ORTHONORMAL EXPANSIONS OF INVARIANT

DENSITIES FOR EXPANDING MAPS

Oliver Jenkinson and Mark Pollicott

Queen Mary University of London and Manchester University

July 2003

Abstract. We give a novel way of constructing the density function for the absolutely continuous invariant measure of piecewise expanding C^{ω} Markov maps. This is a classical problem, with one of the standard approaches being Ulam’s method [U] of phase space discretisation. Our method hinges instead on the expansion of the density function with respect to an L^{2} orthonormal basis, and the computation of the expansion coefficients in terms of the periodic orbits of the expanding map. The efficiency of the method, and its extension to C^{k} expanding maps, are also discussed.

1. Introduction

Let T be either a C^{ω} (i.e. real analytic) expanding map of the circle, or a piecewise C^{ω} expanding Markov map (see §2 for the definition) of the interval. It is well known that there exists a T -invariant probability measure μ which is absolutely continuous with respect to Lebesgue measure (cf. [LY]). If T is topologically mixing^{1} then this absolutely continuous invariant measure is unique and ergodic, with strictly positive C^{ω} density function ϱ.

A central problem in ergodic theory is the computation of time averages of the form lim_{n→∞} \frac{1}{n} ∑_{r=0}^{n−1} g(T^{r}x), for integrable functions g. The straightforward evaluation of \frac{1}{N} ∑_{r=0}^{N−1} g(T^{r}x) for large N is known to be an inefficient way of approximating the limit (cf. [ER]). An application of Birkhoff’s ergodic theorem reduces the problem, for Lebesgue almost every x, to a computation of the space average ∫ g dμ = ∫ g(x)ϱ(x)dx. However for general nonlinear expanding maps the density ϱ is not explicitly known (i.e. is not available in closed form), so that an approximation scheme for evaluating ∫ g dμ is required.

In [JP] we gave an efficient method for computing such integrals, provided the function g is real analytic. The periodic points of period up to M are used to construct a sequence of approximations to ∫ g dμ, whose rate of convergence to ∫ g dμ is superexponential in M. Thus we have a very rapidly convergent algorithm for computing the time average of a given (real analytic) function g.

Typeset by AMS-TEX

We are grateful to O. F. Bandtlow for a growth estimate used in §8, and to T. Driscoll and L. N.

Trefethen for advice on crowding for conformal maps.

1Mixing is automatic if T is an expanding circle map.

If, however, we want to compute time averages for many functions g, the required high number of iterations of our algorithm may outweigh its inherent efficiency. In this case it would be preferable to accurately approximate, once and for all, the density function ϱ. Thereafter, for all functions g, which now may be chosen as merely L^{1} rather than C^{ω}, the integral ∫ g dμ = ∫ g(x)ϱ(x)dx can be approximated directly, without recourse to an algorithm. That is, each integral may be evaluated approximately by either a symbolic or a numerical method, and the time required for such evaluations is comparatively very small. The classical method for approximating density functions, first proposed by Ulam [U], is to discretise the system by partitioning the phase space. The density is then approximated by the dominant eigenvector of a finite matrix. Recent work on Ulam’s method includes [DS], [Froy], [Hunt], [KMY], [Liv], [Mu].

In the present paper we will describe how to perform a one-off computation of ϱ using periodic points, and give error estimates in terms of the largest period used. The starting point is to view the analytic function ϱ as an element of a suitable L^{2} space, and expand ϱ = ∑_{n}(ϱ, f_{n})f_{n} in terms of an orthonormal basis {f_{n}}. If the interval is parametrised as I = [−1, 1], then a natural expansion is in terms of normalised Legendre polynomials f_{n} = (n + \frac{1}{2})^{1/2}P_{n}, for n ≥ 0, where P_{n} is the n^{th} Legendre polynomial (see §3 for the definition). In this case

(ϱ, f_{n}) = ∫_{−1}^{1} ϱ(x)\overline{f_{n}(x)} dx = ∫ \overline{f_{n}} dμ

is the inner product in the Hilbert space L^{2}(I, dx).

If T is a map of the circle then the density ϱ will be analytic, and hence L^{2}, on the circle. Equivalently, ϱ can be considered as a continuous function on the interval^{2} I = [−1, 1] admitting an extension as a function of period two on the real line. A natural complete orthonormal set for L^{2}(𝕋, dx) consists of Fourier characters f_{n}(x) = 2^{−1/2}e^{πnix}, and we have the Fourier series expansion

ϱ = ∑_{n=−∞}^{∞}(ϱ, f_{n})f_{n} ,

where the Fourier coefficients (ϱ, f_{n}) are given by

(ϱ, f_{n}) = 2^{−1/2} ∫_{−1}^{1} ϱ(x)e^{−πnix} dx = 2^{−1/2} ∫ e^{−πnix} dμ(x) .

For our purposes the key feature of L^{2} space is that the inner product is defined in terms of integration. In particular the expansion coefficients (ϱ, f_{n}) are given by integrating complex conjugates of basis functions with respect to the invariant measure μ. If we require that every conjugate basis element \overline{f_{n}} be real analytic, then each coefficient (ϱ, f_{n}) can be approximated by means of the periodic point algorithm of [JP]. That is, there is a sequence (ϱ, f_{n})^{(M)} of approximations to (ϱ, f_{n}), where each number (ϱ, f_{n})^{(M)} is explicitly computable in terms of the points of period at most M.

2For consistency of notation we retain the parametrisation I = [−1, 1]. It is of course more common to use I = [0, 1], in which case the Fourier basis is f_{n}(x) = e^{2πinx}. See also §4.

This suggests a natural means of approximating the density function ϱ: first choose positive integers N and M = M(N), next compute the numbers (ϱ, f_{n})^{(M)} for 0 ≤ n ≤ N, then finally combine them to define

ϱ_{N}^{(M)} := ∑_{n=0}^{N}(ϱ, f_{n})^{(M)}f_{n} .

The function ϱ_{N}^{(M)} is a closed form approximation of the true density ϱ.

This approximation scheme works exceedingly well in practice. In the first half of the article we explain in detail the mechanics of this scheme, as well as its theoretical underpinning (thermodynamic formalism and determinants of trace class operators). The approximation scheme is most successful in providing moderate estimates (double precision, say) in reasonable time. If a lower precision (eg. four correct decimal digits) is sufficient, this can be achieved essentially instantaneously, using orbits of very low period (up to period 5, say); this is illustrated by an explicit example in §4.

If much higher precision estimates on ϱ are required then the choice of N and M = M(N) becomes an issue. The choice of N depends on the decay rate of the coefficients (ϱ, f_{n}) as n → ∞. For suitable orthonormal bases (including the Fourier and Legendre bases, cf. Lemma 7.1) we do have some a priori control over this decay rate: it is O(α_{ϱ}^{n}), for some 0 < α_{ϱ} < 1, by virtue of the analyticity of ϱ. This means that we can use ϱ_{N}^{(M)} to uniformly approximate ϱ, with an error which is O(α_{ϱ}^{N}), provided M = M(N) is chosen sufficiently large. It is desirable, therefore, to have an idea of how large M should be chosen, and this motivates the following theorems.

First we estimate M(N) in the case where {f_{n}} is the normalised Legendre basis.

Theorem 1. (Normalised Legendre basis)

Suppose that T is a topologically mixing piecewise C^{ω} expanding Markov map on the interval, with absolutely continuous invariant measure μ, and corresponding density function ϱ. Let {f_{n}}_{n=0}^{∞} denote the set of normalised Legendre polynomials.

There exists C > 0 and a sequence of computable numbers (ϱ, f_{n})^{(M)}, 0 ≤ n ≤ N, depending on the points of period at most M := M(N) = exp(CN^{2}), such that the function ϱ_{N,Legendre}^{(M)} := ∑_{n=0}^{N}(ϱ, f_{n})^{(M)}f_{n} is an approximation to the density ϱ satisfying the following error bound:

There exists 0 < α_{ϱ} < 1 and E > 0 such that, for all sufficiently large N,

||ϱ − ϱ_{N,Legendre}^{(M)}||_{∞,I} ≤ Eα_{ϱ}^{N}.

Next we consider circle expanding maps. In this case the Fourier basis can be used, and an improved estimate on M(N) is possible.

Theorem 2. (Fourier basis)

Suppose that T is a C^{ω} expanding map of the circle, with absolutely continuous invariant measure μ, and corresponding density function ϱ. Let f_{n}(x) = 2^{−1/2}e^{πnix} for n ∈ ℤ. There exists C > 0 and a sequence of computable numbers (ϱ, f_{n})^{(M)}, −N ≤ n ≤ N, depending on the points of period at most M := M(N) = exp(CN), such that the function ϱ_{N,F ourier}^{(M)} := ∑_{n=−N}^{N}(ϱ, f_{n})^{(M)}f_{n} is an approximation to the density ϱ satisfying the following error bound:

There exists 0 < α_{ϱ} < 1 and E > 0 such that, for all sufficiently large N,

||ϱ − ϱ_{N,F ourier}^{(M)}||_{∞,𝕋} ≤ Eα_{ϱ}^{N}.

The proof of these results occupies most of the second half of the article, from §5 onwards. The two key factors influencing the efficiency of the algorithm are the amount of expansion in the dynamics and the growth properties of the holomorphically extended basis functions.

In §5 we present refined estimates on the decay rates of Taylor coefficients for certain dynamical determinants, improving on previous estimates in [Ru1], [Fried], [JP]. This decay rate is related to the expansiveness of T: the more expanding T is, the faster the decay rate, and, ultimately, the more efficient the algorithm. In §6 these estimates are applied to particular determinants Δ_{n} associated to the basis functions f_{n}, yielding concrete bounds on the differences |(ϱ, f_{n})^{(M)} − (ϱ, f_{n})|. A sequence of conformal coordinate changes is introduced so as to control the growth of the f_{n} in the complex plane (the need for this is illustrated by Proposition 6.1). In §7 this leads to effective bounds on the norms of associated transfer operators. In §8 these various ingredients are combined to prove the above theorems.

§9 consists of some concluding remarks about the application of the approximation scheme in the case where T is smooth but not real analytic; the algorithm can be followed verbatim, though its efficiency is somewhat reduced.

Remarks.

(1) The Legendre polynomial expansion of ϱ suggests the possibility of computing its expansion with respect to other systems of polynomials (cf. [Sz], [BB]). In general such expansions need not even exist, however. Even when they do, the corresponding expansion coefficients need not be computable by integration along I, in which case their calculation is not amenable to our method. Such difficulties are present even for the case of the Taylor expansion of ϱ.

(2) Other authors have used estimates for the density to derive estimates on the metric entropy h(μ) via the Rohlin-Pesin equality [Pe]. However, we recall that a more efficient method is given in [JP], working directly with the periodic point data.

2. Integrals of individual functions

Suppose the interval is partitioned as I = I_{1} ∪ … ∪ I_{d}, d ≥ 2, where I_{i} = [x_{i−1}, x_{i}], and −1 = x_{0} < x_{1} < … < x_{d} = 1. Given T : I → I, we shall always assume that T |_{I_{i}} is real analytic, for each 1 ≤ i ≤ d. We say T is piecewise expanding if there exists Λ > 1 such that |T′| |_{I_{i}} ≥ Λ for all 1 ≤ i ≤ d. We say that T is Markov if there exists a d × d matrix A (the transition matrix for T) with each entry either 0 or 1, such that T (I_{i}) = ∪_{j:A(i,j)=1}I_{j} for each 1 ≤ i ≤ d. The collection {I_{i}}_{i=1}^{d} is called the Markov partition for T. T is topologically mixing if some power of the transition matrix A is a strictly positive matrix.

