of its arguments. Daróczy introduced the fundamental equation for general \(\alpha >0\), and showed that it can be deduced from an axiomatic characterization analogue to that of Tverberg, that again supposed invariance under permutations along with a deformed chain rule akin to (1.8), see [3, Thm. 5].

For \(\alpha = 1\), Tverberg [9] showed that, if \(u:[0,1]\rightarrow {\mathbb {R}}\) is symmetric, Lebesgue integrable and satisfies (1.12), then it must be a multiple of \(s_1(x)\). In [5], Kannappan and Ng weakened the regularity condition, showing that all measurable solutions of (1.12) have the form \(u(x) = As_1(x) + Bx\) (where *A* and *B* are arbitrary real constants), which reduces to \(u(x) = As_1(x)\) when *u* is symmetric. In fact, they solved some generalizations of the fundamental equation, proving among other things that, when \(\alpha =1\), the only measurable solutions of (1.1) are multiples of \(s_1(x)\).

For \(\alpha \ne 1\), Daróczy [3] established that any \(u:[0,1]\rightarrow {\mathbb {R}}\) that satisfies (1.12) and \(u(0)=u(1)\) has the form^{Footnote 2}

*without any hypotheses on the regularity of **u*. The proof starts by proving that any solution of (1.12) must satisfy \(u(0)=0\) (setting \(x=0\)), and hence be symmetric (setting \(y=1-x\)). Since we are able to prove symmetry of the solutions of (1.1) *restricted to rational arguments* without any regularity hypothesis, we also get the following result.

### Corollary 1.4

For any \(\alpha \in (0,\infty )\setminus \{1\}\), the only functions \(u:{\mathbb {Q}}\cap [0,1]\rightarrow {\mathbb {R}}\) that satisfy equation (1.1) are multiples of \(s_\alpha \).

### Proof

Set \(x=0\) in (1.1) to conclude that \(u(1)=0\), and \(y=0\) to obtain \(u(0)=0\). Moreover, *u* must be symmetric (Theorem 1.3), hence it must fulfill (1.12) when the arguments are rational. Given these facts, Daróczy’s proof in [3, p. 39] applies with no modifications when restricted to \(p,q\in {\mathbb {Q}}\). \(\square \)

More details on the characterization of information functions by means of functional equations can be found in the classical reference [1], which gives a detailed historical introduction. Reference [2] summarizes more recent developments in connection with homological algebra.

It is quite remarkable that Theorem 1.1 serves as a fundamental result to prove that, up to a multiplicative constant, \(\{S_\alpha [X]\}_{X\in {\mathcal {S}}}\) is the only collection of measurable functionals (not necessarily invariant under permutations) that satisfy the corresponding \(\alpha \)-chain rule, for any generic set of random variables \({\mathcal {S}}\). In order to do this, one introduces an adapted cohomology theory, called information cohomology [2], where the chain rule corresponds to the 1-cocycle condition and thus has an algebro-topological meaning. The details can be found in the dissertation [10].

## The modular group

The group \(G= SL_2({\mathbb {Z}})/\{\pm I\}\) is called the **modular group**; it is the image of \(SL_2({\mathbb {Z}})\) in \(PGL_2({\mathbb {R}})\). We keep using the matrix notation for the images in this quotient. We make *G* act on \(P^1({\mathbb {R}})\) as follows: an element

acting on \([x:y]\in P^1({\mathbb {R}})\) (homogeneous coordinates) gives

Let *S* and *T* be the elements of *G* defined by the matrices

The group *G* is generated by *S* and *T* [6, Ch. VII, Th. 2]; in fact, one can prove that \(\langle S,T;S^2, (ST)^3\rangle \) is a presentation of *G*.

## Regularity: proof of Theorem 1.2

Lemma 3 in [5] implies that *u* is locally bounded on (0, 1) and hence locally integrable. Their proof is for \(\alpha =1\), but the argument applies to the general case with almost no modification, just replacing

where *x*, *y* are such that \(u(1-x)\le N\), \(u \left( \frac{y}{1-x}\right) \le N\) and \(u\left( \frac{1-x-y}{1-y}\right) \le N\), by

which is evidently valid too whenever \(x,y\in (0,1)\).

To prove the differentiability, we also follow the method in [5]—already present in [9]. Let us fix an arbitrary \(y_0\in (0,1)\); then, it is possible to chose \(s,t\in (0,1)\), \(s<t\), such that

for all *y* in certain neighborhood of \(y_0\). We integrate (1.1) with respect to *x*, between *s* and *t*, to obtain

The continuity of the right-hand side of (3.1) as a function of *y* at \(y_0\), implies that *u* is continuous at \(y_0\) and therefore on (0, 1). The continuity of *u* in the right-hand side of (3.1) implies that *u* is differentiable at \(y_0\). An iterated application of this argument shows that *u* is infinitely differentiable on (0, 1).

