Submitted exclusively to the London Mathematical Society doi:10.1112/S0000000000000000

VISIBLE MEASURES OF MAXIMAL ENTROPY IN DIMENSION ONE

NEIL DOBBS

Abstract

A real one-dimensional analogue of Zdunik’s dichotomy is proven giving dynamical conditions for a multimodal map to have a measure of maximal entropy of dimension one.

1. Introduction

In [12], Zdunik shows the following dichotomy: for every rational map of the Riemann sphere either the measure of maximal entropy is of Hausdorff dimension strictly smaller than the dimension of the Julia set or the map is critically finite with parablic orbifold.

In the complex case there is a unique ergodic measure of maximal entropy, positive on open subsets of the Julia set. In the real case matters are more complicated with potentially the presence of numerous ergodic measures of maximal entropy.

The real one-dimensional analogue of Zdunik’s dichotomy is as follows. For Cmaps (see below) f with positive entropy, an ergodic measure of maximal entropy μ has dimension strictly smaller than 1 or f is conjugate to a continuous piecewiselinear map on some restrictive interval, this conjugacy is smooth outside the (finite) post-critical set and μ is supported on the forward orbit of the restrictive intervals. Interestingly, the conjugacy is C^{r} if f is, for r ≥ 2, so the result is optimal.

A point c is called critical for smooth map f if the derivative Df(c) = 0 (note that inflection points are critical); for continuous piecewise-linear maps turning points are called critical. We denote by Crit the collection of critical points. The post-critical set P C for a map f is defined by P C = ⋃_{i≥1} f^{i}(Crit).

2000 Mathematics Subject Classification 37E05, 37A35 (primary).

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Definition 1.1. Following [7] we say f is a C-map if f : I → I is a continuously differentiable map of the closed interval I and its derivative Df has the following properties:

– Df satisfies a Hölder condition of order ϵ, ϵ > 0.

– There are only a finite number of critical points a_{i}, 1 ≤ i ≤ n, Df(a_{i}) = 0.

– There exist positive numbers k_{i}^{−}(k_{i}^{+}) such that

|log \frac{|Df(x)|}{|x − a_{i}|^{k_{i}^{−(+)}}} |

is bounded in a left (right) neighbourhood of a_{i} - this is a non-flatness condition.

C-maps which are also of class C^{r} shall be called C^{r} C-maps. We state our results for C^{r} C-maps. If f is C^{1+ϵ} (i.e. just a C-map) then the same results hold but the respective conjugacies may have different Hölder constants.

The Hausdorff dimension [10] of a probability measure ν with support X is defined as

HD(ν) = inf{HD(Y) : Y ⊂ X, ν(Y) = 1}.

We say f has a (non-trivial) restrictive interval J ⊂ I of period n > 0 if there exists an interval J(≠ I) disjoint from periodic attractors such that J, f(J), …, f^{n−1}(J) are disjoint, f^{n}(∂J) ⊂ ∂J and f^{n}(J) ⊂ J. Also, f is renormalisable of type n if f has a non-trivial restrictive interval J of period n containing at least one critical point. The map f_{|J}^{n} : J → J is then a renormalisation of f of type n.

Theorem 1. Let f be a C^{r} C-map, r ≥ 2, with positive topological entropy. Exactly one of the following statements holds:

– HD(μ) < 1 for all measures of maximal entropy μ.

– There exists a restrictive interval J of period k such that f_{|J}^{k} has finite postcritical set P C, f^{k} is C^{r} conjugate to a continuous piecewise linear map of constant slope on J \ P C and μ(J) > 0 for some measure of maximal entropy μ.

The singularities of the conjugacy are of root type. If f has no inflection points (this is typical) then the post-critical set must coincide with the boundary, P C = ∂J.