It is well known (see [LY]) that any topologically mixing piecewise C^{ω} expanding Markov map admits a unique ergodic absolutely continuous invariant probability measure. Our results are valid for all such maps, though to simplify the exposition we shall always assume that T is a full branch expanding map. In other words, each T |_{I_{i}} is assumed to be a surjection onto I, or, equivalently, every entry of the corresponding transition matrix A is a 1.

Let μ denote the unique ergodic absolutely continuous T -invariant probability measure, and let ϱ ∈ C^{ω} (cf. [JP, Lemma 1]) denote the associated density i.e., dμ(x) = ϱ(x)dx.

Any C^{ω} expanding (i.e. |T′| > 1) map of the circle is a covering map (see [Sh]). If d ≥ 2 is the absolute value of its degree, then there is a natural identification with a full branch piecewise expanding interval map whose Markov partition has cardinality d. In particular, both maps have the same density function ϱ. If ϱ is regarded as a map defined on the interval, then ϱ(−1) = ϱ(1). Later (see §3 in particular) we shall exploit this by expanding ϱ with respect to an orthonormal basis for L^{2}(𝕋, dx) (rather than for the larger space L^{2}(I, dx)). Usually, however, we shall always assume that T is a piecewise C^{ω} full branch expanding map on the interval.

We recall that in [JP] a sequence of invariant signed measures ν_{M} was defined by

ν_{M} = ∑_{\begin{matrix}(n_{1},... ,n_{m}) \\ n_{1}+...+n_{m}≤M \end{matrix}} \frac{(−1)^{m}}{m!} ∑_{i=1}^{m} ∑_{x∈Fix(n_{i})} (∏_{{j=1 \atop j≠i}}^{m} ∑_{y∈Fix(n_{j})} r(y, n_{j})) \frac{δ_{x}}{|(T^{n_{i}})′(x) − 1|}

where the first summation is over ordered m-tuples of positive integers whose sum is not greater than M, where Fix(n) denotes the set of fixed points of T^{n}, and where

r(x, n) = \frac{1}{n|(T^{n})′(x) − 1|}.

The measure ν_{M} is supported on those periodic points of period at most M. Introducing the normalisation constant

I_{M} = ∑_{\begin{matrix}(n_{1},... ,n_{m}) \\ n_{1}+...+n_{m}≤M \end{matrix}} \frac{(−1)^{m}}{m!} ∑_{i=1}^{m} ∑_{x∈Fix(n_{i})} (∏_{{j=1 \atop j≠i}}^{m} ∑_{y∈Fix(n_{j})} r(y, n_{j})) \frac{1}{|(T^{n_{i}})′(x) − 1|} ,

we then define the invariant signed probability measures μ_{M} = I_{M}^{−1}ν_{M}, and prove [JP]:

Proposition 2.1. Suppose that T : I → I is a topologically mixing piecewise C^{ω} expanding Markov map of the interval, with absolutely continuous invariant probability measure μ. There is a sequence of T -invariant signed probability measures μ_{M}, supported on the points of period at most M, such that for every C^{ω} function g : I → ℝ, there exists 0 < θ < 1 and C > 0 with

|∫ g dμ_{M} − ∫ g dμ| ≤ Cθ^{M(M+1)/2}.

An alternative way to view the formulae defining μ_{M}, ν_{M} is in terms of the Taylor coefficients of certain entire functions Δ : ℂ → ℂ, Δ_{g} : ℂ^{2} → ℂ, defined by

Δ(z) = exp − ∑_{n=1}^{∞} \frac{z^{n}}{n} ∑_{x∈Fix(n)} \frac{1}{|(T^{n})′(x) − 1|} ,  (2.1)

Δ_{g}(z, t) = exp − ∑_{n=1}^{∞} \frac{z^{n}}{n} ∑_{x∈Fix(n)} \frac{exp (tg^{n}(x))}{|(T^{n})′(x) − 1|}  (2.2)

for sufficiently small values of z, and then by analytic continuation. Here g^{n}(x) denotes the sum ∑_{r=0}^{n−1} g(T^{r}x). If

Δ(z) = 1 + ∑_{m=1}^{∞} c_{m}z^{m} , Δ_{g}(z, t) = 1 + ∑_{m=1}^{∞} c_{m,g}(t)z^{m}

are the corresponding Taylor expansions, then (see [JP], or Proposition 2.2 below) we have

∫ g dν_{M} = ∑_{m=1}^{M} c′_{m,g}(0) , I_{M} = ∑_{m=1}^{M} mc_{m} ,

so that

∫ g dμ_{M} = \frac{∑_{m=1}^{M} c′_{m,g}(0)}{∑_{m=1}^{M} mc_{m}} .

The functions Δ, Δ_{g} are both determinants of trace class transfer operators acting on a suitable Hilbert space of analytic functions (see [Sim] for the general theory of determinants associated to certain Hilbert space operators). For any C^{ω} function f : I → ℝ, the corresponding transfer operator L_{f} is defined (cf. [Ru2]) by

L_{f}v(x) = ∑_{T y=x} v(y)e^{f(y)}.

As is well known, this operator preserves various function spaces. In particular we will consider such operators as acting on the Hilbert Bergman space A_{2}(U) corresponding to appropriate complex domains U. Recall (see for example [HKZ]) that A_{2}(U) is the space of functions analytic on U and such that ∫_{U} |f|^{2} dA < ∞, where A is two-dimensional Lebesgue measure on U. When equipped with inner product^{3} ⟨f, g⟩ = ∫_{U} fḡ dA this is a Hilbert space. The reason we work with this particular Hilbert space is its relation to L^{2}(I): if U_{0} ⊃ U_{1} ⊃ … is a sequence of complex domains, shrinking to I, and g ∈ A_{2}(U_{n}) for all sufficiently large n, then A(U_{n})^{−1}||g||_{A_{2}(U_{n})} → ||g||_{L^{2}(I)}. We will occasionally also work with the space A_{∞}(U) of those functions which are analytic on U and bounded on its closure \overline{U}, equipped with the supremum norm (which makes it a Banach, but not a Hilbert, space).

3We use ⟨·, ·⟩ and (·, ·) to denote the inner product in A_{2}(U) and L^{2}(I) respectively

The action of transfer operators on Hilbert Bergman spaces has hitherto not been considered in the literature, so we prove (Lemma 5.1) that L_{f} : A_{2}(U) → A_{2}(U) is a trace class operator, for suitable domains U. This ensures that L_{f} has a well-defined determinant det(I − zL_{f}) = exp [tr (log(I − zL_{f}))], which extends to an entire function of z (cf. [Sim, Thm. 3.3]). Now tr (log(I − zL_{f})) = − ∑_{n=1}^{∞} \frac{z^{n}}{n} tr(L_{f}^{n}) for small z, and each L_{f}^{n} is a sum of weighted composition operators, so its trace may be expressed (cf. [May1], [May2]) as tr(L_{f}^{n}) = ∑_{x∈Fix(n)} \frac{e^{f^{n}(x)}}{1−[(T^{n})′(x)]^{−1}} , yielding the formula

det(I − zL_{f}) = exp − ∑_{n=1}^{∞} \frac{z^{n}}{n} ∑_{x∈Fix(n)} \frac{exp (f^{n}(x))}{1 − [(T^{n})′(x)]^{−1}} .  (2.3)

It is then clear that Δ(z) = det(I − zL_{− log |T′|}) and Δ_{g}(z, t) = det(I − zL_{tg−log |T′|}). The transfer operator corresponding to the first of these choices, f = − log |T′|, is the

Perron-Frobenius operator for the map T, and will be denoted by L.

On the other hand these are spectral determinants: we have (see [Sim, Thm. 3.7]) the equality det(I − zL_{f}) = ∏_{i=1}^{∞}(1 − zλ_{i}(L_{f})), where the infinite product is absolutely convergent and {λ_{i}(L_{f})}_{i=1}^{∞} is an enumeration of the non-zero eigenvalues of L_{f}, each repeated according to its algebraic multiplicity. (L_{f} is compact since it is trace class, so its spectrum is countable, with zero the only point of accumulation, and all non-zero eigenvalues of finite algebraic multiplicity).

The leading (i.e. largest in modulus) eigenvalue of any transfer operator L_{f} is well known to be simple, and equal to e^{P (f)}, where P (f) = P (f, T) = lim_{n→∞} \frac{1}{n} log ∑_{x∈Fix(n)} e^{f^{n}(x)} is the pressure of f (cf. [Ru2]). The leading eigenvalue of the Perron-Frobenius operator L equals 1, since it is easily checked that P (− log |T′|) = 0. The corresponding eigenfunction is precisely the invariant density ϱ (cf. [CE]).

Our motivation for introducing the functions Δ_{g}(z, t) is the following result (see [JP]), whose proof we provide for completeness.

Proposition 2.2. For any real analytic function g : I → ℝ,

∫ g dμ = \frac{∂_{t}Δ_{g}(1, 0)}{∂_{z}Δ_{g}(1, 0)} = \frac{∂_{t}Δ_{g}(1, 0)}{Δ′(1)} .

Truncating the Taylor expansion for both numerator and denominator of the righthand side yields the sequence

∫ g dμ_{M} = \frac{∫ g dν_{M}}{I_{M}} = \frac{∑_{m=1}^{M} c′_{m,g}(0)}{∑_{m=1}^{M} mc_{m}}

of approximations to ∫ g dμ. The rate of convergence of these approximations is superexponential in M, as described in Proposition 2.1.

Proof. The maximal eigenvalue of the operator L_{tg−log |T′|} is exp P (tg − log |T′|). Its reciprocal z(t) := exp [−P (tg − log |T′|)] is then a zero of Δ_{g}(z, t), so that Δ_{g}(z(t), t) = 0 for all t. Moreover, we have z(0) = exp [−P (− log |T′|)] = 1, since 1 is the maximal eigenvalue of the Perron-Frobenius operator L.

The derivative formula for the pressure functional (see [Ru2, p.133], [PP, p.60]) gives

∫ g dμ = \frac{d}{dt}P (tg − log |T′|)|_{t=0} .

Since z(0) = 1, this gives

∫ g dμ = −\frac{d}{dt} log z(t)|_{t=0} = −\frac{z′(0)}{z(0)} = −z′(0).  (2.4)

The implicit function theorem then gives

z′(0) = −\frac{∂_{t}Δ_{g}(z(0), 0)}{∂_{z}Δ_{g}(z(0), 0)} = −\frac{∂_{t}Δ_{g}(1, 0)}{∂_{z}Δ_{g}(1, 0)} = −\frac{∂_{t}Δ_{g}(1, 0)}{Δ′(1)},  (2.5)

the last equality following fom the observation that Δ_{g}(z, 0) = Δ(z) and hence ∂_{z}Δ_{g}(z, 0) = Δ′(z). The result follows. □

3. Series expansion of the density ϱ

We will regard the density function ϱ as an L^{2} function on either the interval or the circle, as appropriate, and consider its expansion with respect to a suitable orthonormal basis (i.e. complete orthonormal set). A minimum requirement will be that the conjugate (see below for the definition) of every basis function is real analytic. Since we wish to perform computations involving these functions, they will in practice be chosen to be as elementary as possible (eg. polynomials or trigonometric functions), and in particular will be entire.

