A Nonlinear Transfer Operator Theorem

In recent papers, Kenyon et al. (Ergod Theory Dyn Syst 32:1567–1584 2012), and Fan et al. (C R Math Acad Sci Paris 349:961–964 2011, Adv Math 295:271–333 2016) introduced a form of non-linear thermodynamic formalism based on solutions to a non-linear equation using matrices. In this note we consider the more general setting of Hölder continuous functions.


Introduction
We first recall a classical result for matrices dating back to work of Perron (1907) and Frobenius (1912) (cf. [5], p. 53). A k × k matrix A is called non-negative if all the entries are non-negative real numbers and aperiodic if there exists n > 0 such that all entries of the nth power A n are strictly positive.

Theorem 1.1 (Perron-Frobenius Theorem) Let A be a non-negative aperiodic k × k-matrix.
There exists a unique positive maximal eigenvalue λ > 0 and a unique positive eigenvector v such that Av = λv.
The following result of Ruelle is a cornerstone of the classical theory of thermodynamic formalism (cf. [10]).
It is also known that, aside from the maximal eigenvalue λ, the rest of the spectrum of In particular, part 3 of Theorem 1.3 then follows from part 1 by standard perturbation theory.
Recently, several authors introduced a particular non-linear version of Theorem 1.1 for matrices which is useful in the study of the dimension of certain sets in the theory of non standard ergodic averages (see section 2). Theorem 1.4 (Kenyon-Peres-Solomyak, Fan-Schmeling-Wu) Let B be a non-negative irreducible k × k matrix. There exists a unique positive vector v such that Bv = v 2 , where the entries of v 2 are the square of those of v, i.e., (v) i 2 = v i for i = 1, · · · , k.
In the special case that A has entries which are natural numbers, this appears as Lemma 1.2 in [6]. A version of this for more general positive matrices appears as 4.1 in [2] (cf. also [3]) under very modest assumptions on the matrix. Other types of non-linear Perron-Frobenius Theorem appear in [7] and [8].
The following is our main result, which can be viewed either as a non-linear version of Theorem 1.3, or a generalisation of the Theorem 1.4 (at least for aperiodic matrices) from matrices to functions.
In the particular case that the function φ(x) = φ(x 0 , x 1 ) depends on only finitely many coordinates then Theorem 1.4 can be recovered as a corollary to Theorem 1.5.
and observe that φ ∈ F θ for any 1/2 < θ < 1. By Theorem 1.5 (with the choice q = 2) there is a function ψ such that L φ ψ = ψ 2 . In Fig. 1 we plot a realisation of ψ using the dyadic expansion on the unit interval.
where ψ is the positive eigenfunction in Theorem 1.3, we can assume without loss of generality that ψ φ 1 (x) = 1 is the constant function taking the value 1, i.e., In particular, for such special normalized functions φ 1 the function ψ Theorem 1.5 can easily be identified as ψ = ψ φ 1 = λ1, then we see that L φ 1 ψ = ψ 2 . Furthermore, the hypothesis for analyticity in part 3 of Theorem 1.5 automatically holds.
I am grateful to the referees and the editors for their patience and help with this short note.

Background to Theorem 1.4
Although our main result (Theorem 1.5) is of independent interest, for the reader's benefit we will now give a brief description of the original application of its precursor (Theorem 1.4) which provided the motivation for its introduction. Following [6] and [2,3] given a probability measure μ on we can define a so-called multiplicative measure ν on = {1, . . . , k} Z + , say, by writing = j odd j where j = {1, · · · , k} j and j = { j2 n : n ≥ 0}, for j = 1, 3, 5, · · · , which form a natural partition of N by N = ∪ j odd j . We can then define ν = j μ, in a natural sense. In [6] and [3] the authors consider the measure μ to be a (generalised) Markov measure defined in terms of the entries in the vector v in Theorem 1.4. The measure ν will typically not be σ -invariant but is still useful in studying the Hausdorff dimension of certain sets.
We can define the pointwise dimension of ν by Finally, by Proposition 2.3 of [6] we have that for any σ -ergodic measure ν the pointwise dimension is constant and takes the explicit value } is the standard partition into cylinders of length one; ∨ n−1 i=0 σ −i α is the usual refinement to a partition by cylinders of length n; and H μ (·) is the entropy for partitions [11].
The pointwise dimension is particularly useful in estimating the Hausdorff Dimension of sets (especially lower bounds via the usual mass distribution principle cf. [1], §4.2) associated to multiple ergodic theorems, as the following example illustrates.
In this case one can consider the matrix B = 1 1 1 0 and the solution to We then have that μ is a Markov measure for which is strictly less than the Minkowski dimension dim M (X ) = 0 · 82429 . . . [4,6].

Proof of Theorem 1.5
The proof of the existence of the fixed point is the more interesting part of the problem. The uniqueness and analyticity are then relatively easy to establish.