VISIBLE MEASURES OF MAXIMAL ENTROPY 3

The condition that f has positive topological entropy is necessary. Indeed a Cmap which is the identity on [−1, 0] and a Feigenbaum map on [0, 1] has zero topological entropy and an absolutely continuous invariant measure.

For non-renormalisable maps a more dynamical version is possible.

Corollary 1. Let f be a C^{r} C-map, r ≥ 2, without non-trivial restrictive intervals. Exactly one of the following statements holds:

– HD(μ) < 1 for measure of maximal entropy μ.

– f has finite post-critical set P C and f is C^{r} conjugate to a continuous piecewise linear map of constant slope on I \ P C.

The (symmetric) tent map T_{s} : [0, 1] → [0, 1], for 0 < s ≤ 2, is defined by

T_{s}(x) = sx,  0 ≤ x ≤ \frac{1}{2},
T_{s}(x) = s(1 − x), \frac{1}{2} < x ≤ 1.

If 1 < s ≤ \sqrt{2}, the tent map T_{s} provides a simple example of a map which is renormalisable of type 2.

In the unimodal case we achieve a complete dynamical classification of maps where the Hausdorff dimension of the measure of maximal entropy is one.

Definition 1.2. A C-map f is unimodal if it has exactly one critical (turning) point, exactly two fixed points and f(∂I) ⊂ ∂I. A unimodal C-map f is called C^{r} pre-Chebyshev if

– f is exactly k times renormalisable, for some 0 ≤ k < ∞, and each renormalisation is of type 2 (i.e. of Feigenbaum type);

– Let f_{|J}^{2^{k}} : J → J be the resultant non-renormalisable map. Then there is a C^{r} conjugacy h on the interior of J from f_{|J}^{2^{k}} to the full tent map T_{2};

If f is pre-Chebyshev then the C^{r} conjugacy h extends to a topological conjugacy on I from f to a symmetric tent map T_{s} of slope ±s, where the entropy of f is log s. Then if f is k times renormalisable, s = 2^{(2^{−k})}. Also remark that the full tent map T_{2} is analytically conjugate to the Chebyshev quadratic polynomial 4x(1 − x).

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Theorem 2. Suppose f is a C^{r} unimodal C-map, r ≥ 2, and that f has postive topological entropy and measure of maximal entropy μ. Exactly one of the following statements holds:

– HD(μ) < 1.

– f is C^{r} pre-Chebyshev.

An interval U is regularly returning if f^{n}(∂U) ∩ U = ∅ for all n > 0. Then if f^{i}(A) = f^{j}(B) = U either A ∩ B = ∅, A ⊂ B or A ⊃ B.

An important principle in our considerations is the existence of induced expansion for maps with absolutely continuous invariant probability measures (acips) (Proposition 2.12). The induced maps we construct have nice “C^{r}” properties, necessary to get good conjugacy results. This principle can be applied, for example, to extend the results of [9] to ergodic C-maps with acips with optimal smoothness results for the conjugacies.

In the following section we shall assemble various results to be used in the proofs.

2. Preliminaries

2.1. Topological results

We start with the result of Milnor and Thurston (1977), also Parry (1966), see [1] II.8.

Fact 2.1. Assume that f : I → I is a continuous, piecewise (strictly) monotone map with positive topological entropy h_{t}(f) and let s = exp(h_{t}(f)). Then there exists a continuous, piecewise linear map T : [0, 1] → [0, 1] with slope ±s, and a continuous, monotone increasing map h : I → [0, 1] which is a semi-conjugacy between f and T, i.e.

h ∘ f = T ∘ h.

Proposition III.4.3 in the same book [1] adds the following properties for C^{1} multimodal maps without wandering intervals (absence of wandering intervals for C-maps is stated in [11]):

VISIBLE MEASURES OF MAXIMAL ENTROPY 5

Fact 2.2. Let X be the set of points which are eventually mapped into a restrictive interval or into the basin of a periodic attractor.

(i) The semi-conjugacy h can collapse an interval only if it is contained in X. Basins of periodic attractors are certainly collapsed.