A natural basis for L^{2}([−1, 1], dx), consists of normalised Legendre polynomials. These are commonly used in mathematical physics, and are generated by applying the Gram-Schmidt orthogonalisation process to the monomials {x^{n}}_{n=0}^{∞}. The n^{th} Legendre polyonomial P_{n} is defined to be

P_{n}(x) = \frac{(2n − 1)(2n − 3) ⋯ 1}{n!} \{x^{n} − \frac{n(n − 1)}{2(2n − 1)}x^{n−2} + \frac{n(n − 1)(n − 2)(n − 3)}{2.4(2n − 1)(2n − 3)}x^{n−4} − … \} ,

so that P_{0}(x) = 1, P_{1}(x) = x, P_{2}(x) = (3x^{2} − 1)/2, P_{3}(x) = (5x^{3} − 3x)/2, P_{4}(x) = (35x^{4} − 30x^{2} + 3)/8, etc. An orthonormal basis is obtained by setting f_{n} = (n + \frac{1}{2})^{1/2}P_{n} (cf. [WW], 15.14). We then have the expansion ϱ = ∑_{n=0}^{∞}(ϱ, f_{n})f_{n}, where (ϱ, f_{n}) = ∫_{−1}^{1} ϱ(x)\overline{f_{n}(x)} dx = ∫_{−1}^{1} f_{n}(x)ϱ(x) dx = ∫ f_{n} dμ.

In the case where T is an expanding map of the circle, its density function belongs to L^{2}(𝕋, dx), which we can regard as a subspace of L^{2}(I, dx). The natural orthonormal basis for L^{2}(𝕋, dx) is given by f_{n}(x) = 2^{−1/2}e^{πinx}, n ∈ ℤ. In this case we can expand

ϱ = ∑_{n=−∞}^{∞}(ϱ, f_{n})f_{n} , where (ϱ, f_{n}) = \overline{(f_{n}, ϱ)} = 2^{−1/2} ∫_{−1}^{1} ϱ(x)e^{−πnix} dx = ∫ \overline{f_{n}} dμ .

More generally, let us suppose that {f_{n}}_{n=0}^{∞} is an orthonormal basis for either of the Hilbert spaces L^{2}(I, dx) or L^{2}(𝕋, dx). Since any function in L^{2}(𝕋, dx) can be considered as a function in L^{2}(I, dx), we shall consider each basis element as a function f_{n} : I → ℂ. By \overline{f_{n}} : I → ℂ we denote the complex conjugate function given by \overline{f_{n}}(x) = \overline{f_{n}(x)}. We shall always require that for each basis function f_{n}, its conjugate \overline{f_{n}} is real analytic (i.e. has a holomorphic extension to some neighbourhood of I in ℂ). For example if f_{n}(x) = 2^{−1/2}e^{πnix} then \overline{f_{n}}(x) = 2^{−1/2}e^{−πnix}, which extends to an entire function 2^{−1/2}e^{−πniz} of the complex variable z. Note that this differs from first extending f_{n} to all of ℂ and then conjugating, as this yields the antiholomorphic function 2^{−1/2}e^{−πniz̄}. Of course in the case where each f_{n} is real-valued on I (as for the Legendre polynomials) then our requirement is simply that each f_{n} itself be real analytic.

The coefficients (ϱ, f_{n}) in the expansion ϱ = ∑_{n=0}^{∞}(ϱ, f_{n})f_{n} are given by (ϱ, f_{n}) = ∫_{−1}^{1} ϱ(x)\overline{f_{n}}(x) dx = ∫ \overline{f_{n}} dμ. Therefore any particular (ϱ, f_{n}) may be accurately computed by setting g = \overline{f_{n}} and applying the periodic point algorithm described in §2.

The first step in such a computation is to let L_{t,n} denote the transfer operators L_{t\overline{f_{n}}−log |T′|} and define the corresponding sequence of determinants

Δ_{n}(z, t) = det(I − zL_{t,n}) = exp − ∑_{k=1}^{∞} \frac{z^{k}}{k} ∑_{x∈Fix(k)} \frac{exp (t\overline{f_{n}}^{k}(x))}{|(T^{k})′(x) − 1|} .  (3.1)

Of course each Δ_{n} enjoys the same properties as the more general determinants Δ_{g}, and in particular is an entire function of z. Let us write the Taylor expansion around zero as

Δ_{n}(z, t) = 1 + ∑_{m=1}^{∞} _{n}c_{m}(t)z^{m} .

The following is a reformulation of Proposition 2.2, giving an exact formula for the n^{th} expansion coefficient (ϱ, f_{n}), as well as a computational scheme for its approximation.

Proposition 3.1. Let T be a topologically mixing C^{ω} expanding Markov map on either the interval I or the circle 𝕋, and let ϱ be the density function for the unique absolutely continuous T -invariant probability measure μ. Let {f_{n}} be an orthonormal basis for either L^{2}(I, dx) or L^{2}(𝕋, dx) as appropriate, where the conjugate \overline{f_{n}} of each basis element is real analytic.

Let ϱ = ∑_{n=0}^{∞}(ϱ, f_{n})f_{n} be the expansion of the density function with respect to this basis. Then for each n,

(ϱ, f_{n}) = \frac{∂_{t}Δ_{n}(1, 0)}{∂_{z}Δ_{n}(1, 0)} = \frac{∂_{t}Δ_{n}(1, 0)}{Δ′(1)},  (3.2)

where Δ and Δ_{n} are the functions defined by (2.1) and (3.1) respectively.

Truncating the Taylor expansion for both numerator and denominator of the righthand side yields the sequence

(ϱ, f_{n})^{(M)} := ∫ \overline{f_{n}} dμ_{M} = \frac{∫ \overline{f_{n}} dν_{M}}{I_{M}} = \frac{∑_{m=1}^{M} _{n}c′_{m}(0)}{∑_{m=1}^{M} mc_{m}}  (3.3)

of approximations to (ϱ, f_{n}) = ∫ \overline{f_{n}} dμ, whose rate of convergence is superexponential in M.

In practice we can of course only compute finitely many of the expansion coefficients (ϱ, f_{n}). It is therefore advantageous if they decay to zero rapidly as n → ∞. For suitable bases {f_{n}} the real analyticity of ϱ is enough to ensure that this decay is exponential. Both the Legendre and Fourier bases are suitable in this sense (see Lemma 7.1), though for general bases this will be an additional hypothesis. The following result, which will be useful later, follows immediately from Proposition 3.1.

Lemma 3.2. Let T be a topologically mixing C^{ω} expanding Markov map on either the interval I or the circle 𝕋, and let ϱ be the density function for the unique absolutely continuous T -invariant probability measure μ. Let {f_{n}} be an orthonormal basis for either L^{2}(I, dx) or L^{2}(𝕋, dx) as appropriate, where the conjugate \overline{f_{n}} of each basis element is real analytic. Suppose in addition that there exist 0 < α_{ϱ} < 1 and K_{ϱ} > 0 such that |(ϱ, f_{n})| ≤ K_{ϱ}α_{ϱ}^{n} for all n ≥ 0.

Then ∂_{t}Δ_{n}(1, 0) = O(α_{ϱ}^{n}) as n → ∞.

4. An example

We have described the way in which each expansion coefficient (ϱ, f_{n}) of the density function ϱ may be approximated by a sequence (ϱ, f_{n})^{(M)} of computable numbers. We now illustrate, by means of an explicit example, the way in which this leads to a uniform approximation of ϱ.

Our expanding map will be T (x) = 2x + \frac{1}{4π} sin 2πx (mod 1). The density function ϱ is not known explicitly, though it has been approximated in [Mur] (see also [JP] for related calculations). As T is an expanding map of the circle^{4}, it is natural to expand ϱ with respect to the Fourier basis f_{n}(x) = e^{2πinx}, n ∈ ℤ. We clearly have (ϱ, f_{0}) = ∫ ϱ(x) dx = 1, since ϱ is a density function.

Now |T′(x)| = 2 + \frac{1}{2} cos 2πx is symmetric about the point x = 1/2. Therefore the Perron-Frobenius operator preserves the space of functions which are symmetric about 1/2, and its fixed point ϱ will belong to this space. Consequently its Fourier coefficients are symmetric: (ϱ, f_{n}) = (ϱ, f_{−n}) for all n ∈ ℤ. In other words, ϱ can be expanded in terms of {cos 2πnx}_{n=0}^{∞}.

Suppose we wish to approximate ϱ by computing its Fourier coefficients (ϱ, f_{n}) for −2 ≤ n ≤ 2. By the above discussion it suffices to approximate the coefficients (ϱ, f_{1}) and (ϱ, f_{2}). These approximations will be derived from the periodic orbit data of T. Clearly T has a single fixed point, at zero. The other periodic orbits can be located numerically. For example there is a single period-2 orbit, {0.308584 , 0.691415}, and two period-3 orbits, {0.098318 , 0.242727 , 0.564949} and {0.435050 , 0.901681 , 0.757272} .

4with circumference 1, in contrast to the parametrisation used in other sections

Define

a_{k} = \frac{1}{k} ∑_{x∈Fix(k)} \frac{1}{|(T^{k})′(x) − 1|} ,  (4.1)

so that Δ(z) = exp − ∑_{k=1}^{∞} a_{k}z^{k}. We previously wrote Δ(z) = 1 + ∑_{m=1}^{∞} c_{m}z^{m}, so comparing coefficients in the two expressions yields

\begin{matrix}c_{1} = − & a_{1} , c_{2} = −a_{2} + \frac{1}{2}a_{1}^{2} , c_{3} = −a_{3} + a_{1}a_{2} − \frac{1}{6}a_{1}^{3} \\ c_{4} = −a_{4} − \frac{1}{2}a_{1}^{2}a_{2} + a_{1}a_{3} + \frac{1}{2}a_{2}^{2} + \frac{1}{24}a_{1}^{4} , etc. \end{matrix}  (4.2)

Define

nb_{k} = \frac{1}{k} ∑_{x∈Fix(k)} \frac{\overline{f}_{n}^{k}(x)}{|(T^{k})′(x) − 1|} ,  (4.3)

and observe that

∂_{t}Δ_{n}(z, 0) = (− ∑_{k=1}^{∞} \frac{z^{k}}{k} ∑_{x∈Fix(k)} \frac{\overline{f}_{n}^{k}(x)}{|(T^{k})′(x) − 1|}) exp (− ∑_{k=1}^{∞} \frac{z^{k}}{k} ∑_{x∈Fix(k)} \frac{1}{|(T^{k})′(x) − 1|})

= (− ∑_{k=1}^{∞} _{n}b_{k}z^{k}) exp − ∑_{k=1}^{∞} a_{k}z^{k}

= (− ∑_{k=1}^{∞} _{n}b_{k}z^{k}) (1 + ∑_{m=1}^{∞} c_{m}z^{m}) .

However ∂_{t}Δ_{n}(z, 0) = ∑_{m=1}^{∞} _{n}c′_{m}(0)z^{m} as well, so comparing coefficients we derive the useful relation

nc′_{m}(0) = − ∑_{i=1}^{m} _{n}b_{i} c_{m−i} .  (4.4)

We are now ready to approximately compute the Fourier coefficients (ϱ, f_{1}) and (ϱ, f_{2}). Suppose we choose to use the four periodic orbits of period ≤ 3 to make these approximations. From (4.1), (4.2) we obtain the values c_{1}, c_{2}, c_{3}, then from (4.3), (4.4) we obtain 1c′_{1}(0), _{1}c′_{2}(0), _{1}c′_{3}(0) and _{2}c′_{1}(0), _{2}c′_{2}(0), _{2}c′_{3}(0). By Proposition 3.1 we therefore have

(ϱ, f_{1}) ≈ (ϱ, f_{1})^{(3)} = \frac{∑_{m=1}^{3} _{1}c′_{m}(0)}{∑_{m=1}^{3} mc_{m}} = \frac{−0.02922276}{−1.36108207} = 0.021470

and

(ϱ, f_{2}) ≈ (ϱ, f_{2})^{(3)} = \frac{∑_{m=1}^{3} _{2}c′_{m}(0)}{∑_{m=1}^{3} mc_{m}} = \frac{0.05752538}{−1.36108207} = −0.042264 .

Our approximate density function is therefore

ϱ_{2}^{(3)}(x) = 1 + 0.04294 cos 2πx − 0.084528 cos 4πx .

