Topological pressure of simultaneous level sets

0.33 MB·English

by  Vaughn Climenhaga

TOPOLOGICAL PRESSURE OF SIMULTANEOUS LEVEL
SETS
1
VAUGHNCLIMENHAGA
1
0
2 Abstract. Multifractalanalysisstudieslevelsetsofasymptoticallyde-
l fined quantities in a topological dynamical system. We consider the
u topological pressure function on such level sets, relating it both to the
J
pressureontheentirephasespaceandtoaconditionalvariationalprin-

0 ciple. Weusethistorecoverinformationonthetopologicalentropyand
3 Hausdorff dimension of the level sets.

Our approach is thermodynamic in nature, requiring only existence
]
S and uniqueness of equilibrium states for a dense subspace of potential
D functions. UsinganideaofHofbauer,weobtainresultsforallcontinuous
potentials by approximating them with functions from thissubspace.
.
h Thistechniqueallowsustoextendanumberofpreviousmultifractal
t results from the C1+ε case to the C1 case. We consider ergodic ratios
a
m Snϕ/Snψ where the function ψ need not be uniformly positive, which
letsusstudydimensionspectrafornon-uniformlyexpandingmaps. Our
[
results also cover coarse spectra and level sets corresponding to more

1 general limiting behaviour.
v
3
6
0
1. Introduction
0
.
8 A general framework for multifractal analysis of dynamical systems was
0
laid out in [BPS97, Pes98]. Broadly speaking, one begins with the following
1
elements:
1
: (1) a topological dynamical system f: X → X;
v
i (2) a local asymptotic quantity φ(x) that depends on x ∈ X and takes
X
values in Rd, usually in a highly discontinuous manner (typically,
r
a each level set is dense);
(3) a global dimensional quantity that assigns to each level set of φ a
“size” or “complexity”, such as its topological entropy or Hausdorff
dimension.
ThelevelsetsK(α) = {x ∈ X |φ(x) = α}formamultifractal decomposition,
and the function α 7→ dimK(α) is a multifractal spectrum.
There are two approaches to computing multifractal spectra: the ther-
modynamic approach and the saturation approach [Cli11b]. We use the
thermodynamic approach, in which the dimension of the level set K(α) is
evaluated by producing an f-invariant measure µ supported on K(α) with
α
dimµ = dimK(α) as an equilibrium state for the appropriate potential
α
Date: August 2, 2011.
1
2 VAUGHNCLIMENHAGA
function. It was observed in [BS01] that the key tool for this approach is
differentiability of the pressure function. In particular, if the pressure func-
tion is differentiable on the subspace of potentials spanned by the functions
appearinginthedefinitionsofφanddim, thenevery level setK(α) hassuch
a measure µ associated to it.
α
In [BSS02], Barreira, Saussol, and Schmeling extended this approach to
higher-dimensional multifractal spectra, where the local quantity φ takes
values in Rd for some d > 1. More precisely, they considered functions
Φ ={ϕ ,...,ϕ },Ψ = {ψ ,...,ψ } ∈ C(X)d with ψ > 0 and examined the
1 d 1 d i
level sets
ϕ (x)+ϕ (f(x))+···+ϕ (fn(x))
i i i
K(α) = x ∈ X lim = α for all i
(cid:26) (cid:12) n→∞ψi(x)+ψi(f(x))+···+ψi(fn(x)) i (cid:27)
(cid:12)
(cid:12)
for α∈ Rd, obtaining the following result.
Theorem 1.1 ([BSS02, Theorem 8]). Let X be a compact metric space and
f: X → X a continuous map with upper semi-continuous metric entropy.
Suppose ϕ ,ψ ∈ C(X) are such that ψ > 0 for every i and every potential
i i i
in span{ϕ ,...,ϕ ,ψ ,...,ψ } has a unique equilibrium state.