(ii) The union of X and the set of points which are mapped eventually into an interval V ≠ I with f(V) ⊂ V is dense in I. Moreover f is conjugate to T on the complement of X.

(iii) If T has a periodic turning point c then h^{−1}(c) is a restrictive interval.

Proposition III.4.4 [1] will also be useful:

Fact 2.3. Let T : I → I be a piecewise linear l-modal map with slope ±s, s > 1. Then one has the following properties.

(i) T has sensitive dependence on initial conditions. (ii) T is only finitely often renormalizable. (iii) The non-wandering set of T contains a finite number of intervals which are

permuted by T and on each of these intervals T is transitive. (The permutation of the intervals may split into disjoint cycles.) On the complement of these intervals, the non-wandering set of T consists of a finite number of periodic points and the remainder is a subshift of finite type.

(iv) The only attractors of T are intervals.

Another well-known fact [5] is:

Fact 2.4. For the symmetric tent map T with slope ±s and turning point c

(i) If \sqrt{2} < s ≤ 2 then for every open interval J there exists an n such that T^{n}(J) ⊃ [T^{2}(c), T (c)]. In particular T has no restrictive intervals.

(ii) If integer k is such that \sqrt{2} < s^{2^{k}} ≤ 2 then T has exactly k restrictive intervals I_{k} ⊂ I_{k−1} ⊂ … ⊂ I_{0} = I containing the turning point. I_{i} has period 2^{i} for i = 0, 1, …, k. For each x ∈ I there exists an integer n such that either T^{n}(x) ∈ I_{k} or such that T^{n}(x) is one of the k periodic orbits that remain outside of the forward orbit of I_{k}.

Lemma 2.5. Let f be a transitive, non-renormalisable continuous map of the interval I. Then for any subinterval U there exists n such that f^{n}(U) = I.

Proof. Fact 2.1 implies we can assume f is an expanding continuous picewise linear map. Consider periodic point p ∈ U of period k and small subinterval V with p ∈ ∂V. Now p ∈ f^{2ki}(V) ⊂ f^{2k(i+1)}(V) for all i ≥ 0. Set W_{i} = f^{2k(i+1)}(V) \ f^{2k(i)}(V). For i sufficiently large one has W_{i} = ∅. This follows from the finiteness of the number of critical points, that the W_{i} are disjoint and that the map is expanding: If W_{i} ∩ Crit = ∅ then |W_{i+1}| > |W_{i}|.

Let k-periodic V_{∗} = ⋃_{i≥1} f^{2ki}(V). By transitivity I ⊂ ⋃_{j=0}^{k−1} f^{j}(V_{∗}). Thus some iterate of U contains a fixed point p′. One restarts the process to find 2-periodic V′_{∗} and concludes by non-renormalisability that V′_{∗} = I.

2.2. Absolutely continuous measures and induced expansion

From Ledrappier [7] we have that for C-maps an ergodic invariant measure ν of positive entropy satisfies the Rokhlin formula

h_{ν}(f) = ∫ log |Df| dν

if and only if ν is absolutely continuous with respect to the Lebesgue measure λ.

Combined with the Dynamical Volume Lemma [4], which for piecewise strictly monotone C^{1+ϵ} maps f with an ergodic, invariant, probability measure ν states that

HD(ν) = \frac{h_{ν}}{∫ log |Df|},

this implies that

Lemma 2.6. HD(ν) = 1 if and only if ν is absolutely continuous with respect to Lebesgue measure.

This holds for ν an ergodic, invariant, probability measure for the C-map f.

Fact 2.7. [6] A piecewise linear map S with slope strictly greater than 1 has an absolutely continuous invariant measure.