A better approximation to ϱ is obtained if we use the six periodic orbits of period ≤ 4. Here we have

(ϱ, f_{1}) ≈ (ϱ, f_{1})^{(4)} = \frac{∑_{m=1}^{4} _{1}c′_{m}(0)}{∑_{m=1}^{4} mc_{m}} = \frac{−0.03574251}{−1.36128967} = 0.026256

and

(ϱ, f_{2}) ≈ (ϱ, f_{2})^{(4)} = \frac{∑_{m=1}^{4} _{2}c′_{m}(0)}{∑_{m=1}^{4} mc_{m}} = \frac{−0.00231526}{−1.36128967} = 0.001700 ,

so the corresponding approximate density is

ϱ_{2}^{(4)}(x) = 1 + 0.052512 cos 2πx + 0.0034 cos 4πx .

Figure 1. Approximate densities ϱ_{2}^{(3)} (dashed) and ϱ_{2}^{(4)}

The two approximate densities ϱ_{2}^{(3)}, ϱ_{2}^{(4)} are graphed in Figure 1. In fact ϱ_{2}^{(4)} is already a good approximation; its graph is indistinguishable, in the resolution of Figure 1, from that of the next approximation

ϱ_{2}^{(5)}(x) = 1 + 0.052458 cos 2πx + 0.004252 cos 4πx .

The above computations require a few minutes with a pocket calculator, or a fraction of a second with a modern computer.

The fact that each ϱ_{n}^{(M)} is in closed form means we can then rapidly evaluate ∫ g dμ = ∫ g(x)ϱ(x) dx for other integrable functions g. Of particular interest is the choice g(x) = log |T′(x)| = log(2 + \frac{1}{2} cos 2πx), since h(μ) = ∫ log |T′| dμ is the metric entropy of the absolutely continuous invariant measure μ, by the Rohlin-Pesin identity [Pe]. Using the approximate density ϱ_{2}^{(5)} we find that h(μ) ≈ ∫ g(x)ϱ_{2}^{(5)}(x) dx = 0.683778 , of which the first five digits are correct (cf. Table 3 in [JP], where a more direct means of approximating h(μ) is employed).

5. Contraction ratios and trace class estimates

We now begin an investigation into the efficiency of our approximation scheme. The first step is to consider transfer operators L_{f} with arbitrary C^{ω} weight functions f. More precisely, we shall regard such L_{f} as acting on suitable Hilbert Bergman spaces, and prove results about the associated determinant det(I − zL_{f}). In §6 these results will be interpreted for weight functions whose complex conjugates belong to our orthonormal basis for L^{2}(I, dx).

Our piecewise C^{ω} map T : I → I is assumed to be a full branch piecewise expanding map. In particular, each T |_{I_{i}} : I_{i} → I is a C^{ω} diffeomorphism, so there is a C^{ω} inverse branch T_{i} : I → I_{i} such that T_{i} ∘ T is the identity on I, and moreover ||T′_{i}||_{∞,I} < 1 for each 1 ≤ i ≤ d.

Definition. (Admissible domain)

The real analyticity hypothesis means we can choose a simply connected complex domain U ⊃ I such that each inverse branch T_{i} has a holomorphic extension (also denoted T_{i}) to U, such that ||T′_{i}||_{∞,U} < 1 for each i = 1, …, d. Such a domain U will be called admissible. If U is an admissible domain then ∪_{i=1}^{d}T_{i}(U) is compactly contained in U.

If f : I → ℂ is real analytic, and such that exp(f) has a holomorphic extension to some admissible domain U for T, then we say that U is an admissible domain for the pair (T, f).

We note that any sufficiently small complex neighbourhood of I is admissible in both of the above senses.

From Proposition 2.2 we know that the rate at which the approximations ∫ g dμ_{M} approach ∫ g dμ depends upon the speed with which the Taylor coefficients c_{m} and c′_{m,g}(0) decay to zero. Roughly speaking, the decay rate of these coefficients can be estimated by comparing the size of an admissible domain U to its image ∪_{i=1}^{d}T_{i}(U) under the inverse branches of T. This is most clear cut in the case where U can be chosen as a round (i.e. Euclidean) disc containing I. Let Ũ denote the smallest disc, concentric with U, which contains ∪_{i=1}^{d}T_{i}(U). If θ < 1 denotes the ratio of the radii of Ũ and U, then this turns out to be the same θ as in Proposition 2.1.

In general, however, it is not possible to choose U to be a round disc: there may be singularities (of either e^{f} or of an inverse branch T_{i}) close to I in the complex plane. Nevertheless, for technical reasons (see the proof of Lemma 8 in [JP], or of Lemma 5.1 below) it is important to work with round discs. In [JP] this problem was circumvented by working with a sufficiently refined Markov partition, then choosing a small complex disc around each partition piece. In this article we shall use a different approach, working with general (simply connected) domains U but then mapping them biholomorphically onto the open unit disc.

Definition (Contraction ratio). We say that τ ∈ (0, 1) is a contraction ratio associated to the admissible domain U for the map T if

∪_{i=1}^{d}T_{i}(U) ⊂ τU ,

where τU = {τu : u ∈ U}.

Now suppose U_{0} ⊃ U_{1} ⊃ … is a sequence of admissible domains, with vertical height decreasing to zero, and horizontal semi-width strictly decreasing to some W > 1. We say that τ ∈ (0, 1) is a contraction ratio for this sequence of domains if it is a contraction ratio for each U_{n}. The fact that each T_{i} is a strict contraction ensures that each union ∪_{i=1}^{d}T_{i}(U_{n}) is compactly contained in U_{n}. Therefore every such sequence of domains has a contraction ratio τ which is strictly smaller than 1.

Definition (Disc contraction ratio). We say that θ ∈ (0, 1) is a disc contraction ratio associated to the admissible domain U for the map T if there is a holomorphic diffeomorphism φ : 𝔻 → U sending the open unit disc onto U such that φ^{−1}(∪_{i=1}^{d}T_{i}(U)) is contained in the open disc 𝔻_{θ} of radius θ centred at 0.

The Riemann mapping φ is only unique up to pre-composition with a disc automorphism. The smallest possible disc contraction ratio for the domain U is obtained by choosing θ = θ_{φ} as small as possible for each Riemann mapping φ, and then minimising over all φ. The smallest possible disc contraction ratio for the map T is then obtained by minimising this quantity over all admissible domains U. Such optimality in the choice of disc contraction ratio can be useful in concrete estimates for particular maps T, though in the sequel we shall not aim for such sharpness.

Given a compact operator L : H → H on a Hilbert space H, its i^{th} approximation number (or singular value) s_{i}(L) is defined as

s_{i}(L) = inf{||L − K|| : rank(K) ≤ i − 1}.  (5.1)

For p > 0, define the trace ideal S_{p}(H) to consist of those compact operators L : H → H for which ∑_{i} s_{i}(L)^{p} < ∞. S_{1}(H) is known as the trace class ideal, and operators in S_{1}(H) are called trace class. Any trace class operator L has a determinant, defined for small z by exp [tr (log(I − zL))], and admitting an analytic continuation as an entire function of z. Its zeros are the reciprocals of the non-zero eigenvalues of L, the order of any zero being the algebraic multiplicity of the corresponding eigenvalue.

The next lemma gives useful bounds on approximation numbers for transfer operators acting on Hilbert Bergman spaces, and in particular implies that such operators belong to every trace ideal S_{p}(A_{2}), p > 0.

For economy of notation we shall write || · ||_{θ,1} to denote || · ||_{A_{2}(𝔻_{θ}),A_{2}(𝔻)}, the operator norm of operators A_{2}(𝔻_{θ}) → A_{2}(𝔻), and || · ||_{θ,θ} to denote || · ||_{A_{2}(𝔻_{θ}),A_{2}(𝔻_{θ})}, the operator norm of operators A_{2}(𝔻_{θ}) → A_{2}(𝔻_{θ}), We also use (·, ·)_{A_{2}(𝔻)} to denote the inner product in A_{2}(𝔻).

Lemma 5.1. Let T : I → I be a topologically mixing piecewise C^{ω} expanding Markov map on the interval I. Let U ⊂ ℂ be an admissible domain for the pair (T, f).

The transfer operator L_{f} preserves the space A_{2}(U), and L_{f} : A_{2}(U) → A_{2}(U) is a trace class operator. In fact if 0 < θ < 1 is a disc contraction ratio for (T, U), then the j^{th} singular value s_{j}(L_{f}) satisfies

s_{j}(L_{f}) ≤ \frac{||L_{f}||_{θ,1}}{1 − θ} θ^{j},

for all j ∈ ℕ.

Proof. We may suppose that conjugation by the Riemann mapping φ : 𝔻 → U has already been performed. That is, we suppose that S = φ^{−1} ∘ T ∘ φ is a full branch expanding map on φ^{−1}(I), with local inverse branches S_{1}, …, S_{d}, such that ∪_{i=1}^{d}S_{i}(𝔻) ⊂ 𝔻_{θ}, the disc of radius θ centred at 0. We therefore wish to estimate the singular values of L_{f} : A_{2}(𝔻_{θ}) → A_{2}(𝔻_{θ}).

Suppose v ∈ A_{2}(𝔻_{θ}). Since v is holomorphic on 𝔻_{θ} then it is also holomorphic on ∪_{i=1}^{d}S_{i}(𝔻). It follows that L_{f}v is holomorphic on an open neighbourhood of the unit disc 𝔻, and in particular belongs to A_{2}(𝔻). We may write

L_{f}v = ∑_{k=0}^{∞} l_{k}(v)p_{k},

where p_{k}(z) = z^{k}, and l_{k}(v) = \frac{1}{k!}(L_{f}v)^{(k)}(0).

An easy calculation gives

||p_{k}||_{A_{2}(𝔻_{θ})} = \sqrt{\frac{π}{k + 1}}θ^{k+1},  (5.2)

and similarly

||p_{k}||_{A_{2}(𝔻)} = \sqrt{\frac{π}{k + 1}} .  (5.3)

We now seek to estimate |l_{k}(v)|. The monomials {p_{j}} are an orthogonal family in A_{2}(𝔻), so we have ⟨L_{f}v, p_{k}⟩_{A_{2}(𝔻)} = l_{k}(v)||p_{k}||_{A_{2}(𝔻)}^{2}. But |⟨L_{f}v, p_{k}⟩_{A_{2}(𝔻)}| ≤ ||L_{f}v||_{A_{2}(𝔻)}||p_{k}||_{A_{2}(𝔻)}, by the Cauchy-Schwarz inequality, so from (5.3) we have

|l_{k}(v)| ≤ ||L_{f}v||_{A_{2}(𝔻)} ||p_{k}||_{A_{2}(𝔻)}^{−1} = ||L_{f}v||_{A_{2}(𝔻)}\sqrt{\frac{k + 1}{π}}.  (5.4)

Let us define the rank-j operator L_{f}^{(j)} by L_{f}^{(j)} = ∑_{k=0}^{j−1} l_{k}(v)p_{k}. For any v ∈ A_{2}(𝔻_{θ}) we can use (5.2) and (5.4) to estimate

|| (L_{f} − L_{f}^{(j)}) (v)||_{A_{2}(𝔻_{θ})} ≤ ∑_{k=j}^{∞} |l_{k}(v)| ||p_{k}||_{A_{2}(𝔻_{θ})}

≤ ||L_{f}v||_{A_{2}(𝔻)} ∑_{k=j}^{∞} θ^{k+1}

≤ ||L_{f}||_{θ,1}||v||_{A_{2}(𝔻_{θ})} \frac{θ^{j+1}}{1 − θ} .

