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
S
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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NEIL DOBBS
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 , > 0.
– There are only a finite number of critical points ai , 1 ≤ i ≤ n, Df (ai ) = 0.
– There exist positive numbers ki− (ki+ ) such that
|Df (x)| log
−(+) |x − ai |ki
is bounded in a left (right) neighbourhood of ai - 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+ (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(6= 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
n
point. The map f|J
: 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.
k
– There exists a restrictive interval J of period k such that f|J
has finite post-
critical 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 Ts : [0, 1] → [0, 1], for 0 < s ≤ 2, is defined by
0≤x≤
Ts (x) = sx,
1
< x ≤ 1.
2
Ts (x) = s(1 − x),
If 1 < s ≤
1
,
2
√
2, the tent map Ts 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);
k
2
– Let f|J
: J → J be the resultant non-renormalisable map. Then there is a C r
k
2
conjugacy h on the interior of J from f|J
to the full tent map T2 ;
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 Ts 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 T2 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 ht (f ) and let s = exp(ht (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 6= 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
√
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.
√
k
(ii) If integer k is such that 2 < s2 ≤ 2 then T has exactly k restrictive
intervals Ik ⊂ Ik−1 ⊂ . . . ⊂ I0 = I containing the turning point. Ii has
period 2i for i = 0, 1, . . . , k. For each x ∈ I there exists an integer n such
that either T n (x) ∈ Ik or such that T n (x) is one of the k periodic orbits
that remain outside of the forward orbit of Ik .
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NEIL DOBBS
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 Wi = f 2k(i+1) (V ) \
f 2k(i) (V ). For i sufficiently large one has Wi = ∅. This follows from the finiteness
of the number of critical points, that the Wi are disjoint and that the map is
expanding: If Wi ∩ Crit = ∅ then |Wi+1 | > |Wi |.
S
Sk−1
Let k-periodic V∗ = i≥1 f 2ki (V ). By transitivity I ⊂ j=0 f j (V∗ ). Thus some
iterate of U contains a fixed point p0 . One restarts the process to find 2-periodic V∗0
and concludes by non-renormalisability that V∗0 = 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
Z
hν (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+ maps f with an ergodic, invariant, probability measure ν states
that
HD(ν) = R
hν
,
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.
7
VISIBLE MEASURES OF MAXIMAL ENTROPY
Definition 2.8. Given a C 1+ function f , we say a map F :
S
i
Ui → U is a
Markov map induced by f if U is a subinterval of I, Ui ⊂ U are disjoint intervals
and
– λ(U \
S
i
Ui ) = 0
– for all i, F (Ui ) = U and F|Ui = f ni for some ni
– 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 ,
DF n (x)
<K
DF n (y)
(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:
i
1
D log |DF (x)| < L,
i
|DF (x)|
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 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
R
µ with positive Lyapunov exponent. Suppose 0 < χ < log |Df |dµ. Then there exist
on Y four measurable functions α, β, γ, γ and a constant K such that:
– α > 0, 1 < β < ∞, 0 < γ < γ < ∞, µ-almost everywhere;
– let y = (x, z) and |t| < α(y); the point yt = (x + t, z) lies in Y ;
– |Π(f −n y) − Π(f −n yt )| ≤ β(y) exp(−χn)|t|;
– for any n,
Df (Π(f −n y))
1
≤
≤ K;
K
Df (Π(f −n yt ))
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–
γ(y) ≤ Π∞
n=1
Df (Π(f −n y))
≤ γ(y).
Df (Π(f −n yt ))
Proposition 2.11. Supplementarily to Fact 2.10, there exists measurable γ1 <
∞ µ-almost everywhere such that for all n
| log Df n (Πf −n y) − log Df n (Πf −n yt )| < γ1 (y)|t| ,
( being the Hölder exponent). If f is also C r , for some integer r ≥ 2, then there
exists a measurable function γr with γr (y) < ∞ µ-almost everywhere such that for
all n and all i = 1, 2, . . . , r − 1,
1
|Df n (Πf −n y)|
i
D log |Df n (Πf −n y)| ≤ γr (y).
(2.2)
Proof.
log Df n (Πf −n y) =
n−1
X
log Df (Πf −n+j y) =
j=0
n
X
log Df (Πf −j y)
j=1
and
0 Df (x) − Df (x0 ) ≤ C |x − x | ,
| log Df (x) − log Df (x )| = log 1 + Df (x0 )
|Df (x0 )|
0
for some constant C. Thus
| log Df n (Πf −n y) − log Df n (Πf −n yt )| ≤ C
n
X
|Π(f −j y) − Π(f −j yt )|
j=1
|Df (Πf −j y)|
Integrability of the Lyapunov exponent and ergodicity imply that
.
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 yt )| ≤ β(y) exp(−χj)|t|; thus almost everywhere we are
summing an exponentially decreasing sequence and there exists a function γ1 , γ1 <
∞ almost everywhere, such that
n
X
|Π(f −j y) − Π(f −j yt )|
j=1
|Df (Πf −j y)|
≤
∞
X
|Π(f −j y) − Π(f −j yt )|
j=1
|Df (Πf −j y)|
≤ γ1 (y)|t| ,
completing the first part of the proof.
For i ≥ 1 we can expand D i log |Df n | as a polynomial
n−1
X D2 f (f j x) D3 f (f j x)
Di f (f j x)
,
,
.
.
.
,
Di log |Df n (x)| =
Pi
Df (f j x) Df (f j x)
Df (f j x)
j=0
where Pi is some (calculable) polynomial of order i.