## Symmetry: proof of Theorem 1.3

Define the function \(h:[0,1]\rightarrow {\mathbb {R}}\) through

Observe that *h* is anti-symmetric around 1/2, that is, we have

Let now \(z\in \left[ \frac{1}{2}, 1\right] \) be arbitrary and use the substitutions \(x=1-z\) and \(y=1-z\) in (1.1) to derive the identity

Using the anti-symmetry of *h* to modify the right-hand side of the previous equation, we also deduce that

Setting \(x=0\) (respectively \(y=0\)) in (1.1), we to conclude that \(u(1)=0\) (resp. \(u(0)=0\)). Hence, the function *h* is subject to the boundary conditions \(h(0)=h(1)=0\). From (4.3), it follows that \(h(1/2)=h(0)/2^\alpha = 0\). If the domain of *h* is extended to the whole real line imposing 1-periodicity:

a similar argument can be used to determine the value of *h* at any rational argument. To that end, it is important to establish first that (4.3) and (4.4) hold for the extended function.

### Theorem 4.1

The function *h*, extended periodically to \({\mathbb {R}}\), satisfies the equations

We establish first the anti-symmetry around 1/2 of the extended *h* (Lemma 4.2), which implies that (4.7) follows from (4.6); the latter is a consequence of Lemmas 4.3–4.7.

### Lemma 4.2

### Proof

We write \(x = [x]+\{x\}\), where \(\{x\}:= x-[x]\). Then,

\(\square \)

### Lemma 4.3

### Proof

For *h* is periodic, (4.8) is equivalent to

and the change of variables \(u=x-1\) gives

Note that \(1 - \frac{u}{u+1} = \frac{1}{u+1} \in [1/2,1]\) whenever \(u\in [0,1]\). Therefore,

This establishes (4.10). \(\square \)

### Lemma 4.4

### Proof

If \(x\in [2,\infty [\), then \(1 - \frac{1}{x} \in \left[ \frac{1}{2}, 1\right] \) and we can apply Eq. (4.3) to obtain

We prove (4.11) by recurrence. The case \(x\in [1,2]\) corresponds to Lemma 4.3. Suppose it is valid on \([n-1,n]\), for certain \(n\ge 2\); for \(x\in [n,n+1]\),

\(\square \)

### Lemma 4.5

### Proof

The previous lemma and periodicity imply that \(h(x-1) = x^\alpha h(1-x^{-1})\) for all \(x\ge 2\), i.e.

Then, for \(u\ge 1\),

We set \(y=(u+1)^{-1}\in \left( 0,\frac{1}{2}\right] \). Equation (4.15) reads

Since \(h(0) = 0\), the lemma is proved. \(\square \)

### Lemma 4.6

### Proof

Immediately deduced from the previous lemma using the anti-symmetric property in Lemma 4.2. \(\square \)

### Lemma 4.7

### Proof

On the one hand, periodicity implies that \(h(x) = h(x+1) \overset{{{(\text {Lem. }4.2)}}}{=} -h(1-(x+1)) = -h(-x)\). On the other, for \(x\le 0\), the preceding results imply that \(h(-x) = (-x)^\alpha h(2-(-x)^{-1})= |x|^\alpha h(2-(-x)^{-1})\). Therefore,

\(\square \)

The transformations \(x\mapsto \frac{2x-1}{x}\) and \(x\mapsto \frac{1-x}{x}\) in Eqs. (4.6) and (4.7) are homographies of the real projective line \(P^1({\mathbb {R}})\), that we denote respectively by \(\alpha \) and \(\beta \). They correspond to elements

in *G*, that satisfy

This last matrix corresponds to \(x\mapsto 1-x\).

### Lemma 4.8

The matrices *A* and \(B^2\) generate *G*.

### Proof

Let

One has

and

Therefore, \(PAP^{-1} = T^{-1}\) and \(S=T^{-3} P B^{-2} P^{-1}\). Inverting these relations, we obtain

Let *X* be an arbitrary element of *G*. Since \(Y=PXP^{-1}\in G\) and *G* is generated by *S* and *T*, the element *Y* is a word in *S* and *T*. In consequence, *X* is a word in \(P^{-1}SP\) and \(P^{-1}TP\), which in turn are words *A* and \(B^2\). \(\square \)