It follows that

||L_{f} − L_{f}^{(j)}||_{θ,θ} ≤ \frac{||L_{f}||_{θ,1}}{1 − θ} θ^{j+1}.

Since L_{f}^{(j)} is of rank j then we can estimate the (j + 1)^{st} approximation number by

s_{j+1}(L_{f}) ≤ \frac{||L_{f}||_{θ,1}}{1 − θ} θ^{j+1},

and the result follows. □

The importance of the exponential decay of the singular values of L_{f} is that we can deduce the superexponential decay of the Taylor coefficients c_{m}(f) of the corresponding determinant det(I −zL_{f}), as described in the following lemma. This estimate is a sharpening of the inequality (3.7) in [JP].

Lemma 5.2. Let T : I → I be a topologically mixing piecewise C^{ω} expanding Markov map on the interval I, and suppose f : I → ℂ is real analytic. Let U ⊂ ℂ be an admissible domain for the pair (T, f).

The determinant det(I − zL_{f}) of the trace class operator L_{f} : A_{2}(U) → A_{2}(U) is an entire function of z. Indeed if we write det(I − zL_{f}) = 1 + ∑_{m=1}^{∞} c_{m}(f)z^{m}, then

|c_{m}(f)| ≤ B(θ) (\frac{||L_{f}||_{θ,1}}{1 − θ} )^{m} θ^{m(m+1)/2}  (5.5)

where 0 < θ < 1 is a disc contraction ratio for (T, U), and B(θ) = ∏_{m=1}^{∞}(1 − θ^{m})^{−1} < ∞.

Proof. By [Sim, Lemma 3.3] we have c_{m}(f) = ∑_{i_{1}<...<i_{m}} λ_{i_{1}}(L_{f}) ⋯ λ_{i_{m}}(L_{f}), the summation being over all m-tuples (i_{1}, …, i_{m}) of positive integers satisfying i_{1} < … < i_{m}. Therefore

|c_{m}(f)| = | ∑_{i_{1}<...<i_{m}} λ_{i_{1}}(L_{f}) ⋯ λ_{i_{m}}(L_{f})|

≤ ∑_{i_{1}<...<i_{m}} s_{i_{1}}(L_{f}) ⋯ s_{i_{m}}(L_{f})

≤ (\frac{||L_{f}||_{θ,1}}{1 − θ} )^{m} ∑_{i_{1}<...<i_{m}} θ^{i_{1}+...+i_{m}}

= (\frac{||L_{f}||_{θ,1}}{1 − θ} )^{m} \frac{θ^{m(m+1)/2}}{(1 − θ)(1 − θ^{2}) ⋯ (1 − θ^{m})}.

≤ B(θ) (\frac{||L_{f}||_{θ,1}}{1 − θ} )^{m} θ^{m(m+1)/2}

The first inequality is the content of Corollary VI.2.6 of [GGK]. The second inequality is by Lemma 5.1, while the penultimate step is an m-fold geometric summation. □

Remark. As already mentioned, Lemma 5.2 sharpens the inequality (3.7) of [JP], which was itself based on [Ru1, p. 236] (cf. [Fr, p. 505–507], [Ru3, Remark 6.7] for corrections to the estimate in higher dimensions). The most significant improvement is the absence of any m^{m/2} term. The reason for this is that here the operator L_{f} is acting on a Hilbert, rather than simply a Banach, space.

There are other ways in which our method here differs from that of [JP]. For example there we used a sufficiently refined Markov partition and a complexification consisting of small round discs, whereas here we use the original Markov partition and a complexification to any admissible domain, followed by a Riemann mapping to the unit disc. In practice this usually leads to an improved estimate on the disc contraction ratio θ.

6. Conformal coordinate changes

Suppose a piecewise C^{ω} expanding Markov map T : I → I is given. As mentioned previously, in general it is not possible to choose a Euclidean disc as an admissible domain. In fact from a technical viewpoint it may be preferable to choose the admissible domain U to be a rather thin region. This will often give superior estimates on the operator norm ||L_{f}||_{A_{2}(U)}, which is roughly proportional to the uniform norm ||f||_{A_{∞}(U)} (cf. Lemmas 7.2 and 7.3). If f grows rapidly in the imaginary direction, say, then choosing a thin “cigar-shaped” domain U will reduce ||L_{f}||_{A_{2}(U)}, and may give improved bounds on the righthand side of (5.5).

In fact in this article we are not just considering a single function f, but rather a sequence {f_{n}} of analytic functions forming an L^{2}-orthonormal basis. To simultaneously control the uniform norms of these functions, and hence the operator norms of L_{t,n}, it is helpful to use a decreasing sequence U_{n} of admissible domains. The rationale for this is given by the following observation.

Proposition 6.1. Let {f_{n}} be an orthonormal basis for L^{2}(I, dx). Suppose there is some open complex neighbourhood U of the interval I such that each f_{n} extends holomorphically to U. Then the sequence ||f_{n}||_{A_{∞}(U)} is unbounded.

Proof. Let us suppose that the sequence ||f_{n}||_{A_{∞}(U)} is bounded.

Choose K ⊂ U to be compact, with non-empty interior, and containing the interval I. By Montel’s theorem, the sequence f_{n}|_{K} has a subsequence f_{n_{i}}|_{K} which converges uniformly on K to a function f which is holomorphic on the interior of K. In particular f is real analytic on I, and so f ∈ L^{2}(I, dx).

Suppose the integer k ≥ 0 is given. Then for all n_{i} > k we have

|(f, f_{k})|^{2} = |(f, f_{k}) − (f_{n_{i}}, f_{k})|^{2}
= |(f − f_{n_{i}}, f_{k})|^{2}
≤ ||f − f_{n_{i}}||_{L^{2}(I)}||f_{k}||_{L^{2}(I)}
= ||f − f_{n_{i}}||_{L^{2}(I)}
≤ length(I)||f − f_{n_{i}}||_{∞,I}
≤ length(I)||f − f_{n_{i}}||_{A_{∞}(K)} → 0 as i → ∞ .

But |(f, f_{k})| is independent of n_{i}, therefore it must be identically zero. So (f, f_{k}) = 0 for all k ≥ 0. But {f_{k}} is a complete orthonormal set, hence f|_{I} ≡ 0. So f_{n_{i}} → 0 in L^{2}(I, dx) as i → ∞. But this is impossible because ||f_{n_{i}}||_{L^{2}(I)} = 1 for all i, giving the required contradiction. □

The penalty we pay for working with thin cigar-shaped domains U is that the corresponding disc contraction ratio approaches 1 at a rate which is exponential in the height of U. The fact that we necessarily pay this penalty follows from estimates for the Grötzsch modulus in conformal geometry (cf. [LV, §II.2], [Dub]). In the numerical conformal mapping literature this phenomenon is known as crowding (cf. [DeL], [DT, §2.6], [How, Ch. 2]). For our purposes it is helpful to specify a particular family of biholomorphisms φ_{ε,L} : 𝔻 → U_{ε,L}, whose vertical height shrinks to zero as ε ↘ 0, and whose horizontal axis is [−L, L] for some fixed L > 1 chosen small enough to ensure that each U_{ε,L} is an admissible domain.

Our family φ_{ε,L} will be related to the family ψ_{r} : 𝔻 → ℂ, 0 < r < 1, defined by ψ_{r}(z) = tanh^{−1}(rz). The images ψ_{r}(𝔻) resemble flattened ellipses, symmetric with respect to both real and imaginary axes. Each ψ_{r}(𝔻) is contained in the vertical strip −1 < Im(w) < 1, and its horizontal width tends to infinity as r ↗ 1.

If γ_{a}(t) = ae^{it}, t ∈ [0, 1), then

ψ_{r}(γ_{a}(t)) = \frac{1}{2} tanh^{−1} (\frac{2ar cos t}{1 + a^{2}r^{2}} ) + \frac{i}{2} tan^{−1} (\frac{2ar sin t}{1 − a^{2}r^{2}} ) .

In particular, setting a = 1, we see that the semi-height of ψ_{r}(𝔻) approaches 1 as r ↗ 1, while the semi-width is asymptotic to | log(1 − r)|/2.

Now define Φ_{ε,L}(z) = tanh^{−1}(rz) = ψ_{r}(z), where r = r(ε, L) is chosen so that

\frac{L}{ε} = \frac{semi-width(ψ_{r}(𝔻))}{semi-height(ψ_{r}(𝔻))}.

Consequently \frac{L}{ε} ∼ \frac{1}{2}| log(1 − r)| as r ↗ 1. That is,

r(ε, L) ∼ 1 − e^{−2L/ε} as ε ↘ 0 .

Now define φ_{ε,L}(z) = (semi-width(ψ_{r(ε,L)}(𝔻)))^{−1}Φ_{ε,L}(z), a homothetic scaling of Φ_{ε,L}(z) so that φ_{ε,L}(𝔻) is a thin cigar-shaped region contained in [−L, L]×[−ε, ε], with semi-width L and semi-height ε. Define U_{ε,L} := φ_{ε,L}(𝔻) for 0 < ε < ε_{0}, where ε_{0} is chosen small enough to ensure that each U_{ε,L} in the family is admissible. We then have the following estimate on the disc contraction ratios of the domains U_{ε,L} as ε ↘ 0.

Lemma 6.2. (Distortion Estimate) Let T : I → I be a topologically mixing piecewise C^{ω} expanding Markov map on the interval I. Let U_{ε,L} = φ_{ε,L}𝔻 ⊃ I be the family of admissible complex domains defined above. Suppose 0 < τ < 1 is a contraction ratio for each U_{ε,L}, 0 < ε < ε_{0}. Then there is a disc contraction ratio θ_{ε} ∈ (0, 1) associated to the domain U_{ε,L} (for the map T), with the asymptotic

θ_{ε} ∼ 1 − 2^{1−τ} e^{−2τL/ε} as ε ↘ 0

Proof. Let θ_{ε} be the smallest disc contraction ratio associated to the Riemann mapping φ_{ε,L} : 𝔻 → U_{ε,L}. If r = r(ε, L) is as in the definition of Φ_{ε,L} then

τ = \frac{φ_{ε,L}(θ_{ε})}{φ_{ε,L}(1)} = \frac{Φ_{ε,L}(θ_{ε})}{Φ_{ε,L}(1)} = \frac{tanh^{−1}(rθ_{ε})}{tanh^{−1}(r)} = \begin{matrix}log & (\frac{1+rθ_{ε}}{1−rθ_{ε}} ) \\ log & (\frac{1+r}{1−r} ) \end{matrix} .

It follows that

θ_{ε} = \frac{(\frac{1+r}{1−r} )^{τ} − 1}{r [1 + (\frac{1+r}{1−r} )^{τ} ]} = \frac{1}{r} − \frac{2}{r [1 + (\frac{1+r}{1−r} )^{τ} ]} .

However r = r(ε, L) ∼ 1 − e^{−2L/ε} as ε ↘ 0, so that

θ_{ε} ∼ \frac{1}{1 − e^{−2L/ε}} − \frac{2}{(1 − e^{−2L/ε}) [1 + (2e^{2L/ε})^{τ} ]} ∼ 1 − 2^{1−τ} e^{−2τL/ε} ,

as required. □

For the remainder of this section we shall fix the following:

• An orthonormal basis \{f_{n}\} for L^{2}(I), with the property that every \overline{f_{n}} : I → ℂ is real analytic, and such that there exist 0 < α_{ϱ} < 1 and K_{ϱ} > 0 with |(ϱ, f_{n})| ≤ K_{ϱ}α_{ϱ}^{n} for all n ≥ 0.

• A decreasing sequence U_{0} ⊃ U_{1} ⊃ … of complex domains, where U_{n} is an admissible domain for the pair (T, \overline{f_{n}}).