1 d 1 d
LetI(Φ,Ψ) = Rϕ1dµ,..., Rϕddµ |µ ∈ Mf(X) , where Mf(X) isthe
Rψ1dµ Rψddµ
space of f-invarinan(cid:16)t Borel probability(cid:17)measures on Xo. Then K(α) = ∅ for
α∈/ I(Φ,Ψ), while for α∈ intI(Φ,Ψ), we have
d
htopK(α) = inf P qi(ϕi −αiψi) q ∈Rd
( ! )
Xi=1 (cid:12)
(1.1) = max h (f) µ ∈ Mf(X) an(cid:12)(cid:12)d ϕidµ = α for all i
µ i
ψ dµ
(cid:26) (cid:12) R i (cid:27)
= max{h (f)|(cid:12)µ ∈ Mf(X) and µ(K(α)) = 1}.
µ (cid:12) R
Remark 1.2. Infact, Theorem 1.1 is aspecialisation of theresultin[BSS02],
which considers the more general u-dimension introduced in [BS00] in place
of the topological entropy. We state the simpler version here for ease of
exposition.
We prove the following generalisation of Theorem 1.1.
Theorem A. Let X be a compact metric space and f: X → X a continuous
map such that the entropy map Mf(X) → R is upper semi-continuous and
h (f) < ∞. Suppose that there is a dense subspace D ⊂ C(X) such that
top
every φ∈ D has a unique equilibrium state.
Let Φ,Ψ ∈ C(X)d be such that ψ dµ ≥ 0 for all µ ∈ Mf(X) and
i
1 ≤ i ≤ d, with equality only permitted if ϕ dµ 6= 0. Then K(α) = ∅ for
i
R
R
TOPOLOGICAL PRESSURE OF SIMULTANEOUS LEVEL SETS 3
every α ∈/ I(Φ,Ψ), while for every α∈ intI(Φ,Ψ), we have
d
htopK(α) = inf P qi(ϕi −αiψi) q ∈ Rd
( ! )
Xi=1 (cid:12)
(1.2) = sup h (f) µ ∈ Mf(X) and(cid:12)(cid:12) ϕidµ = α for all i
µ i
ψ dµ
(cid:26) (cid:12) R i (cid:27)
= sup{h (f)|(cid:12)µ ∈Mf(X) and µ(K(α)) = 1}.
µ (cid:12) R
Theorem A generalises Theorem 1.1 in two ways.
(1) The result applies to all continuous functions, not just those whose
span lies inside the collection of potentials with unique equilibrium
states. In Theorems 3.5–3.8 in §3.2.2, we use this added generality
to extend multifractal results for various dimension spectra from the
C1+ε case to the C1 case.
(2) We weaken the hypothesis that ψ > 0. In §3.2.3, this lets us obtain
i
results for non-uniformly expanding maps that had previously only
been shown for uniformly expanding maps.
In most previous multifractal literature, the added generality of treating
all continuous functions has only been available using the saturation ap-
proach, which relies on a version of the specification property. Using an
idea of Hofbauer [Hof95, Hof10], we obtain this generality with the thermo-
dynamic approach, provided there is a dense subspace of C(X) comprising
potentials with unique equilibrium states. Broadly speaking, this hypoth-
esis is satisfied for systems satisfying a version of the specification prop-
erty [Bow75, CT11], and so our result should apply to a similar class of
systems as the saturation approach does.
The advantage of the thermodynamic approach over the saturation ap-
proach is that it establishes the conditional variational principle that is the
final equality in (1.2); the saturation approach gives no information on in-
variant measures supported on the level sets [Cli11b].
We observe that the weakening of the hypothesis forces us to replace
the maxima in the last two lines of (1.1) with suprema, which may not be
achieved.
Theorem A is a special case of our main result, Theorem C, which adds
several additional generalisations.
(1) Instead of topological entropy, we study the topological pressure
P (ξ) for ξ ∈ C(X), which carries enough information to de-
K(α)
termine both the topological entropy and the u-dimension (the full
statement of Theorem 1.1 in [BSS02] treats u-dimension). When
ξ = 0, we recover the topological entropy, while for ξ = −tu with
t ∈ R and u ∈ C(X), we can use Bowen’s equation to obtain the
u-dimension and offer a proper generalisation of [BSS02, Theorem
8]; see Theorem 3.3 in §3.1. In particular, when f is conformal this
can be used to compute the Hausdorff dimension.
