A Nonlinear Transfer Operator Theorem

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1 J Stat Phys 207) 66: DOI 0.007/s A Nonlinear Transfer Operator Theorem Mark Pollicott Received: 22 June 206 / Accepted: 8 October 206 / Published online: 9 November 206 The Authors) 206. This article is published with open access at Springerlink.com Abstract In recent papers, Kenyon et al. Ergod Theory Dyn Syst 32: ), and Fan et al. C R Math Acad Sci Paris 349: , Adv Math 295: ) introduced a form of non-linear thermodynamic formalism based on solutions to a nonlinear equation using matrices. In this note we consider the more general setting of Hölder continuous functions. Keywords Ruelle operator theorem Transfer operator Thermodynamic formalism Introduction We first recall a classical result for matrices dating back to work of Perron 907) and Frobenius 92) 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. 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. We next recall a generalisation of the Perron Frobenius Theorem to Banach spaces of functions. Let σ : be a one-sided mixing subshift of finite type with alphabet F ={,, k} i.e., there exists an aperiodic k k matrix B with 0 entries such that ={x = x n ) n=0 n=0 F : x n F, Bx n, x n+ ) =, n 0} and σ x) n = x n+. Given 0 <θ<, let F θ be the space of functions f : R for which the semi-norm Dedicated to David Ruelle on the occasion of his 80th Birthday. B Mark Pollicott mpollic@maths.warwick.ac.uk Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

2 A Nonlinear Transfer Operator Theorem 57 var n f ) f θ = sup n 0 θ n < + is finite, where var n f ) = sup{ f x) f x ) : x i = x i for i = 0,, n }. In particular, F θ is a Banach space with respect to the norm f = f θ + f. Definition.2 Let φ F θ. We can define a transfer operator L φ : F θ F θ by L φ ψx) = e φy) ψy), where ψ F θ and x. σ y=x The following result of Ruelle is a cornerstone of the classical theory of thermodynamic formalism cf.[0]). Theorem.3 Ruelle Perron Frobenius Theorem) Let φ F θ.. There exists λ = λ φ > 0 and ϕ = ϕ φ F θ with ψ>0 such that L φ ϕ = λϕ; 2. any ψ F θ with L φ ψ = λψ is necessarily a multiple of ψ; and 3. the dependences F θ φ λ φ R + and F θ φ ϕ φ F θ are analytic. It is also known that, aside from the maximal eigenvalue λ, the rest of the spectrum of L φ : F θ F θ is contained in {z C : z <λ φ }. In particular, part 3 of Theorem.3 then follows from part by standard perturbation theory. The value Pφ) := log λ φ is called the pressure of the function φ F θ [0]. In the case that φx) = φx 0, x ) depends only on the first two terms in x = x n ) n=0 then Theorem.3 reduces to Theorem., bytakingai, j) = exp φi, j). In this case, ψx) = ψx 0 ) depends only on the first term and we can set v = ψ),...,ψk)). Recently, several authors introduced a particular non-linear version of Theorem. 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.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 =,, k. In the special case that A has entries which are natural numbers, this appears as Lemma.2 in [6]. A version of this for more general positive matrices appears as 4. 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.3, or a generalisation of the Theorem.4 at least for aperiodic matrices) from matrices to functions. Theorem.5 Main Theorem) Let φ F θ.. There exists ψ = ψ φ F θ with ψ>0 such that L φ ψ = ψ 2 ; 2. for any ψ F θ with L φ ψ = ψ 2 and ψ > 0 then ψ = ψ; and 3. the dependence F θ φ ψ φ F θ is analytic providing ψ φ is sufficiently close to the constant function λ φ in norm.