VISIBLE MEASURES OF MAXIMAL ENTROPY 7

Definition 2.8. Given a C^{1+ϵ} function f, we say a map F : ⋃_{i} U_{i} → U is a Markov map induced by f if U is a subinterval of I, U_{i} ⊂ U are disjoint intervals and

– λ(U \ ⋃_{i} U_{i}) = 0

– for all i, F (U_{i}) = U and F_{|U_{i}} = f^{n_{i}} for some n_{i}

– there exists δ > 1 such that |DF | ≥ δ

– there exists a bound on distortion K > 1 such that, for all n > 0, on each connected component V of the domain of F^{n},

\frac{DF^{n}(x)}{DF^{n}(y)} < K  (2.1)

for all x, y ∈ V.

Definition 2.9. If f is C^{r}, for integer r ≥ 2, then we say F is a C^{r} Markov map induced by f if in addition to being Markov the following condition is satisfied:

\frac{1}{|DF (x)|^{i}} |D^{i} log |DF (x)|| < L,

for some L > 0, for all x in the domain of F and for i = 1, 2, …, r − 1.

For existence of Markov maps, we shall need to extend a theorem of Ledrappier [7]. Summarily, here (Y, f) is the extension or inverse system for (I, f), f being a C-map of the interval I with ergodic invariant measure μ. An element of Y is a point x ∈ I and a sequence z which says which inverse branch of f to take (coded by the partition defined by critical points). (Y, f) is invertible and if Π is the projection on I then there exists a unique ergodic invariant measure \overline{μ} on Y whose image by Π is μ.

Fact 2.10. [7] Let f be a C-map with an invariant ergodic probability measure μ with positive Lyapunov exponent. Suppose 0 < χ < ∫ log |Df|dμ. Then there exist on Y four measurable functions α, β, γ, \overline{γ} and a constant K such that:

– α > 0, 1 < β < ∞, 0 < γ < \overline{γ} < ∞, \overline{μ}-almost everywhere;

– let y = (x, z) and |t| < α(y); the point y_{t} = (x + t, z) lies in Y;

– |Π(f^{−n}y) − Π(f^{−n}y_{t})| ≤ β(y) exp(−χn)|t|;

– for any n,

\frac{1}{K} ≤ \frac{Df(Π(f^{−n}y))}{Df(Π(f^{−n}y_{t}))} ≤ K;

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–

γ(y) ≤ Π_{n=1}^{∞} \frac{Df(Π(f^{−n}y))}{Df(Π(f^{−n}y_{t}))} ≤ \overline{γ(y)}.

Proposition 2.11. Supplementarily to Fact 2.10, there exists measurable γ_{1} < ∞ \overline{μ}-almost everywhere such that for all n

| log Df^{n}(Πf^{−n}y) − log Df^{n}(Πf^{−n}y_{t})| < γ_{1}(y)|t|^{ϵ},

(ϵ being the Hölder exponent). If f is also C^{r}, for some integer r ≥ 2, then there exists a measurable function γ_{r} with γ_{r}(y) < ∞ \overline{μ}-almost everywhere such that for all n and all i = 1, 2, …, r − 1,

\frac{1}{|Df^{n}(Πf^{−n}y)|} |D^{i} log |Df^{n}(Πf^{−n}y)|| ≤ γ_{r}(y).  (2.2)

Proof.

log Df^{n}(Πf^{−n}y) = ∑_{j=0}^{n−1} log Df(Πf^{−n+j}y) = ∑_{j=1}^{n} log Df(Πf^{−j}y)

and

| log Df(x) − log Df(x′)| = log (1 + |\frac{Df(x) − Df(x′)}{Df(x′)}|) ≤ C \frac{|x − x′|^{ϵ}}{|Df(x′)|},

for some constant C. Thus

| log Df^{n}(Πf^{−n}y) − log Df^{n}(Πf^{−n}y_{t})| ≤ C ∑_{j=1}^{n} \frac{|Π(f^{−j}y) − Π(f^{−j}y_{t})|^{ϵ}}{|Df(Πf^{−j}y)|} .