4 VAUGHNCLIMENHAGA
(2) Weshowthatthequantities in(1.2)arealsoequaltothecoarse mul-
tifractal spectra in (2.10) below, which have not been studied before
in this context. When µ is a Gibbs measure for ξ, the coarse spectra
associated to P (ξ) are intimately related to large deviations es-
K(α)
timates for µ, which were studied in [Kif90] underhypotheses nearly
identical to those in our main theorem.
(3) We replace the level sets K(α) with more general sets K(A) for
A ⊂ Rd (see (2.5)). This allows us to consider points whose asymp-
totic behaviour has limit points inside the fixed set A, but does not
necessarily converge to any fixed α.
With appropriate choices of ϕ and ψ , the results of this paper can be
i i
used to study various well-known multifractal decompositions. We mention
two of the most important examples, which are discussed further in §3.
(1) With ψ ≡ 1, the sets K(α) are level sets for Birkhoff averages of the
i
functionsϕ . Ifµ isaweakGibbsmeasureforϕ ,thentheseBirkhoff
i i i
averages determine the local entropies of µ . If f is conformal and
i
ϕ = logkDfk, we obtain the decomposition in terms of Lyapunov
i
exponents.
(2) For a conformal map f, with ψ = logkDfk and µ a weak Gibbs
i i
measureforϕ ,theratioS ϕ (x)/S ψ (x)convergestothepointwise
i n i n i
dimensionof µ atx, andwe obtain thedecomposition into level sets
i
for pointwise dimension.
Because we consider simultaneous level sets, one can easily consider decom-
positions into sets on which various combinations of the above quantities
take specified values. For the sake of simplicity in exposition, however, we
focus on the case d = 1, where only one quantity is specified. Various new
phenomenaoccur in the case d > 1; thesehave beenwell studiedin [BSS02],
and are not our principal concern here.
Acknowledgments. Part of this research was carried out during a visit to
the Pennsylvania State University; I am grateful for the hospitality of the
mathematics department and my host, Yakov Pesin.
2. Definitions and results
2.1. Topological pressure. LetX beacompactmetricspaceandf: X →
X be continuous. We assume throughout this paper that h (f) < ∞.
top
We recall the definition of the topological pressure P (φ) for φ ∈ C(X)
Z
andZ ⊂ X. For agiven δ > 0and N ∈ N, let P(Z,N,δ) bethecollection of
countablesets {(x ,n )} ⊂ Z×{N,N+1,...}suchthat Z ⊂ B(x ,n ,δ),
i i i i i
where
S
B(x,n,δ) = {y ∈ X |d(fk(x),fk(y)) < δ for all 0≤ k ≤ n}
is the Bowen ball of order n and radius δ centred at x. For each s ∈ R, let
(2.1) m (Z,s,φ,δ) = lim inf exp(−n s+S φ(x )),
P i ni i
N→∞P(Z,N,δ)
(xXi,ni)
TOPOLOGICAL PRESSURE OF SIMULTANEOUS LEVEL SETS 5
where S ϕ(x) = ϕ(x)+ϕ(f(x))+···+ϕ(fn−1(x)). The function m takes
n P
values ∞ and 0 at all but at most one value of s. Writing
PZ(φ,δ) = inf{s ∈ R |mP(Z,s,φ,δ) = 0}
= sup{s ∈ R | mP(Z,s,φ,δ) =∞},
the topological pressure of ϕ on Z is
P (φ) = limP (φ,δ) = supP (φ,δ).
Z Z Z
δ→0 δ>0
In the particular case φ = 0, this definition yields the topological entropy
h (Z) = P (0), as defined by Bowen for non-compact sets [Bow73].
top Z
Remark 2.1. Formally, this definition differs slightly from the one given by
Pesin and Pitskel’ [PP84] (see also [Pes98]). However, as shown in [Cli11a,
Proposition 5.2], the two definitions yield the same value.
Remark 2.2. WeadopttheconventionthatP (φ) = −∞foreveryφ ∈ C(X).