3 58 M. Pollicott Fig. Aplotofψ φ x) using the dyadic expansion 0 n=0 x n ) 2 n+ on the unit interval to represent x 2 The result easily generalises to L φ ψ = ψ q, for any natural number q 2. We consider only the case q = 2 to avoid introducing additional notation. In the particular case that the function φx) = φx 0, x ) depends on only finitely many coordinates then Theorem.4 can be recovered as a corollary to Theorem.5. Example.6 Let 2 ={, 2} Z + correspond to a full shift with F ={, 2}. We can define a function φ : 2 R by ) x n ) φx) = 4sin 2π 2 n+ and observe that φ F θ for any /2 <θ<. By Theorem.5 with the choice q = 2) there is a function ψ such that L φ ψ = ψ 2.InFig. we plot a realisation of ψ using the dyadic expansion on the unit interval. n=0 Remark.7 If φ F θ then, as usual, by replacing φ by φ = φ + log ψ φ log ψ φ σ F θ,whereψ is the positive eigenfunction in Theorem.3, we can assume without loss of generality that ψ φ x) = is the constant function taking the value, i.e., L φ = λ, where λ = λ φ = λ φ. In particular, for such special normalized functions φ the function ψ Theorem.5 can easily be identified as ψ = ψ φ = λ, thenweseethatl φ ψ = ψ 2. Furthermore, the hypothesis for analyticity in part 3 of Theorem.5 automatically holds. I am grateful to the referees and the editors for their patience and help with this short note. 2 Background to Theorem.4 Although our main result Theorem.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.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 ={,...,k} Z+, say, by writing = j odd j where j ={,, k} j and j = { j2 n : n 0}, for j =, 3, 5,, which form a natural

4 A Nonlinear Transfer Operator Theorem 59 partition of N by N = j odd j. We can then define ν = j μ,inanaturalsense.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.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 dim H ν) = lim n + n log ν [x 0, x,, x n ]) for a.e.ν)x = x n ) where [x 0, x,, x n ]={y = y n ) : x i = y i, 0 i n } is a cylinder set. 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 dim H ν) = n= ) 2 n+ H μ i=0 n σ i α where α ={[],...,[k]} is the standard partition into cylinders of length one; n i=0 σ i α is the usual refinement to a partition by cylinders of length n; andh μ ) is the entropy for partitions []. The pointwise dimension is particularly useful in estimating the Hausdorff Dimension of sets especially lower bounds via the usual mass distribution principle cf. [], 4.2) associated to multiple ergodic theorems, as the following example illustrates. Example 2. Golden Mean Example [4,6]) Fan Liao Ma and Kenyon Peres Solomyak considered the golden mean example: X = { } x n ) {0, } N : x n x 2n = 0, n, with the usual metric d θ x, x ) = { θ Nx,x ) if x = x 0 if x = x. where Nx, x ) = sup{n 0:x i = x i for 0 i n} and ) Nx, x ) = 0ifx 0 = x 0 ). In this case one can consider the matrix B = and the solution to Bv = v 0 2, i.e., ) v v = satisfies v v 2 = v + v 2 and v2 2 = v. We then have that μ is a Markov measure for 2 ) p p P =,wherep 0 3 = p) 2, and finally dim H X) = log 2 p = which is strictly less than the Minkowski dimension dim M X) = [4,6]. 3 Proof of Theorem.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.

5 520 M. Pollicott 3. Existence of the Fixed Point The existence can be shown by looking for a fixed point of a suitable map in the space { c := u : [0, ] : ux) ux )e cd θ x,x ) } for those x, x with x 0 = x 0 where c > 0andd θ, ) is as defined in Example 2.. We first note that c F θ since for u c and x, x we can bound ux) ux ) u e cd θ x,x ) ) Cd θ x, x ) 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: Definition 3. For each n we can associate to u c a new function N n u) : R defined by Lφ u + N n u)x) = n ) ) x) 2 L φ u + n ) where n represents the function taking the constant value n. Lemma 3.2 We have that L φ c ) c for c = c + φ θ )θ. Proof Let x, x with x 0 = x 0. Assume that d θ x, x ) = θ N,forsomeN 0, then x i = x i for 0 i N and x N+ = x N+.Ify σ x then we denote by y σ x the corresponding sequence for which y 0 = y 0, and thus we have that d θ y, y ) = θ N+.Let u c then we have that L φ ux) = σ y=x eφy) uy) σ y=x eφy )+ φ θ θ N+ uy )e e c+ φ θ )θd θ x,x ) σ y =x eφy ) uy ) = e c+ φ θ )θd θ x,x ) L φ ux ) cθ N+) wherewehaveusedthatd θ y, y ) = θ N+ and then since u c we have that uy) uy )e cθ N+.Inparticular,wehavethatL φ ux) e c d θ x,x ) L φ ux ), i.e., L φ u c. Remark 3.3 By definition of c, we see that if c < c then c c thus, providing c is sufficiently large, Lemma 3.2 gives L φ c ) c. We can use the above lemma to deduce the following. Lemma 3.4 For c > 0 sufficiently large we have that N φ c ) c. Proof Let u c. For each n the constant function n c and so by applying Lemma 3.2 to the new function u + n we see that L φ u + ) n x) e c d θ x,x ) L φ u + ) n x ), 3.) for all x, x with x 0 = x 0. Dividing both sides of 3.) by L u + n ) > 0we have that L φ u + n ) x) L u + n ) e c d θ x,x ) L φ u + n ) x ) L u + n ). 3.2)