Integrability of the Lyapunov exponent and ergodicity imply that \frac{1}{|Df(Πf^{−j}y)|} is bounded by a subexponentially increasing sequence. From Fact 2.10 we also have that |Π(f^{−j}y) − Π(f^{−j}y_{t})| ≤ β(y) exp(−χj)|t|; thus almost everywhere we are summing an exponentially decreasing sequence and there exists a function γ_{1}, γ_{1} < ∞ almost everywhere, such that

∑_{j=1}^{n} \frac{|Π(f^{−j}y) − Π(f^{−j}y_{t})|^{ϵ}}{|Df(Πf^{−j}y)|} ≤ ∑_{j=1}^{∞} \frac{|Π(f^{−j}y) − Π(f^{−j}y_{t})|^{ϵ}}{|Df(Πf^{−j}y)|} ≤ γ_{1}(y)|t|^{ϵ},

completing the first part of the proof.

For i ≥ 1 we can expand D^{i} log |Df^{n}| as a polynomial

D^{i} log |Df^{n}(x)| = ∑_{j=0}^{n−1} P_{i} (\frac{D^{2}f(f^{j}x)}{Df(f^{j}x)}, \frac{D^{3}f(f^{j}x)}{Df(f^{j}x)}, …, \frac{D^{i}f(f^{j}x)}{Df(f^{j}x)})

where P_{i} is some (calculable) polynomial of order i.

VISIBLE MEASURES OF MAXIMAL ENTROPY 9

Since f is C^{r} there is a constant C such that |Df^{i}(x)| < C for all x and all i ≤ r. Let δ(x) = \frac{1}{C} min(1, |Df(x)|), and set δ_{n}(x) = min_{j=0}^{n−1} δ(f^{j}x). If K_{i} is the product of the number of terms of the polynomial P_{i} with the maximum of moduli of the coefficients, then

|D^{i} log |Df^{n}(x)|| ≤ ∑_{j=0}^{n−1} |P (\frac{1}{δ(f^{j}x)}, \frac{1}{δ(f^{j}x)}, …, \frac{1}{δ(f^{j}x)})|
≤ ∑_{j=0}^{n−1} K_{i}δ_{n}(x)^{−i}
≤ nK_{i}δ_{n}(x)^{−i}.

As before, \overline{μ}-almost everywhere δ_{n}(Πf^{−n}y) is bounded by a subexponentially decreasing sequence, so \overline{μ}-almost everywhere |D^{i} log |Df^{n}(Πf^{−n}y)|| is bounded by a subexponentially increasing sequence for each i = 1, 2, …, r−1. We have that |Df^{n}(Πf^{−n}y)| increases exponentially almost everywhere, hence for every y in a set of full measure there exists a γ_{r}(y) such that (2.2) holds for all i = 1, 2, …, r − 1.

Proposition 2.12. Let f be a C-map with an ergodic absolutely continuous invariant probability measure μ with positive Lyapunov exponent. Then there exists a Markov map induced by f. If in addition f is C^{r} then there exists a C^{r} Markov map induced by f.

Proof. We apply Fact 2.10 and Proposition 2.11 to fix α_{0} > 0, β_{0} < ∞, K < ∞ such that A = {y ∈ Y : α(y) > α_{0}, β(y) < β_{0}, γ(y) < K, γ_{1}(y) < K, γ_{r}(y) < K} has positive measure, \overline{μ}(A) > 0. Let x ∈ I be a density point of (Π_{|A})_{∗}\overline{μ}, and set A_{x} = A ∩ Π^{−1}x. Let U ∋ x be a regularly returning interval with |U| < \frac{α_{0}}{2} .