∅
We willalsoneed toconsider thecapacity pressure,whichweintroducein
a slightly more general formulation than is standard. To wit, fix a potential
φ∈ C(X) and a sequence of subsets Zn ⊂ X. For each n∈ N and δ > 0, let
Λ (Z ,φ,δ) = inf eSnφ(x) B(x,n,δ) ⊃ Z .
n n n
( )
xX∈E (cid:12) x[∈E
(cid:12)
Then the lower and upper capacity pressu(cid:12)res of φ on the sequence (Zn) are
1
CP (φ) = lim lim logΛ (Z ,φ,δ),
(Zn) δ→0n→∞n n n
1
CP (φ) = lim lim logΛ (Z ,φ,δ).
(Zn) δ→0n→∞n n n
When the sequence (Z ) is constant – that is, when there is Z ⊂ X such
n
that Z = Z for every n – we recover the capacity pressure as considered
n
in [Pes98]. In particular, when Z = X for all n, we have
n
(2.2) P (φ) = CP (φ) = CP (φ) = P∗(φ),
X X X X
where we write
P∗(φ) = sup h (f)+ φdµ µ ∈ Mf(Z)
Z µ
(cid:26) Z (cid:12) (cid:27)
for the variational pressure on Z ⊂ X. (Here(cid:12)Mf(Z) is the space of all f-
(cid:12)
invariant Borel probability measures supportedon Z.) Whenthe pressureis
evaluated on the whole space X, we will write merely P(φ) for the common
value of the four quantities in (2.2).
For arbitrary Z ⊂ X, one has
(2.3) P∗(φ) ≤ P (φ) ≤ CP (φ) ≤ CP (φ).
Z Z Z Z
The last inequality becomes equality when Z is f-invariant, and all the in-
equalitiesbecomeequalitieswhenZ isbothf-invariantandcompact[Pes98].
6 VAUGHNCLIMENHAGA
For non-compact Z, the second inequality is generally strict, whereas it is
a question of great interest to determine for which sets Z ⊂ X we have
P∗(φ) = P (φ).
Z Z
2.2. Multifractal decompositions and spectra. Fix continuous func-
tions ϕ ,...,ϕ and ψ ,...,ψ ∈ C(X). Write Φ = (ϕ ,...,ϕ ) ∈ C(X)d,
1 d 1 d 1 d
and similarly for Ψ. Given x ∈ X and n ∈N, let
S ϕ (x) S ϕ (x)
(2.4) An(x) = n 1 ,..., n d ∈ Rd.
S ψ (x) S ψ (x)
(cid:18) n 1 n d (cid:19)
Let A (x) denote the set of limit points of the sequence (A (x)), and given
∞ n
A⊂ Rd, let
(2.5) K(A) = {x ∈ X | A (X) ⊂ A}.
∞
Given α∈ Rd, write K(α) = K({α}) for the case when A is the single point
α, and observe that
S ϕ (x)
n i
(2.6) K(α) = x ∈ X lim = α for all i .
i
(cid:26) (cid:12) n→∞Snψi(x) (cid:27)
Furthermore, we have K(A′) ⊂(cid:12) K(A) whenever A′ ⊂ A, and in particular
(cid:12)
K(α) ⊂ K(A) whenever α ∈A.
Remark 2.3. We do not consider in this paper the exceedingly interesting
question of studying the level sets K′(A) = {x |A (x) = A} when A is not
∞
a singleton. These sets supportno invariant measures and hence require the
saturation approachratherthan thethermodynamicapproachof thispaper.
The (fine) pressure spectrum for simultaneous level sets is the function
F: P(Rd)×C(X) → R∪{−∞} given by
(2.7) F(A,ξ) = P (ξ),
K(A)
whereP(Rd)denotesthecollectionofallsubsetsofRd. WewillwriteF(α,ξ)
for the case when A is a singleton.
We also consider the coarse pressure spectrum for simultaneous level sets,
defined as follows. Given an open set U ⊂ Rd, consider the sets
(2.8) G (U) = {x ∈ X | A (x) ∈ U}.
n n
Thus for n ∈ N, δ > 0, and ξ ∈ C(X), we have
(2.9) Λ (G (U),ξ,δ) = inf eSnξ(x) B(x,n,δ) ⊃ G (U) .
n n n
( )
xX∈E (cid:12) x[∈E
(cid:12)
Then the lower and upper coarse spectra ar(cid:12)e
1
F(A,ξ) = inf CP (ξ) = inf lim lim logΛ (G (U),ξ,δ),
U⊃A (Gn(U)) U⊃Aδ→0n→∞n n n
(2.10)
1
F(A,ξ) = inf CP (ξ) = inf lim lim logΛ (G (U),ξ,δ),
U⊃A (Gn(U)) U⊃Aδ→0n→∞n n n
TOPOLOGICAL PRESSURE OF SIMULTANEOUS LEVEL SETS 7
where the infima are taken over all open sets U containing A. As with the
fine spectrum, we will write F(α,ξ) = F({α},ξ), and similarly for F.