It is possible to find explicit formulas for *S* and *T* in terms of *A* and \(B^2\). Since \(P=S^{-1}T^{-1}\), we deduce that \(PSP^{-1}= S^{-1}T^{-1}STS\) and \(PTP^{-1} = S^{-1}T^{-1}TTS = S^{-1}TS\). Hence, in virtue of (4.22),

and

To finish our proof of Proposition 1.3, we remark that the orbit of 0 by the action of *G* on \(P^1({\mathbb {R}})\) is \({\mathbb {Q}}\cup \{\infty \}\), where \({\mathbb {Q}}\cup \{\infty \}\) has been identified with \(\{[p:q] \in P^1({\mathbb {R}}) \mid p,q\in {\mathbb {Z}}\}\subset P^1({\mathbb {R}})\). This is a consequence of Bezout’s identity: for every point \([p:q]\in P^1({\mathbb {R}})\) representing a reduced fraction \(\frac{p}{q} \ne 0\) (\(p,q \in {\mathbb {Z}}\setminus \{0\}\) and coprime), there are two integers *x*, *y* such that \(xq - yp = 1\). Therefore

is an element of *G* and \(g'[0:1] = [p:q]\). The case \(q=0\) is covered by

The extended Eqs. (4.6) and (4.7) are such that

- 1.
For all \(x\in {\mathbb {R}}\), if \(h(x) = 0\) then \(h(\alpha ^{-1} x) = 0\) and \(h(\beta ^{-1} x) = 0\);

- 2.
For all \(x\in {\mathbb {R}}\setminus \{0\}\), if \(h(x) = 0\) then \(h(\alpha x) = 0\) and \(h(\beta x) = 0\).

Since \(h(1/2)=0\), the following lemma is the missing piece to establish that the extended *h* vanishes on \({\mathbb {Q}}\) (and hence the original *h* necessarily vanishes on \([0,1]\cap {\mathbb {Q}})\).

### Lemma 4.9

For any \(r\in {\mathbb {Q}}\setminus \{0\}\), there exists a finite sequence

such that \(r=w_n\circ \cdots \circ w_1(1/2)\) and, for all \(i\in \{1,...,n\}\), the iterate \(x_i:=w_i\circ \cdots \circ w_1(1/2)\) does not equal 0 or \(\infty \).

### Proof

Since the orbit in \(P^1({\mathbb {R}})\) of 1/2 by the group of homographies generated by *A* and \(B^2\) (i.e. *G* itself) contains the whole set of rational numbers \({\mathbb {Q}}\), there exists a *w* such that \(r=w_n \circ \cdots \circ w_1(1/2)\), where each \(w_i\) equals \(\alpha \), \(\beta \) or one of their inverses.

If some iterate equals 0 or \(\infty \), the sequence *w* can be modified to avoid this. Let \(i\in \{0,...,n\}\) be the largest index such that \(x_i\in \{0,\infty \}\); in fact, \(i<n\) because \(r\ne 0,\infty \).

If \(x_i = 0\), then \(x_{i+1} \in \{1/2,1\}\) (the possibility \(x_{i+1}=\infty \) is ruled out by the choice of

*i*). In the case \(x_{i+1}=1/2\), the equality \(r=w_n\circ \cdots \circ w_{i+2}(1/2)\) holds, and when \(x_{i+1}=1\), we have \(r=w_n\circ \cdots \circ w_{i+2}\circ \beta (1/2)\).If \(x_i = \infty \), then \(x_{i+1}\in \{2,-1\}\) (again, \(x_{i+1}=0\) is ruled out). When \(x_{i+1}=2\), we have \( r=w_n\circ \cdots \circ w_{i+2} \circ \beta \circ \alpha \circ \beta ^{-1}\circ \beta ^{-1}(1/2)\), and when \(x_{i+1}=-1\), it also holds that \(r=w_n\circ \cdots \circ w_{i+2} \circ \alpha \circ \alpha \circ \beta ^{-1}\circ \beta ^{-1}(1/2)\).

\(\square \)

## Notes

- 1.
Assumption A can be deduced from B if one identifies

*X*with (*X*,*X*) through the diagonal map \(E_X \rightarrow E_X\times E_X, \;x\mapsto (x,x)\) and then evaluates (1.9) at \(Y=X\) and \(p=\delta _{x_0}\), for any \(x_0\in E_X\). - 2.
In fact, he supposes \(u(1/2)=1\), but the argument works in general.

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### Cite this article

Bennequin, D., Vigneaux, J.P. A functional equation related to generalized entropies and the modular group.
*Aequat. Math.* (2020). https://doi.org/10.1007/s00010-020-00717-2

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### Keywords

- Generalized entropies
- Shannon entropy
- Tsallis entropy
- Modular group
- Functional equation
- Information cohomology

### Mathematics Subject Classification

- Primary 97I70
- 94A17