By ergodicity almost every point y ∈ Y accumulates on A_{x}. Thus for almost every y there exist iterates f^{n}y = (x′, z) with x′ ∈ U and (x, z) ∈ A_{x} with n arbitrarily large. By Fact 2.10, since α((x, z)) ≥ α_{0} the interval U can be pulled back along the branch defined by z with bounded distortion to an interval containing Πy, i.e. for \overline{μ}-almost every y there exist n_{y} and interval V_{y} ∋ Πy such that f^{n_{y}} maps V_{y} diffeomorphically onto U with bounded distortion. Taking n_{y} large (and thus V_{y} small) gives expansion: |Df^{n_{y}}| > δ > 1 on V_{y}.

For y, y′ their respective intervals V_{y}, V_{y′} are disjoint or nested since U is regularly returning. A set of full \overline{μ}-measure projects to a set of full μ-measure, so μ-almost

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every point in U is contained in some V_{y}. Take a disjoint collection of full measure of intervals V_{y} and define F on ⋃ V_{y} ∋ x by

F_{|V_{y}}(x) = f^{n_{y}}(x).

Now \frac{DF (x)}{DF (x′)} < K^{2} \frac{|x−x′|^{ϵ}}{|V_{y}|} if x, x′ ∈ V_{y} and a standard summation argument gives the distortion bound (2.1) so F is a Markov map induced by f.

In the C^{r} case we have \frac{1}{|DF (x)|} |D^{i} log |DF (x)|| < K for all x ∈ V_{y} for all connected components V_{y} of the domain of F, and for i = 1, 2, …, r − 1. Since |DF (x)| > 1, we conclude that

\frac{1}{|DF (x)|^{i}} |D^{i} log |DF (x)|| < K

as required.

Proposition 2.13. Let f be a C-map possessing an ergodic absolutely continuous invariant measure μ with positive Lyapunov exponent. Then μ is equivalent to Lebesgue measure λ on a finite union of intervals U. Moreover there exist ε > 0 and measurable density function ρ > ε on U such that dμ = ρdλ.

Proof. Proposition 3.2 of [7] constructs the Pesin partition η of the natural extension Y. Proposition 3.6 of [7] gives a formula for the Rohlin decomposition for \overline{μ}. The proof shows that, on the partition element η(y) containing y, the density of the Rohlin decomposition is bounded inside [\frac{1}{θ(y)}, θ(y)] say, where θ is a measurable function with 1 < θ(y) < ∞ on a set (the set A_{s} of [7]) of positive measure.

There exist δ > 0, C > 1 such that the set B = {y ∈ Y : |Π(η(y))| > δ, θ(y) < C} is of positive measure. Let x be a density point of (Π_{|B})_{∗}\overline{μ}. Then at least one of \{y ∈ B : Π(η(y)) ⊃ (x− \frac{δ}{2}, x)\}, \{y ∈ B : Π(η(y)) ⊃ (x, x+ \frac{δ}{2})\} is of positive measure ν > 0 say. Projecting, on at least one of (x − \frac{δ}{2}, x), (x, x + \frac{δ}{2}) the density ρ of μ is bounded from below by \frac{2ν}{δC} almost everywhere.

Thus there exists a small interval, which we now denote by V, on which the density ρ of μ is greater than some strictly positive constant ε_{0} > 0. By ergodicity, f is transitive on the support of μ. An application of Lemma 2.5 provides an n such that ⋃_{i=0}^{n} f^{i}(V) coincides with the support of μ. The invariance of μ implies that the density of μ is greater than some strictly positive constant on its support, since |Df^{n}| is bounded. This completes the proof of Proposition 2.13. Note that this result is analagous to the stated Theorem 6 of [8] which treats rational maps.

Fact 2.14. ([1] V, Theorem 2.2) For any Markov map G with corresponding Markov partition ⋃ U_{i}, there exists a unique absolutely continuous (w.r.t. Lebesgue measure λ) G-invariant probability measure ν. The density is Hölder continuous and uniformly bounded from above and from below away from zero.