6 A Nonlinear Transfer Operator Theorem 52 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., N n u)x) = L φ u+ n )x) L φ u+ n ) ) 2 e c /2)d θ x,x ) L φ u+ n ) = e c /2)d θ x,x ) N n u)x ) ) x ) 2 L φ u+ n ) i.e., N n u) c /2. Providing c is sufficiently large that c > c /2 = c + φ θ )θ/2wehave 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, there exists a non-trivial fixed point ψ n N n ψ n ) = ψ n. c such that Proof By the Arzela Ascoli Theorem the space c is compact with respect to the norm. For each n andc > 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) ) = and by construction ψ n x n) ) = N n ψ n )x n) ) =. This completes the proof. We can now use the compactness of c with respect to to deduce that ψ n ) n= has a -convergent subsequence. We denote the limit point by ψ 0 and observe that we have that L φ ψ 0 ) = λψ 2 0,whereλ = L φψ 0 ). Moreover, since we observed that by construction that ψ n =, for each n, we can deduce that ψ 0 = and thus, in particular, ψ 0 is non-zero. If we replace ψ 0 by ψ = λψ 0 then we finally get L φ ψ) = ψ 2, as required. To see that ψ>0, assume for a contradiction that there exists x 0 such that ψx 0 ) = 0. Since ψ 0 we see from the identity L φ ψ)x 0 ) = ψ 2 x 0 ) = 0, which implies that ψy) = 0 whenever σ y = x 0. Proceeding iteratively, we have that ψ vanishes on the set n=0 σ n x 0, which is dense by the mixing hypothesis on σ : corresponding to the aperiodicity assumption on A). However, this contradicts that ψ = 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 > 0andψ = ψ. We can then associate ξ := inf{t > 0:tψ ψ 0} and thus, in particular, ξψ ψ. Observe that L φ ξψ ψ ) = ξψ 2 ψ 2 0, since L φ preserves positive functions. Since ξψ 2 ψ 2 = ξψ + ψ ) ξψ ψ ) 0 we deduce that ξψ ψ 0. In particular, this implies that ξ, otherwise it contradicts the original definition of ξ. Interchanging the roles of ψ and ψ we can define ξ := inf{t > 0:tψ ψ 0} and andthus,inparticular,ξ ψ ψ 0. A similar argument to the above shows that ξ. However, since we can then write ξ ξ)ψ ξψ ψ this implies that ξ = ξ =. For the definition of ξ we can choose x 0 with ψx 0 ) = ψ x 0 ). We can then write L φ ψ ψ )x 0 ) = ψ 2 x 0 ) ψ 2 x 0 ) = ψx 0 ) + ψ x 0 ))ψx 0 ) ψ x 0 )) = 0which implies that ψy) = ψ y) whenever σ y = x 0. Proceeding inductively we deduce that ψy) = ψ y) on the dense set of y n= σ n x 0, and this ψ = ψ.