9
VISIBLE MEASURES OF MAXIMAL ENTROPY
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) =
1
C
n−1
min(1, |Df (x)|), and set δn (x) = minj=0
δ(f j x). If Ki is the product
of the number of terms of the polynomial Pi with the maximum of moduli of the
coefficients, then
X 1
i
n−1
1
1
D log |Df n (x)| ≤
P
,
,
.
.
.
,
j
j
j
δ(f x) δ(f x)
δ(f x) j=0
≤
n−1
X
Ki δn (x)−i
j=0
≤ nKi δn (x)−i .
As before, µ-almost everywhere δn (Πf −n y) is bounded by a subexponentially decreas
ing sequence, so µ-almost everywhere Di log |Df n (Πf −n y)| is bounded by a subex-
ponentially 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 γr (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 α0 > 0, β0 < ∞, K < ∞
such that A = {y ∈ Y : α(y) > α0 , β(y) < β0 , γ(y) < K, γ1 (y) < K, γr (y) < K}
has positive measure, µ(A) > 0. Let x ∈ I be a density point of (Π|A )∗ µ, and set
Ax = A ∩ Π−1 x. Let U 3 x be a regularly returning interval with |U | <
α0
2 .
By ergodicity almost every point y ∈ Y accumulates on Ax . Thus for almost every
y there exist iterates f n y = (x0 , z) with x0 ∈ U and (x, z) ∈ Ax with n arbitrarily
large. By Fact 2.10, since α((x, z)) ≥ α0 the interval U can be pulled back along
the branch defined by z with bounded distortion to an interval containing Πy, i.e.
for µ-almost every y there exist ny and interval Vy 3 Πy such that f ny maps Vy
diffeomorphically onto U with bounded distortion. Taking ny large (and thus Vy
small) gives expansion: |Df ny | > δ > 1 on Vy .
For y, y 0 their respective intervals Vy , Vy0 are disjoint or nested since U is regularly
returning. A set of full µ-measure projects to a set of full µ-measure, so µ-almost
10
NEIL DOBBS
every point in U is contained in some Vy . Take a disjoint collection of full measure
S
of intervals Vy and define F on Vy 3 x by
F|Vy (x) = f ny (x).
Now
DF (x)
DF (x0 )
0 |
< K 2 |x−x
if x, x0 ∈ Vy and a standard summation argument gives
|Vy |
the distortion bound (2.1) so F is a Markov map induced by f .
In the C r case we have |DF1(x)| Di log |DF (x)| < K for all x ∈ Vy for all
connected components Vy of the domain of F , and for i = 1, 2, . . . , r − 1. Since
|DF (x)| > 1, we conclude that
as required.
i
1
D log |DF (x)| < K
i
|DF (x)|
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
µ. The proof shows that, on the partition element η(y) containing y, the density of
1
the Rohlin decomposition is bounded inside [ θ(y)
, θ(y)] say, where θ is a measurable
function with 1 < θ(y) < ∞ on a set (the set As 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 )∗ µ. Then at least one of
{y ∈ B : Π(η(y)) ⊃ (x− δ2 , x)}, {y ∈ B : Π(η(y)) ⊃ (x, x+ 2δ )} is of positive measure
ν > 0 say. Projecting, on at least one of (x − 2δ , x), (x, x + δ2 ) the density ρ of µ is
bounded from below by
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 ε0 > 0. By ergodicity,
f is transitive on the support of µ. An application of Lemma 2.5 provides an n
Sn
such that i=0 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
VISIBLE MEASURES OF MAXIMAL ENTROPY
11
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
S
Markov partition Ui , 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
k
period k on which f|J
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
Sk−1
support i=0 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 0 , on which f is transitive.
Let X be a component of X 0 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 = sn the entropy of g is log t.
12
NEIL DOBBS
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 µ by Fact 2.7, thus HD(µ) = 1.
For the map T the Lyapunov exponent is log t. By the Dynamical Volume Lemma
the measure-theoretical entropy hµ = log t and so µ maximises entropy. By unicity
h∗ µ = 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 F for T by the conjugacy h. Now Fact 2.14 gives unique absolutely
continuous invariant probability measures ρ and ρ for F, F with densities Hölder
continuous and bounded away from 0 and ∞. By unicity h∗ ρ = ρ thus
Dh =
dρ
dρ
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) = tn · Dh(x),
so if Dg n (x) 6= 0 then
Dh(g n (x)) =
tn
Dh(x)
Dg n (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) 6= 0 for all finite i ≤ r.
We claim that the post-critical set P C =
S
i≥0
g i (g(Crit)) is finite and h is
C r on X \ P C. Indeed, by Lemma 2.5 there exists n0 such that g n0 (U ) = X.
VISIBLE MEASURES OF MAXIMAL ENTROPY
13
Thus h is C r (and indeed strictly positive) on X \ 0≤i≤n0 g i (g(Crit)). Because
S
Dh(g i (g(Crit))) = ∞ (upon taking limits) P C ⊂ 0≤i≤n0 g i (g(Crit)) as required.
S
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
Sk−1
the support of µ is i=0 f k (J). The measure µ|J when normalised is the measure
k
of maximal entropy for f|J
: 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 ks
such that
−(+) log |Dh(x) · |x − s|ks |
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 0 : I → I, for each point y in the interior of I, there exists an infinite
sequence y = y0 , y1 , . . . satisfying f 0 (yi+1 ) = yi , Df 0 (yi ) 6= 0 and yi 6= yj for all
i 6= j.
Proof. Firstly note that there exists x in the interior of I with f 0 (x) = y
and Df 0 (x) 6= 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
14
NEIL DOBBS
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(yi ) = ∞ for all i, where
yi 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 T2 . 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
[email protected]