TheutilityofthecoarsespectraF andF,whichareoftenneglected inthe
multifractal literature, is immediately demonstrated by the following result.
Proposition 2.4. If A ⊂ Rd is compact, then for every ξ ∈ C(X) we have
(2.11) F(A,ξ) = supF(α,ξ).
α∈A
Remark 2.5. Proposition 2.4 is reminiscent of the standard result on count-
ablestabilityofpressuregivenin[Pes98,Theorem11.2(3)], whichstatesthat
P (ξ) = sup P (ξ) for any countable union. However, the present re-
SnZn n Zn
sult applies to F, not to F; furthermore, it is stronger than that statement
in two ways:
(1) there may be uncountably many values of α, and
(2) we typically have K(α) $ K(A), due to the existence of points
α
for which A (x) is not a singleton.
∞
S
2.3. Conditional variational principles and predicted spectra. In
light of the classical variational principle and (2.3), it is reasonable to con-
sider the conditional variational pressure
(2.12) T(α,ξ) = P∗ (ξ) = sup h (f)+ ξdµ µ ∈Mf(K(α)) .
K(α) µ
(cid:26) Z (cid:12) (cid:27)
Weobservethatusingtheergodicdecomposition,the(cid:12)valueofthesupremum
(cid:12)
f
is unchanged if we restrict to the collection M (K(α)) of ergodic measures
E
supported on K(α).
WealsowriteT(A,ξ) =P∗ (ξ);bytheaboveobservationandBirkhoff’s
K(A)
ergodic theorem, we see immediately that T(A,ξ) = sup T(α,ξ).
α∈A
Our main results relate both T and the pressure spectra to the pressure
functiononthewholespaceX. Givenβ ∈ Rd andΞ = (ξ1,...,ξd)∈ C(X)d,
we write
d
β ∗Ξ = (β ξ ,...,β ξ )∈ C(X)d, hβ,Ξi = β ξ ∈ C(X),
1 1 d d i i
i=1
X
and consider the predicted spectrum S: Rd×C(X) → R given by
(2.13) S(α,ξ) = inf{P(hq,Φ−α∗Ψi+ξ) |q ∈ Rd}.
As with T, we write S(A,ξ) = sup S(α,ξ).
α∈A
For some values of α ∈ Rd and ξ ∈ C(X), we may have S(α,ξ) = −∞.
Consider the set
(2.14) I′(Φ,Ψ) = {α ∈ Rd | S(α,0) > −∞},
and observe that S(α,0) > −∞ if and only if S(α,ξ) > −∞ for every
ξ ∈ C(X);thisfollowsbecause|P(φ)−P(φ′)| ≤ kφ−φ′kforallφ,φ′ ∈ C(X).
We show in Proposition 2.8 below that under mild conditions on Φ and
Ψ we have I′(Φ,Ψ) = I(Φ,Ψ), where the latter is defined in Theorem 1.1.
8 VAUGHNCLIMENHAGA
2.4. Results. We begin with a series of inequalities reminiscent of (2.3)
that hold quite generally and relate the quantities introduced so far.
Theorem B. For every compact A ⊂ Rd and ξ ∈ C(X), we have
(2.15) T(A,ξ) ≤ F(A,ξ) ≤ F(A,ξ) ≤ F(A,ξ) ≤ S(A,ξ).
In particular, if α ∈ Rd is such that S(α,ξ) = −∞ for some (and hence
every) ξ ∈ C(X), then K(α) = ∅.
Remark 2.6. The requirement that A be compact is only needed for the last
inequality in (2.15). The other inequalities hold for all A⊂ Rd.