7 522 M. Pollicott Remark 3.6 This simple argument doesn t rule out the possibility of another non-positive fixed point. 3.3 Analyticity To show the analytic dependence of the solution we want to use the implicit function theorem applied to the function G : F θ F θ F θ defined by Gφ, ψ) = L φ ψ ψ 2. In order to apply the implicit function theorem at φ 0,ψ 0 ) F θ F θ with ψ 0 > 0, say, satisfying Gφ 0,ψ 0 ) = 0 we need to show that D 2 G)φ 0,ψ 0 ) : F θ F θ is invertible. An easy calculation gives that D 2 G)φ 0,ψ 0 ) = L φ0 2ψ 0 ) : Fθ F θ. 3.3) 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. Lemma 3.7 Ruelle) The operator L φ0 : F θ F θ has spectral radius λ φ0. In particular, we see from Lemma 3.7 that L φ0 2λ φ0 ) : F θ F θ is a bounded linear operator since 2λ φ0 is not in the spectrum of L φ0. We can write Lφ0 2ψ 0 ) = Lφ0 2λ φ0 ) + 2λ φ0 ψ 0 ) ) = L φ0 2λ φ0 ) n=0 2λφ0 ψ 0 )L φ0 2λ φ0 ) ) n ) which exists and is a bounded linear operator provided λ φ0 ψ 0 < 2 L φ0 2λ φ0 ). In particular, by 3.3) we see that D 2 G)φ 0,ψ 0 ) is invertible and thus the implicit function theorem applies. This allows us to deduce the analytic dependence. 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 Gφ 0, ) : ψ L φ0 ψ ψ 2 where ψ = ψ 0 + ɛψ ) + oɛ). A simple calculation gives that the tangent operator Gφ 0,ψ 0 + ɛψ ) ) Gφ 0,ψ 0 ) T φ0 ψ := lim = L φ0 ψ ) 2ψ 0 ψ ). ɛ 0 ɛ For definiteness, we can consider the specific case where we replace φ 0 by φ as in Remark.7, then we see that the spectra spt φ ) and spl φ ) are simply related by spt φ ) = spl φ ) 2. Thus, since λ φ =, by Lemma 3.7 the tangent operator T φ will have its spectra in the disk centred at 2 and of radius and thus outside the unit disk, except for the value ). This suggests that the fixed point ψ φ is locally unstable in a codimension one space under the iteration Gφ, ) n 4 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

8 A Nonlinear Transfer Operator Theorem 523 to the transfer operator L φ satisfying L φ = ) for the associated function φ cf. Remark.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.5, we can consider the linear operator M φ : F θ F θ given by M φ w)x) = ψ e φy) ψ 2 σ σ y=x ) y)wy) which then satisfies M φ = cf. Remark.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.3, and thus of its dual, that there is a unique σ - invariant probability measure μ such that M φ μ = μ, i.e., fdμ = M ψ fdμ for all f C 0 ). This leads to a non-standard version of the variational principle. Lemma 4. Variational Principle for φ and ψ) For any σ -invariant probability measure ν we have that hν) + φdν log ψdν 0 4.) with equality if and only if ν = μ. Proof Let φ 2 := φ + log ψ 2logψ σ then since M φ = L φ2 satisfies L φ2 = we see that Pφ 2 ) = 0andμ is the unique equilibrium state associated to φ 2, by Proposition 3.4 in [9]. Thus by the variational principle Theorem 3.5 in [9]) we have hν) + φ + log ψ 2log ψ) σ ) dν = hν) + φ 2 dν Pφ 2 ) = 0 = hμ) + φ 2 dμ = hμ) + φ + log ψ 2log ψ) σ ) dμ 4.2) with equality if and only if μ = ν.byσ-invariance of the measures we have that log ψ) σ dμ = log ψdμ and log ψ) σ dν = log ψdν and thus 4.) follows from 4.2). We consequently have a particularly simple expression for the entropy hμ). Corollary 4.2 We can write hμ) = log ψ)dμ φdμ. Acknowledgements This work was supported by the Leverhulme Trust RPG ) and EPSRC EP/M00903/). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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