Lemma 2.15. Every conjugacy between two C^{r}, r = 2, 3, … Markov maps which preserves multipliers is a C^{r} diffeomorphism.

Definition 2.8 for Markov maps is more general than that of [9]. Instead of demanding negative Schwarzian we simply have the distortion bounds which suffice - all the proofs of [9] follow through with little modification, including Corollary 2.9 which has the same statement as this lemma. We remark that in this paper we actually use this lemma for r > 2.

3. Proofs

Proposition 3.1. Let f be a C^{r}, r ≥ 2, C-map and μ an ergodic measure of maximal entropy for f. If HD(μ) = 1 then there exists a restrictive interval J of period k on which f_{|J}^{k} has finite post-critical set P C and f^{k} is C^{r} conjugate to a continuous piecewise linear map of constant slope on J \ P C. The measure μ has support ⋃_{i=0}^{k−1} f^{i}(J).

Proof. By Lemma 2.6, μ is absolutely continuous with respect to Lebesgue measure. By Proposition 2.13 the measure is equivalent to Lebesgue and is supported on a cycle of intervals X′, on which f is transitive.

Let X be a component of X′ and n such that f^{n} : X → X, and set g := f^{n}. Then μ (when normalised) is the unique (probability) measure of maximal entropy for g : X → X and is absolutely continuous. Let log s be the entropy of f. Then setting t = s^{n} the entropy of g is log t.

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By Facts 2.1-2.2 and transitivity there is a conjugacy h to a piecewise linear map T : I → I with slope ±t and the same entropy log t,

h ∘ g = T ∘ h.

Note that X could contain a periodic (restrictive) subinterval. Since T is only finitely often renormalisable the same holds for g and there is a smallest restrictive interval J. We suppose (without loss of generality) that X = J, i.e. that g is nonrenormalisable.

The push-forward of μ, h_{∗}μ, is the unique measure of maximal entropy for T. But T has an absolutely continuous invariant measure \overline{μ} by Fact 2.7, thus HD(\overline{μ}) = 1. For the map T the Lyapunov exponent is log t. By the Dynamical Volume Lemma the measure-theoretical entropy h\overline{μ} = log t and so \overline{μ} maximises entropy. By unicity h_{∗}μ = \overline{μ}. This measure is ergodic and thus equivalent to Lebesgue.

It follows that h and h^{−1} are absolutely continuous: Indeed, λ(A) = 0 ⟺ μ(A) = 0 ⟺ h_{∗}μ(h(A)) ⟺ λ(h(A)) = 0.

By Lemma 2.12 there exists a Markov map F for g, which pushes forward to a Markov map \overline{F} for T by the conjugacy h. Now Fact 2.14 gives unique absolutely continuous invariant probability measures ρ and \overline{ρ} for F, \overline{F} with densities Hölder continuous and bounded away from 0 and ∞. By unicity h_{∗}ρ = \overline{ρ} thus

Dh = \frac{dρ}{d\overline{ρ}}

is Hölder continuous and bounded away from 0 and ∞ on the domain of F. The conjugacy being C^{1} implies multipliers are preserved so by Lemma 2.15 the conjugacy is upgraded to C^{j} for all finite j ≤ r and thus for j = r = ∞ if such is the case. Let U be a connected component of the domain of F, and n > 0. On U,

D(h ∘ g^{n})(x) = Dh(g^{n}(x)) · Dg^{n}(x) = D(T^{n} ∘ h)(x) = t^{n} · Dh(x),

so if Dg^{n}(x) ≠ 0 then

Dh(g^{n}(x)) = \frac{t^{n}}{Dg^{n}(x)}Dh(x)  (3.1)

exists and is non-zero. A similar calculation gives existence and continuity of D^{i}h(g^{n}(x)) where Dg^{n}(x) ≠ 0 for all finite i ≤ r.