Existence of the Fixed Point
The existence can be shown by looking for a fixed point of a suitable map in the space where c > 0 and d θ (·, ·) is as defined in Example 2.1. We first note that c ⊂ F θ since for u ∈ c and x, x ∈ we can bound u( for sufficiently large C > 0 and then interchanging x and x gives that u θ ≤ C (cf. [9], p. 22).
We can now introduce a family of non-linear operators defined as follows: where 1 n 1 represents the function taking the constant value 1 n .

Lemma 3.2 We have that
If y ∈ σ −1 x then we denote by y ∈ σ −1 x the corresponding sequence for which y 0 = y 0 , and thus we have that d θ (y, y ) = θ N +1 . Let u ∈ c then we have that where we have used that d θ (y, y ) = θ N +1 and then since u ∈ c we have that u(y) ≤ u(y )e cθ N +1 . In particular, we have that We can use the above lemma to deduce the following.

Lemma 3.4 For c > 0 sufficiently large we have that
Proof Let u ∈ c . For each n ≥ 1 the constant function 1 n 1 ∈ c and so by applying Lemma 3.2 to the new function u + 1 n 1 we see that for all x, x ∈ with x 0 = x 0 . Dividing both sides of (3.1) by L u + 1 n 1 ∞ > 0 we have that Finally, since the values taken by both sides of (3.2) lie in the unit interval, taking square roots preserves this property with c replaced by c /2, i.e., i.e., N n (u) ∈ c /2 . Providing c is sufficiently large that c > c /2 = (c + φ θ )θ/2 we have that c /2 ⊂ c and the result follows.
This now brings us to the existence of the fixed point for each of the operators N n : c → c .

Lemma 3.5
For each n ≥ 1, there exists a non-trivial fixed point ψ n ∈ c such that N n (ψ n ) = ψ n .
Proof By the Arzela-Ascoli Theorem the space c is compact with respect to the norm · ∞ . For each n ≥ 1 and c > 0 sufficiently large the map N n : c → c is a continuous map on a compact convex subspace of C 0 ( ) and we can apply the Schauder fixed point theorem to deduce that there is a fixed point ψ n ∈ c for N n . To see that ψ n is not identically zero we need only observe that by the definition of N n there exists x (n) ∈ with N n (ψ n )(x (n) ) = 1 and by construction ψ n (x (n) ) = N n (ψ n )(x (n) ) = 1. This completes the proof.

Uniqueness of the Positive Fixed Point
Assume for a contradiction that we had a second distinct non-trivial positive fixed point, i.e., L φ (ψ ) = ψ 2 with ψ > 0 and ψ = ψ . We can then associate ξ := inf{t > 0 : tψ − ψ ≥ 0} and thus, in particular, In particular, this implies that ξ ≤ 1, otherwise it contradicts the original definition of ξ .

Remark 3.6
This simple argument doesn't rule out the possibility of another non-positive fixed point.

Analyticity
To show the analytic dependence of the solution we want to use the implicit function theorem applied to the function G : In order to apply the implicit function theorem at An easy calculation gives that The spectral radius of any linear operator is the radius of the smallest disk (centred at the origin) containing the spectrum.
We recall the following result [9] which is also due to Ruelle.
Remark 3.8 (The Tangent Operator) Closely related to this circle of ideas is the use of a standard technique in understanding the iterates of a non-linear operator in a neighbourhood of a fixed point. More precisely, we consider the first order approximation to G(φ 0 , ·) : ψ → L φ 0 ψ − ψ 2 where ψ = ψ 0 + ψ (1) + o( ). A simple calculation gives that the tangent operator (1) .
For definiteness, we can consider the specific case where we replace φ 0 by φ 1 as in Remark 1.7, then we see that the spectra sp(T φ 1 ) and sp(L φ 1 ) are simply related by sp(T φ 1 ) = sp(L φ 1 )−2. Thus, since λ φ 1 = 1, by Lemma 3.7 the tangent operator T φ 1 will have its spectra in the disk centred at −2 and of radius 1 (and thus outside the unit disk, except for the value −1). This suggests that the fixed point ψ φ 1 is locally unstable in a codimension one space under the iteration G(φ 1 , ·) n

Measures
The classical transfer operator L φ plays an important role in the ergodic theory of equilibrium states associated to φ. More precisely, the equilibrium state is a fixed point for the dual L * φ 1 to the transfer operator L φ 1 (satisfying L φ 1 1 = 1) for the associated function φ 1 (cf. Remark 1.6). Although there is no direct analogue of equilibrium states in the context of the nonlinear equations L φ ψ = ψ 2 we have been studying, one can still use this identity to associate to the two functions φ and ψ a natural invariant measure.
Given a solution L φ ψ = ψ 2 as in Theorem 1.5, we can consider the linear operator M φ : F θ → F θ given by which then satisfies M φ 1 = 1 (cf. Remark 1.7). Since M φ is a transfer operator with a Hölder continuous potential, it is a consequence of the simplicity of the maximal positive eigenvalue for the operator in Theorem 1.3, and thus of its dual, that there is a unique σinvariant probability measure μ such that M * φ μ = μ, i.e., f dμ = M ψ f dμ for all f ∈ C 0 ( ). This leads to a non-standard version of the variational principle. with equality if and only if ν = μ.
We consequently have a particularly simple expression for the entropy h(μ).