TheoremBholdswithoutany hypothesesonthedynamicalsystem(X,f)
orthefunctionsΦ,Ψ,ξ beyondcompactnessofX andcontinuityoff,Φ,Ψ,ξ.
InTheoremCbelow,wegiveconditionsunderwhichequalityholdsin(2.15).
Every measure in Mf(K(α)) satisfies the condition (Φ−α∗Ψ)dµ = 0.
For ergodic measures with ψ dµ 6= 0 for all i, the converse is true as well:
i
R
if the integral vanishes then µ(K(α)) = 1. Thus we consider potentials
R
satisfying the following condition:
(Q): ψ dµ ≥ 0 for every µ ∈ Mf(X) and 1 ≤ i ≤ d, and the
i
inequality is strict whenever ψ dµ =0.
i
R
If Φ and Ψ satisfy (Q), then any µ ∈ Mf(X) with (Φ−α ∗Ψ)dµ = 0
R
for some α ∈ Rd must have ψidµ > 0 for all i, whenRce one can show that
µ(K(α))= 1 whenever µ is ergodic.
R
Remark 2.7. Observe that (Q) is automatically satisfied if ψ > 0 for all i;
i
this is the case considered in [BSS02].
We will see that taking a supremum over (not necessarily ergodic) mea-
suressatisfyingtheintegralconditiongivesanotherwayofcomputingS(α,ξ),
with the possible exception of α ∈ ∂I(Φ,Ψ). Given α∈ Rd, let
(2.16) Mf(X) = µ ∈ Mf(X) (Φ−α∗Ψ)dµ = 0 .
α
(cid:26) (cid:12) Z (cid:27)
Let Tˆ: Rd×C(X) → R∪{−∞} be g(cid:12)(cid:12)iven by
(2.17) Tˆ(α,ξ) = sup h (f)+ ξdµ .
µ
µ∈Mfα(X)(cid:18) Z (cid:19)
As with T and S, we write Tˆ(A,ξ) = sup Tˆ(α,ξ).
α∈A
Proposition 2.8. For every α ∈ Rd and ξ ∈ C(X), we have
(2.18) T(α,ξ) ≤ Tˆ(α,ξ) ≤ S(α,ξ).
If Φ and Ψ satisfy (Q), then
ϕ dµ ϕ dµ
(2.19) I′(Φ,Ψ) =I(Φ,Ψ) = 1 ,..., d µ ∈ Mf(X) ,
ψ dµ ψ dµ
(cid:26)(cid:18)R 1 R d (cid:19) (cid:12) (cid:27)
(cid:12)
R R (cid:12)
TOPOLOGICAL PRESSURE OF SIMULTANEOUS LEVEL SETS 9
and for every ξ ∈ C(X) and α∈ Rd\∂I(Φ,Ψ), we have
(2.20) Tˆ(α,ξ) = S(α,ξ).
Furthermore, for every α ∈ intI(Φ,Ψ) there exists R > 0 such that every
kqk ≥ R has P(hq,Φ−α∗Ψi+ξ)> S(α,ξ). In particular, this implies that
the infimum in the definition of S(α,ξ) is achieved for some kqk ≤ R.
Remark 2.9. In light of Proposition 2.8, it is tempting to try to fit Tˆ into
the series of inequalities in (2.15) by conjecturing that F ≤ Tˆ. We show in
§3.4 that for α∈ ∂I(Φ,Ψ), this is not necessarily the case.
All our results up to this point assumed only that X is a compact metric
space, f: X → X is a continuous map with finite topological entropy, and
ϕ ,ψ ,ξ are all continuous. Our main result gives further conditions under
i i
which all the quantities in (2.15) and (2.18) are equal.
Theorem C. Let X be a compact metric space and f: X → X a continuous
map such that the entropy map Mf(X) → R is upper semi-continuous and
h (f) < ∞. Suppose that there is a dense subspace D ⊂ C(X) such that
top
every φ∈ D has a unique equilibrium state.