• An increasing sequence 0 < θ_{0} < θ_{1} < …, where each 0 < θ_{n} < 1 is a disc contraction ratio for the pair (T, U_{n}).

• B(θ_{n}) = ∏_{m=0}^{∞}(1 − θ_{n}^{m})^{−1}, as introduced in Lemma 5.2.

Lemma 6.3. Let T : I → I be a topologically mixing piecewise C^{ω} expanding Markov map. Let {f_{n}}_{n=0}^{∞} be an orthonormal basis for L^{2}(I, dx), where each conjugate function \overline{f_{n}} is real analytic. Let U_{0} ⊃ U_{1} ⊃ … be a sequence of admissible domains for the pairs (T, f_{n}), and let θ_{n} be a corresponding sequence of disc contraction ratios. Set B(θ_{n}) = ∏_{m=1}^{∞}(1 − θ_{n}^{m})^{−1} < ∞. Then

(i) The determinant Δ(z) = 1 + ∑_{m=1}^{∞} c_{m}z^{m} defined in (2.1) is an entire function of z, whose Taylor coefficients satisfy

|c_{m}| ≤ B(θ_{0}) (\frac{||L||_{θ_{0},1}}{1 − θ_{0}} )^{m} θ_{0}^{m(m+1)/2} for all m ≥ 0.

(ii) The determinants Δ_{n}(z, t) = 1 + ∑_{m=1}^{∞} _{n}c_{m}(t)z^{m} defined in (3.1) are each entire functions of z, whose Taylor coefficients satisfy

|_{n}c_{m}(t)| ≤ B(θ_{n}) (\frac{||L_{t,n}||_{θ_{n},1}}{1 − θ_{n}} )^{m} θ_{n}^{m(m+1)/2} for all m ≥ 0.

(iii) Each Δ_{n}(z, t) is an entire function of t.

Proof. Parts (i) and (ii) are immediate from Lemma 5.2. Part (iii) is by [Gr, Prop. 1]. □

We can now show that both numerator and denominator in (3.3) approach the respective numerator and denominator in (3.2) super-exponentially fast in M. An important difference is that while the super-exponential decay rate of the numerator does depend on n, the super-exponential decay rate of the denominator does not.

Lemma 6.4. Let T be a topologically mixing piecewise C^{ω} expanding Markov map on either the interval I or the circle 𝕋, and let ϱ be the density function for the unique absolutely continuous T -invariant probability measure μ. Let {f_{n}} be an orthonormal basis for either L^{2}(I, dx) or L^{2}(𝕋, dx) as appropriate, where the conjugate \overline{f_{n}} of each basis element is real analytic.

Let {U_{n}} be a decreasing sequence of complex domains, where each U_{n} is an admissible domain for the pair (T, \overline{f_{n}}). Let \{θ_{n}\} be an increasing sequence such that each 0 < θ_{n} < 1 is a disc contraction ratio for the pair (T, U_{n}). (i)

|Δ′(1) − ∑_{m=1}^{M} mc_{m}| = | ∑_{m=M+1}^{∞} mc_{m}| ≤ B(θ_{0}) ∑_{m=M+1}^{∞} m (\frac{||L||_{θ_{0},1}}{1 − θ_{0}} )^{m} θ_{0}^{m(m+1)/2}. (6.1)

(ii) If we define L(n, ξ) := max_{|t|=ξ} ||L_{t,n}||_{θ_{n},1} for any ξ > 0, then

|∂_{t}Δ_{n}(1, 0) − ∑_{m=0}^{M−1} _{n}c′_{m}(0)| = | ∑_{m=M}^{∞} _{n}c′_{m}(0)| ≤ ∑_{m=M}^{∞} \frac{B(θ_{n})}{ξ} (\frac{L(n, ξ)}{1 − θ_{n}})^{m} θ_{n}^{m(m+1)/2}.

Proof.

(i) We can bound

\begin{aligned}|Δ′(1) − ∑_{m=1}^{M} mc_{m}| & = | ∑_{m=M+1}^{∞} mc_{m}| \\ & ≤ ∑_{m=M+1}^{∞} m|c_{m}| ≤ ∑_{m=M+1}^{∞} mB(θ_{0}) (\frac{||L||_{θ_{0},1}}{1 − θ_{0}} )^{m} θ_{0}^{m(m+1)/2} \end{aligned}

using the bound |c_{m}| ≤ B(θ_{0}) (\frac{||L||_{θ_{0},1}}{1−θ_{0}} )^{m} θ_{0}^{m(m+1)/2} in Lemma 6.3 (i).

(ii) Let Γ_{ξ} be a circle of radius ξ > 0 centred at 0. Cauchy’s theorem gives

|_{n}c′_{m}(0)| = |\frac{1}{2πi} ∫_{Γ_{ξ}} \frac{_{n}c_{m}(t)}{t^{2}}dt|

≤ \frac{2πξ}{2π} ξ^{−2} max_{t∈Γ_{ξ}} |_{n}c_{m}(t)|

≤ \frac{B(θ_{n})}{ξ} (\frac{L(n, ξ)}{1 − θ_{n}})^{m} θ_{n}^{m(m+1)/2}

using Lemma 6.3 (ii). This leads to the bound

\begin{aligned}|∂_{t}Δ_{n}(1, 0) − ∑_{m=1}^{M−1} _{n}c′_{m}(0)| & = | ∑_{m=M}^{∞} _{n}c′_{m}(0)| \\ & ≤ ∑_{m=M}^{∞} \frac{B(θ_{n})}{ξ} (\frac{L(n, ξ)}{1 − θ_{n}})^{m} θ_{n}^{m(m+1)/2}. □ \end{aligned}

We now arrive at a very explicit estimate on the difference between the expansion coefficients (ϱ, f_{n}) and their approximants (ϱ, f_{n})^{(M)}.

Lemma 6.5. Let T be a topologically mixing piecewise C^{ω} expanding Markov map on either the interval I or the circle 𝕋, and let ϱ be the density function for the unique absolutely continuous T -invariant probability measure μ. Let {f_{n}} be an orthonormal basis for either L^{2}(I, dx) or L^{2}(𝕋, dx) as appropriate, where the conjugate \overline{f_{n}} of each basis element is real analytic. Suppose there exist 0 < α_{ϱ} < 1 and K_{ϱ} > 0 such that |(ϱ, f_{n})| ≤ K_{ϱ}α_{ϱ}^{n} for all n ≥ 0.

Let {U_{n}} be a decreasing sequence of complex domains, with each U_{n} an admissible domain for the pair (T, \overline{f_{n}}). Let θ_{0} < θ_{1} < … be an increasing sequence such that each 0 < θ_{n} < 1 is a disc contraction ratio for the pair (T, U_{n}).

For all ξ > 0, all n ≥ 0, and all sufficiently large M ∈ ℕ,

|(ϱ, f_{n})^{(M)} − (ϱ, f_{n})| ≤ \frac{1}{κ} (\frac{B(θ_{n})}{ξ} [\frac{θ_{n}^{M(M+1)/2} (\frac{L(n,ξ)}{1−θ_{n}})^{M}}{1 − (\frac{L(n,ξ)}{1−θ_{n}}) θ_{n}^{M/2}} ] + λ(M)K_{ϱ} α_{ϱ}^{n}) ,

where κ > 0 and λ(M) = O(β^{M^{2}}) as M → ∞, for some 0 < β < 1.

Proof. Using Proposition 3.1 we can estimate

|(ϱ, f_{n})^{(M)} − (ϱ, f_{n})| = |\frac{∫ \overline{f_{n}} dν_{M}}{I_{M}} − ∫ \overline{f_{n}} dμ|

= |\frac{∑_{m=1}^{M} _{n}c′_{m}(0)}{∑_{m=1}^{M} mc_{m}} − \frac{∂_{t}Δ_{n}(1, 0)}{Δ′(1)}|

= |\frac{∑_{m=1}^{M} _{n}c′_{m}(0)}{∑_{m=1}^{M} mc_{m}} − \frac{∑_{m=1}^{∞} _{n}c′_{m}(0)}{∑_{m=1}^{∞} mc_{m}} |

≤ \frac{|Δ′(1)|| ∑_{m=M+1}^{∞} _{n}c′_{m}(0)| + |∂_{t}Δ_{n}(1, 0)|| ∑_{m=M+1}^{∞} mc_{m}|}{|Δ′(1)| | ∑_{m=1}^{M} mc_{m}|}

≤ \frac{∑_{m=M+1}^{∞} |_{n}c′_{m}(0)|}{| ∑_{m=1}^{M} mc_{m}|} + |(ϱ, f_{n})|\frac{∑_{m=M+1}^{∞} |mc_{m}|}{| ∑_{m=1}^{M} mc_{m}|}

≤ \frac{1}{| ∑_{m=1}^{M} mc_{m}|} ( ∑_{m=M+1}^{∞} |_{n}c′_{m}(0)| + K_{ϱ}α_{ϱ}^{n} ∑_{m=M+1}^{∞} m|c_{m}|) .

(6.2)

By Lemma 6.4 (i) we know that ∑_{m=1}^{M} mc_{m} → Δ′(1) ≠ 0 as M → ∞, so we can choose M_{0} ∈ ℕ such that κ := inf_{M≥M_{0}} ∑_{m=1}^{M} mc_{m} is strictly positive.

If M ≥ M_{0} then (6.2) gives

|(ϱ, f_{n})^{(M)} − (ϱ, f_{n})|

≤ \frac{1}{κ} ( ∑_{m=M+1}^{∞} |_{n}c′_{m}(0)| + K_{ϱ}α_{ϱ}^{n}B(θ_{0}) ∑_{m=M+1}^{∞} m (\frac{||L||_{θ_{0},1}}{1 − θ_{0}} )^{m} θ_{0}^{m(m+1)/2})

≤ \frac{1}{κ} ([ ∑_{m=M+1}^{∞} \frac{B(θ_{n})}{ξ} (\frac{L(n, ξ)}{1 − θ_{n}})^{m} θ_{n}^{m(m+1)/2}] + λ(M)K_{ϱ}α_{ϱ}^{n})

≤ \frac{1}{κ} (\frac{B(θ_{n})}{ξ} [\frac{θ_{n}^{M(M+1)/2} (\frac{L(n,ξ)}{1−θ_{n}})^{M}}{1 − (\frac{L(n,ξ)}{1−θ_{n}}) θ_{n}^{M/2}} ] + λ(M)K_{ϱ}α_{ϱ}^{n}) ,

as required. The first inequality in this final calculation is by Lemma 6.4 (i). For the second inequality we use Lemma 6.4 (ii), and set λ(M) := B(θ_{0}) ∑_{m=M+1}^{∞} m (\frac{||L||_{θ_{0},1}}{1−θ_{0}} )^{m} θ_{0}^{m(m+1)/2}.

It is clear that λ(M) is independent of n, and has the asymptotic λ(M) = O(β^{M^{2}}) as M → ∞ for some β < θ_{0}^{1/2}. The final inequality is a crude bound using summation of a (larger) geometric series. □

7. Transfer operator norms

Our estimates in §6 are valid for a wide class of orthonormal bases {f_{n}} for L^{2}(I, dx) or L^{2}(𝕋, dx). In this section we specialise to the case where the basis for L^{2}(I, dx) consists of normalised Legendre polynomials, and the basis for L^{2}(𝕋, dx) consists of Fourier characters.

The normalised Legendre polynomials f_{n} = \sqrt{n + \frac{1}{2}}P_{n} are naturally indexed by the non-

negative integers. If we write e_{n}(x) = 2^{−1/2}e^{πnix} then these Fourier characters are indexed by the integers. In order to harmonise our notation, we let {f_{n}}_{n=0}^{∞} be the re-ordering of the e_{n} defined by f_{2n} = e_{−n} and f_{2n+1} = e_{n}.