We claim that the post-critical set P C = ⋃_{i≥0} g^{i}(g(Crit)) is finite and h is C^{r} on X \ P C. Indeed, by Lemma 2.5 there exists n_{0} such that g^{n_{0}}(U) = X.

Thus h is C^{r} (and indeed strictly positive) on X \ ⋃_{0≤i≤n_{0}} g^{i}(g(Crit)). Because Dh(g^{i}(g(Crit))) = ∞ (upon taking limits) P C ⊂ ⋃_{0≤i≤n_{0}} g^{i}(g(Crit)) as required.

Proof of Theorem 1. One direction is given by the preceding proposition.

According to [2], [3], measures of maximal entropy form a convex set with a finite number of (ergodic) extremal points. A measure of maximal entropy of Hausdorff dimension one therefore has an ergodic component of dimension one.

Thus given the restrictive interval J of period k and μ such that μ(J) > 0 from the statement, we can assume without loss of generality that μ is ergodic. Then the support of μ is ⋃_{i=0}^{k−1} f^{k}(J). The measure μ_{|J} when normalised is the measure of maximal entropy for f_{|J}^{k} : J → J and coincides with the C^{r} pullback of the absolutely continuous measure of maximal entropy for the piecewise linear map. Thus μ is absolutely continuous and HD(μ) = 1.

We remark that from equation (3.1) and the non-flatness of critical points it follows immediately that the singularities are of root type: For each s ∈ P C there exist numbers k_{s}^{−(+)} such that

|log |Dh(x) · |x − s|^{k_{s}^{−(+)}}||

is bounded for x in left(right) neighbourhoods of s.

We now show that if f (and thus g) has no inflection points, then P C = ∂X. This is equivalent to showing that g(Crit ∪ ∂X) ⊂ ∂X.

Lemma 3.2. For each non-renormalisable transitive C^{1} map without inflection points f′ : I → I, for each point y in the interior of I, there exists an infinite sequence y = y_{0}, y_{1}, … satisfying f′(y_{i+1}) = y_{i}, Df′(y_{i}) ≠ 0 and y_{i} ≠ y_{j} for all i ≠ j.

Proof. Firstly note that there exists x in the interior of I with f′(x) = y and Df′(x) ≠ 0, for otherwise there is a contradiction with transitivity and nonrenormalisability (all critical points are turning points). We can in this way construct a sequence which satisfies the conditions of the lemma unless y is part of a periodic orbit without non-critical (or boundary) preimages (other than itself). The partition induced by this periodic orbit defines restrictive intervals, contradicting non-renormalisability.

Lemma 3.3. Suppose g has no inflection points, then P C = ∂X,

Proof. By transitivity, g(Crit) ∪ g^{2}(Crit) ⊃ ∂X. It remains to show P C ⊂ ∂X. Suppose there exists y ∈ P C \ ∂X. Then Dh(y) = Dh(y_{i}) = ∞ for all i, where

y_{i} is given by Lemma 3.2. But from above, the conjugacy is C^{r} on all of X except the finite post-critical set. The contradiction proves the lemma.

In this case all branches are therefore full and a simple calculation gives that if the slope is ±t, then T has t − 1 turning points.

Proof of Corollary 1. It follows immediately from Theorem 1.

Proof of Theorem 2. A unimodal map has no inflection points. Suppose HD(μ) = 1. Then μ is absolutely continuous so the semi-conjugacy h to the symmetric tent map T given by Fact 2.1 collapses no restrictive intervals and h is a conjugacy on I. By unicity of measures of maximal entropy for unimodal maps, μ is the pullback of the absolutely continuous invariant measure for T. Arguing as above there is a restrictive interval containing the critical point c on which h is a C^{r} conjugacy and on which the renormalisation of T is the full tent map T_{2}. Applying Fact 2.4 gives the remainder of the theorem.

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Neil Dobbs Département de Mathématiques, Université Paris-Sud (Orsay), 91405 ORSAY, France

ndobbs@maths.tcd.ie