Let Φ,Ψ ∈ C(X)d satisfy (Q). Then equality holds in (2.15) and (2.18)
for every compact A ⊂ intI(Φ,Ψ). That is, for such an A the pressure
function P : C(X)→ R is given by
K(A)
P (ξ) = inf CP (ξ) = inf CP (ξ)
K(A) U⊃A (Gn(U)) U⊃A (Gn(U))
= sup inf P(hq,Φ−α∗Ψi+ξ)
α∈Aq∈Rd
(2.21)
= supsup h (f)+ ξdµ µ ∈ Mf(X)
µ α
α∈A (cid:26) Z (cid:12) (cid:27)
(cid:12)
= supsup h (f)+ ξdµ(cid:12)µ ∈ Mf(K(α)) .
µ
α∈A (cid:26) Z (cid:12) (cid:27)
(cid:12)
In particular, we have PK(A)(ξ) = supα∈APK(α)(ξ(cid:12)) for every compact A ⊂
intI(Φ,Ψ) and ξ ∈ C(X).
We emphasise that we do not require ϕ ,ψ ,ξ to lie in D; in particular,
i i
the pressure function may not be differentiable on the span of these func-
tions. This is a stronger result than was obtained in prior thermodynamic
approaches to multifractal analysis, with the exception of Hofbauer’s work
on piecewise monotonic transformations [Hof95, Hof10], from which the key
ideas in the proof of (2.21) for ϕ ,ψ ,ξ ∈/ D are derived. (A similar criterion
i i
was used in [Kif90] to derive large deviations results.)
The key to this strengthening is the following fact. Suppose φ ∈ C(X)
q
is a continuously varying family of potentials. Then using the variational
principle P(φ ) = sup{h (f) + φ dµ | µ ∈ Mf(X)} and the fact that
q µ q
Mf(X) is convex, one can obtain for every δ > 0 a family νδ of measures
R q
10 VAUGHNCLIMENHAGA
such that
(2.22) h (f)+ φ dνδ > P(φ )−δ.
νδ q q q
q
Z
However, themeasuresνδ areingeneralnotergodic. Onecanobtainergodic
q
νδ satisfying (2.22) at the cost of losing continuity in q; the key consequence
q
of the hypothesis on the subspace D ⊂ C(X) in Theorem C is that it
allows us to choose a family of measures νδ satisfying both ergodicity and
q
continuous dependence on q.
Wereiterate thatwhiletherelationshipsin(2.21)arewell-known inmany
cases, the following aspects of Theorem C are new, as discussed in the in-
troduction.
(1) The results apply to all continuous potentials, not just those whose
span lies in D.
(2) The denominators ψ need not be uniformly positive, which allows
i
us to treat non-uniformly expanding systems.
(3) By obtaining a result for the topological pressure, we have enough
information to recover both entropy and Hausdorff dimension for
conformal maps.
(4) Coarse spectra are also included.
(5) The case where A is not a singleton is covered.
Remark 2.10. It is worth pointing out that when d> 1 the domain I(Φ,Ψ)
maynotbetheclosureofitsinterior. Thisphenomenonisdiscussedindetail
in [BSS02, Bar08].
Remark 2.11. A direct corollary of Theorem C is the following. Suppose
(X,f) has upper semi-continuous entropy and finite topological entropy,
and suppose that X ⊂ X are compact f-invariant sets such that
n
(1) lim P (φ) = P (φ) for every φ ∈ C(X);
n→∞ Xn X
(2) there exist dense subspaces D ⊂ C(X ) such that every φ ∈ D
n n n
has a unique equilibrium state on (X ,f| ).
n Xn
Then if Φ,Ψ are as in Theorem C, the result of Theorem C still holds.
Remark 2.12. As mentioned in Remark 2.3, it is a very interesting question
to study what happens when we replace the sets K(A) from (2.5) with the
sets K′(A) = {x ∈ X | A (x) = A}. It is known [Ols03, PS07, GR09]
∞
that in the case ξ = 0, we have h (K′(A)) = inf S(α,0) provided the
top α∈A
system has some specification-like properties and A ⊂ I(Φ,Ψ) is connected
and compact.
BecausethelevelsetK′(A)doesnotsupportanyinvariantmeasureswhen
A has more than one element, the sets K′(A) cannot be fully studied using
the thermodynamic approach in this paper. The desired upper boundcould
perhaps be produced using general thermodynamic arguments, but a lower
bound requires the saturation approach.
We observe that it is not clear what (if anything) plays the role of the
coarse spectra in this situation.