Our first observation is that for either basis, the corresponding expansion coefficients (ϱ, f_{n}) of the density function ϱ decay to zero at an exponential rate.

Lemma 7.1. Let {f_{n}}_{n=0}^{∞} denote either the orthonormal basis for L^{2}(I, dx) consisting of normalised Legendre polynomials, or the orthonormal basis for L^{2}(𝕋, dx) consisting of Fourier characters, re-ordered as above. For any real analytic function ϱ : I → ℂ, there exist 0 < α_{ϱ} < 1 and K_{ϱ} > 0 such that

|(ϱ, f_{n})| ≤ K_{ϱ} α_{ϱ}^{n} for all n ≥ 0 ,

and hence there exists K̃_{ϱ} > 0 such that

||ϱ − ∑_{n=0}^{N}(ϱ, f_{n})f_{n}||_{∞,I} ≤ K̃_{ϱ} α_{ϱ}^{N} for all N ≥ 0.

Proof. The Legendre case follows from [Sz, Thm. 9.1.1], since Legendre polynomials are a special case of Jacobi polynomials (cf. [Sz, Ch. IV]). In this case the optimal choice of α_{ϱ} (namely lim sup_{n→∞} |(ϱ, f_{n})|^{1/n}) is the reciprocal of the sum of the semi-axes of the largest ellipse with foci at ±1 on which ϱ is holomorphic.

For the Fourier case see for example [Ka, p. 26]. □

The supremum norm of any Fourier character e_{n} on 𝕋 is equal to 2^{−1/2}. In particular this bound is independent of n. If, however, we consider any fixed complex neighbourhood U of 𝕋, it is no longer true that the supremum norms are uniformly bounded in n, since ||e_{n}||_{A_{∞}(U)} = exp(nπ sup_{z∈U} |Im(z)|). To obtain uniform bounds on the transfer operator norms L(n, ξ) appearing in §6, we would like a better control of the norms of the e_{n} on complex domains. We therefore consider a decreasing sequence of complex domains U_{n}, chosen so that ||e_{n}||_{A_{∞}(U_{n})} is uniformly bounded in n. Clearly this is possible by ensuring that the vertical height of U_{n} is O(\frac{1}{n}).

Lemma 7.2. Let e_{n}(x) = 2^{−1/2}e^{πnix}, for n ∈ ℤ. Let {f_{n}}_{n=0}^{∞} be the re-ordering of the e_{n} so that f_{2n} = e_{−n} and f_{2n+1} = e_{n}.

Let U_{0} ⊃ U_{1} ⊃ … be a decreasing sequence of complex domains, such that U_{n} is admissible for the pair (T, f_{n}), and sup_{z∈U_{n}} |Im(z)| = O(\frac{1}{n}).

Then for any δ > 0, the sequence of values L(n, δ) = max_{|t|≤δ} ||L_{t,n}||_{A_{2}(U_{n})} is uniformly bounded independently of n.

Proof. As observed above, the restriction on the vertical height of U_{n} is enough to ensure that the supremum norms ||f_{n}||_{A_{∞}(U_{n})} are bounded independent of n, by J > 0 say.

The norms ||L||_{A_{2}(U_{n})} of the Perron-Frobenius operator, as an operator A_{2}(U_{n}) → A_{2}(U_{n}) are clearly decreasing in n, since U_{0} ⊃ U_{1} ⊃ . . .. In particular there exists E > 0 such that ||L||_{A_{2}(U_{n})} ≤ E for all n ≥ 0.

If |t| ≤ δ then we have the estimate

||L_{t,n}||_{A_{2}(U_{n})}^{2} = ∫_{U_{n}} | ∑_{i=1}^{d} e^{tf_{n}(z)}v(T_{i}z)|^{2} dA(z)

≤ ||e^{tf_{n}}||_{A_{∞}(U_{n})} ∫_{U_{n}} | ∑_{i=1}^{d} v(T_{i}z)|^{2} dA(z)

≤ e^{δJ}E^{2}. □

The case of normalised Legendre polynomials is slightly more involved. Each Legendre polynomial P_{n} has supremum norm on I equal to 1. The maximum modulus of P_{n} is achieved at the two endpoints, where P_{n}(1) = 1 and P_{n}(−1) = (−1)^{n} (see [WW], §15.1). The normalised basis functions f_{n} = (n + \frac{1}{2})^{1/2}P_{n} therefore have supremum norm ||f_{n}||_{∞,I} = (n + \frac{1}{2})^{1/2}, which in particular is not uniformly bounded in n. Consequently for any sequence of complex neighbourhoods U_{n} of I we will have ||f_{n}||_{A_{∞}(U)} ≥ (n+ \frac{1}{2})^{1/2}. Nevertheless we would like a uniform bound on operator norms analogous to that of Lemma 7.2. We are forced, therefore, to choose a sequence δ_{n} ↘ 0, and prove that L(n, δ_{n}) = max_{|t|≤δ_{n}} ||L_{t,n}||_{A_{2}(U_{n})} can be bounded independently of n.

Lemma 7.3. Let {f_{n}}_{n=0}^{∞} be the orthonormal basis for L^{2}(I, dx) consisting of Legendre polynomials.

Let U_{0} ⊃ U_{1} ⊃ … be a decreasing sequence of complex domains, such that U_{n} is admissible for the pair (T, f_{n}), and dist(I, ∂U_{n}) = O(n^{−2}).

Let δ_{n} be a real sequence decreasing to zero, with δ_{n} = O(n^{−1/2}).

Then L(n, δ_{n}) = max_{|t|≤δ_{n}} ||L_{t,n}||_{A_{2}(U_{n})} is uniformly bounded independently of n.

Proof. The condition dist(I, ∂U_{n}) = O(n^{−2}) ensures that the norms ||P_{n}||_{A_{∞}(U_{n})} of the Legendre polynomials are bounded independent of n, so that ||f_{n}||_{A_{∞}(U_{n})} = O((n+ \frac{1}{2})^{1/2}). The condition on the decay of δ_{n} then ensures that δ_{n}||f_{n}||_{A_{∞}(U_{n})} is bounded by some constant F > 0, say.

We can now argue as in Lemma 7.2. Firstly there is a uniform bound ||L||_{A_{2}(U_{n})} ≤ E on the norms of the Perron-Frobenius operator L on the spaces A_{2}(U_{n}). Secondly, if |t| ≤ δ_{n} then

||L_{t,n}||_{A_{2}(U_{n})}^{2} = ∫_{U_{n}} | ∑_{i=1}^{d} e^{tf_{n}(z)}v(T_{i}z)|^{2} dA(z)

≤ ||e^{tf_{n}}||_{A_{∞}(U_{n})} ∫_{U_{n}} | ∑_{i=1}^{d} v(T_{i}z)|^{2} dA(z)

≤ e^{F} E^{2}. □

8. Proof of Theorems

We are now ready to combine all our previous estimates to prove Theorems 1 and 2.

Let {f_{n}}_{n=0}^{∞} be our orthonormal basis, consisting of either the Legendre polynomials or the (re-ordered) Fourier characters. If ϱ is our invariant density, then by Lemma 7.1 we have the bound ||ϱ − ∑_{n=0}^{N}(ϱ, f_{n})f_{n}||_{∞,I} ≤ K̃_{ϱ}α_{ϱ}^{N} for constants K̃_{ϱ} > 0 and 0 < α_{ϱ} < 1.

Consequently,

\begin{aligned}|| ∑_{n=0}^{N}(ϱ, f_{n})^{(M)} & f_{n} − ϱ||_{∞,I} \\ & ≤ || ∑_{n=0}^{N}(ϱ, f_{n})^{(M)}f_{n} − ∑_{n=0}^{N}(ϱ, f_{n})f_{n}||_{∞,I} + || ∑_{n=0}^{N}(ϱ, f_{n})f_{n} − ϱ||_{∞,I} \\ & ≤ || ∑_{n=0}^{N}(ϱ, f_{n})^{(M)}f_{n} − ∑_{n=0}^{N}(ϱ, f_{n})f_{n}||_{∞,I} + K̃_{ϱ}α_{ϱ}^{N} , \end{aligned} (8.1)

so it remains to analyse the term ||∑_{n=0}^{N}(ϱ, f_{n})^{(M)}f_{n} − ∑_{n=0}^{N}(ϱ, f_{n})f_{n}||_{∞,I}.

We clearly have

\begin{aligned}|| ∑_{n=0}^{N}(ϱ, f_{n})^{(M)} & f_{n} − ∑_{n=0}^{N}(ϱ, f_{n})f_{n}||_{∞,I} ≤ ∑_{n=0}^{N} ||f_{n}||_{∞,I} |(ϱ, f_{n})^{(M)} − (ϱ, f_{n})| \\ & ≤ \frac{1}{κ} ∑_{n=0}^{N} ||f_{n}||_{∞,I} (\frac{B(θ_{n})}{ξ} [\frac{θ_{n}^{M(M+1)/2} (\frac{L(n,ξ)}{1−θ_{n}})^{M}}{1 − (\frac{L(n,ξ)}{1−θ_{n}}) θ_{n}^{M/2}} ] + λ(M)K_{ϱ}α_{ϱ}^{n}) , \end{aligned}

(8.2)

the latter inequality holding for all ξ > 0 and all sufficiently large M (and hence all sufficiently large N) by Lemma 6.5, where κ > 0 is a constant, independent of both M and N.

We now estimate the disc contraction ratios θ_{n}. In the case of Fourier expansions, by Lemma 7.2 the vertical height of the sequence of complex domains U_{0} ⊃ U_{1} ⊃ … must be bounded by \frac{D}{n}, for some fixed D > 0. Setting ε_{n} = \frac{D}{n} we can choose U_{n} = φ_{ε_{n},L}(𝔻), where the family of biholomorphisms φ_{ε,L} was defined in §6. By Lemma 6.2 we therefore have θ_{N} ∼ 1 − exp(−qN) as N → ∞, where q = 2τLD (recall that 0 < τ < 1 is a contraction ratio valid for all the domains U_{ε,L}).

Choose C > q. Let us suppose that M ∼ exp(CN), as in Theorem 2. We then calculate that

\frac{L(N, ξ)}{1 − θ_{N}}θ_{N}^{M/2} ∼ L(N, ξ)e^{qN} (1 − e^{−qN})^{exp(CN)/2} ∼ L(N, ξ)e^{qN}(1 − e^{−qN})^{exp(qN) exp([C−q]N)/2} ∼ L(N, ξ)e^{qN}(e^{−1/2})^{exp([C−q]N)} ,

which tends to zero as N → ∞. In particular we have

\frac{L(N, ξ)}{1 − θ_{N}}θ_{N}^{M/2} < \frac{1}{2} for sufficiently large N.

For Legendre expansions we use Lemmas 6.2 and 7.3, together with an argument analogous to the one below for the Fourier case, to find q > 0 such that θ_{N} ∼ 1 − exp(−qN^{2}) as N → ∞.

As in Theorem 1, we assume that M ∼ exp(CN^{2}), where again C > q. A calculation analogous to the one above shows that

\frac{L(N, ξ)}{1 − θ_{N}}θ_{N}^{M/2} ∼ L(N, ξ)e^{qN^{2}} (1 − e^{−qN^{2}})^{exp(CN^{2})/2} ∼ L(N, ξ)e^{qN^{2}}(e^{−1/2})^{exp([C−q]N^{2})},

which tends to zero as N → ∞. In particular we again have that

\frac{L(N, ξ)}{1 − θ_{N}}θ_{N}^{M/2} < \frac{1}{2} for sufficiently large N.

Let δ_{n} be a non-increasing sequence of positive real numbers. In the case of normalised Legendre polynomials we assume δ_{n} = O(n^{−1/2}) while in the case of Fourier characters we assume δ_{n} = δ > 0 is a constant sequence.

By Lemmas 7.2 and 7.3 we deduce that L(n, δ_{n}) is uniformly bounded in n. Now θ_{n} is increasing with n, so we can estimate

\frac{1}{1 − \frac{L(n,δ_{n})}{1−θ_{n}}θ_{n}^{M/2}} < \frac{1}{1 − \frac{L(N,δ_{N})}{1−θ_{N}}θ_{N}^{M/2}} < 1.

Setting ξ = δ_{n} in (8.2) we have

|| ∑_{n=0}^{N}(ϱ, f_{n})^{(M)}f_{n} − ∑_{n=0}^{N}(ϱ, f_{n})f_{n}||_{∞,I}

≤ \frac{1}{κ} ∑_{n=0}^{N} ||f_{n}||_{∞,I} (\frac{B(θ_{n})}{δ_{n}} [θ_{n}^{M(M+1)/2} (\frac{L(n, ξ)}{1 − θ_{n}})^{M}] + λ(M)K_{ϱ}α_{ϱ}^{n})

≤ \frac{(N + 1)||f_{N}||_{∞,I}}{κ} (\frac{λ(M)K_{ϱ}}{1 − α_{ϱ}} + \frac{B(θ_{N})θ_{N}^{M(M+1)/2}}{δ_{N}} (\frac{Q}{1 − θ_{N}} )^{M}) .

(8.3)

In the final inequality, Q > 0 denotes the uniform bound for L(n, δ_{n}) which is guaranteed by Lemmas 7.2 and 7.3, and we have used the fact that both 1/δ_{n} and ||f_{n}||_{∞,I} are nondecreasing with n, while both θ_{n} and B(θ_{n}) := ∏_{m=1}^{∞}(1 − θ_{n}^{m})^{−1} are increasing with n.

We now wish to analyse more carefully the growth of B(θ_{N}), and are grateful to Oscar Bandtlow for showing us the following argument, establishing the exponential bound

B(θ_{N}) := ∏_{m=1}^{∞}(1 − θ_{N}^{m})^{−1} ≤ exp (\frac{−π^{2}}{6 log θ_{N}} ) .  (8.4)

In particular, if θ_{N} ∼ 1 − e^{−qN} for q > 0 then this gives

B(θ_{N}) ≤ exp (\frac{π^{2}}{6} e^{qN}) ,  (8.5a)

while if θ_{N} ∼ 1 − e^{−qN^{2}} then

B(θ_{N}) ≤ exp (\frac{π^{2}}{6} e^{qN^{2}}) .  (8.5b)

To prove (8.4) we first recall that the dilogarithm is defined by the power series Li_{2}(x) = ∑_{n=1}^{∞} \frac{x^{n}}{n^{2}} , and has the integral representation Li_{2}(x) = ∫_{x}^{0} \frac{log(1−s)}{s} ds. Then

∑_{m=1}^{k} − log(1 − θ_{N}^{m}) ≤ ∫_{0}^{k} − log(1 − θ_{N}^{t}) dt = ∫_{0}^{k} − log(1 − e^{−γ_{N} t}) dt

where we write γ_{N} = − log θ_{N} (> 0).

Making the change of variable s = e^{−γ_{N} t} we obtain

∫_{0}^{k} − log(1 − e^{−γ_{N} t}) dt = ∫_{s=1}^{exp(−γ_{N} k)} \frac{log(1 − s)}{γ_{N}s} ds

= \frac{1}{γ_{N}} (Li_{2}(1) − Li_{2}(e^{−γ_{N} k})) .

Letting k → ∞ we obtain the estimate

∑_{m=1}^{∞} − log(1 − θ_{N}^{m}) ≤ γ_{N}^{−1}(Li_{2}(1) − Li_{2}(0)) = \frac{−π^{2}}{6 log θ_{N}} ,

and hence

B(θ_{N}) = ∏_{m=1}^{∞}(1 − θ_{N}^{m})^{−1} = exp − ∑_{m=1}^{∞} log(1 − θ_{N}^{m}) ≤ exp (\frac{−π^{2}}{6 log θ_{N}} ) ,

which is the inequality (8.4). If θ_{N} = 1 − x then − log θ_{N} = ∑_{j=1}^{∞} \frac{1}{j}x^{j} ≥ x, and (8.5a), (8.5b) follow.

We now express the righthand side of (8.3) as R(N) = R_{1}(N) + R_{2}(N) (we suppress the dependence on M, since M = M(N)), where

R_{1}(N) = \frac{(N + 1)||f_{N}||_{∞,I}}{κ} \frac{λ(M)K_{ϱ}}{1 − α_{ϱ}}

and

R_{2}(N) = \frac{(N + 1)||f_{N}||_{∞,I}}{κ} \frac{B(θ_{N})θ_{N}^{M(M+1)/2}}{δ_{N}} (\frac{Q}{1 − θ_{N}} )^{M} .

We observe that R_{1}(N) is converging rapidly to zero, due to the λ(M) factor which, as noted in Lemma 5.4, is O(β^{M^{2}}) as M → ∞, for some 0 < β < 1. Since either M ∼ e^{CN} or M ∼ e^{CN^{2}} then λ(M) is certainly O(γ^{N}) as N → ∞, for every 0 < γ < 1. The other factor \frac{(N+1)||f_{N} ||_{∞,I}}{κ} \frac{K_{ϱ}}{1−α_{ϱ}} has comparatively slow growth; it is O(N^{3/2}) in the case of the normalised Legendre basis, and O(N) in the case of the Fourier basis. Consequently

R_{1}(N) = O(γ^{N}) as N → ∞ , for every 0 < γ < 1 .  (8.6)

To analyse R_{2}(N) we write R_{2}(N) = R_{3}(N)R_{4}(N), where

R_{3}(N) = \frac{(N + 1)||f_{N}||_{∞,I}}{κδ_{N}} = O(N^{2})

(and in the Fourier case we have the improved bound R_{3}(N) = O(N)).

In particular we will now show that the modest growth of R_{3}(N) is dominated by the rapid decay of

R_{4}(N) = B(θ_{N}) (\frac{Q}{1 − θ_{N}} )^{M} θ_{N}^{M(M+1)/2}.

It will be convenient to treat the Fourier and Legendre cases separately.

The case of Fourier series. Since θ_{N} ∼ 1 − e^{−qN} and M ∼ exp(CN) for a fixed C > q then

θ_{N}^{M(M+1)/2} ∼ ((1 − e^{−qN})^{e^{qN}})^{e^{(C−q)N} /2} ∼ (e^{−1})^{e^{(C−q)N} /2} = exp (−e^{(C−q)N}/2) .

Combining this with (8.5a) gives

log R_{4}(N) ∼ \frac{π^{2}e^{qN}}{6} + M log Q − qMN − e^{(C−q)N}/2.

Since M ∼ e^{CN} we see that the dominant term is −qMN = −qe^{CN}N if C ≤ 2q, and is −e^{(C−q)N}/2 if C > 2q, so in both cases log R_{4}(N) → −∞ as N → ∞. Therefore in either case we have

R_{4}(N) = O (exp (−qe^{CN}N)) ,

which converges to zero faster than any exponential. Therefore for any 0 < γ < 1 we have R_{2}(N) = R_{3}(N)R_{4}(N) = O(γ^{N}) as N → ∞.

The case of Legendre series. Since θ_{N} ∼ 1 − e^{−qN^{2}} and M ∼ exp(CN^{2}) for a fixed C > q then

θ_{N}^{M(M+1)/2} = ((1 − e^{−qN^{2}})^{e^{qN^{2}}} )^{e^{(C−q)N^{2}} /2} ∼ (e^{−1})^{e^{(C−q)N^{2}} /2} = exp (−e^{(C−q)N^{2}}/2) .

Combining this with (8.5a) gives

log R_{4}(N) ∼ \frac{π^{2}M}{6} + M log Q − qMN^{2} − e^{(C−q)N^{2}}/2.

Setting M = e^{CN^{2}} we see that if C ≤ 2q then the dominant term is −qMN^{2} = −qe^{CN^{2}}N^{2}, while if C > 2q then the dominant term is −e^{(C−q)N^{2}}/2. In both cases we see that log R_{4}(N) → −∞ as N → ∞, and that

R_{4}(N) = O (exp (−qe^{CN^{2}}N^{2})) ,

which converges to zero faster than any exponential. Therefore we have

R_{2}(N) = R_{3}(N)R_{4}(N) = O(γ^{N}) as N → ∞, for any 0 < γ < 1 .  (8.7)

So in both the Fourier and the Legendre cases, combining (8.6) and (8.7) allows us to deduce that

|| ∑_{n=0}^{N}(ϱ, f_{n})^{(M)}f_{n} − ∑_{n=0}^{N}(ϱ, f_{n})f_{n}||_{I,∞}
≤ R(N) = R_{1}(N) + R_{2}(N) = O(γ^{N}) as N → ∞, for every 0 < γ < 1.

Combining the above estimate with (8.1) gives us

|| ∑_{n=0}^{N}(ϱ, f_{n})^{(M)}f_{n} − ϱ||_{∞,I} = O(α_{ϱ}^{N}) as N → ∞ ,

thus completing the proof of Theorems 1 and 2. □

9. C^{k} expanding maps

Suppose that T is a topologically mixing piecewise expanding Markov map, but is only piecewise^{5} C^{k}, for some 2 ≤ k ≤ ∞, rather than piecewise real analytic. In this case there exists an absolutely continuous invariant probability measure, whose density function ϱ is C^{k−1}. The requirement k ≥ 2 can be weakened; if the derivative of T is Hölder continuous, say, then again there exists an absolutely continuous invariant probability measure, though if T is merely C^{1} then this need not be the case (cf. [GS]).

All of the previous formulae for approximating ϱ only involve the first derivative of T, so at least formally they make sense when T is C^{k}. Indeed the algorithm does produce a sequence of approximations to ϱ in this case, though in general their rate of convergence is slower than in the C^{ω} case.

The reasons for this are twofold. Firstly the expansion coefficients (ϱ, f_{n}) are only O(\frac{1}{n^{k−1}} ) as n → ∞. Secondly, the decay of the Taylor coefficients c_{m} and _{n}c′_{m}(0) will only be exponential in n, so that (ϱ, f_{n})^{(M)} defined in Proposition 3.1 will converge to (ϱ, f_{n}) at an exponential, rather than superexponential, rate. This is because the determinants defined by (2.1), (2.2), (3.1) are not entire functions of z. Strictly speaking they are not determinants in the classical sense, since the corresponding transfer operators are not trace class (nor even compact, though they are quasi-compact, cf. [Ru4]). Nevertheless these determinants do have a meromorphic extension to a complex disc centred at the origin and containing the zero of minimum modulus (see [Ru4]), thus ensuring the exponential decay of their Taylor coefficients.

An alternative approach is to first approximate the C^{k} map T by a C^{ω} expanding map S. Such an approximation is possible in various topologies, in particular the Banach C^{k} topology; explicitly this can be achieved by convolving T with a suitable summability kernel (cf. [Kat]). One then works with the map S, applying the algorithm described in §§1–8 to construct an approximation ϱ̃_{S} to the density ϱ_{S} of S. But ϱ_{S} is close to the density ϱ_{T} of T in various topologies, in particular the uniform topology, so ϱ̃_{S} is a uniform approximation to ϱ_{T}, as required.

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Oliver Jenkinson, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS.

   E-mail address: omj@maths.qmul.ac.uk

Mark Pollicott, Department of Mathematics, Manchester University, Oxford Road, Manchester M13 9PL.

ORTHONORMAL EXPANSIONS OF INVARIANT DENSITIES FOR EXPANDING MAPS 31

E-mail address: mp@ma.man.ac.uk