Previous work

We now outline some relevant prior results, including the work of Walters [W] and Kifer, Peres and Weiss [KPW], which are important to the story of the dimension gap problem.

We observe that by the Kolmogorov-Sinai theorem, the measure-theoretic en- tropyh(µp) has the simple formh(µp) =−P∞_n=1pnlogpn. We define theLyapunov

exponent of an ergodic measureµ(with respect to the map T) by

χ(µ) =

Z

log|T0|dµ

which measures the amount of expansion (or contraction) in the system from the point of view of the measureµ.

In 1966 Kinney and Pitcher [KP2] first proved that the dimension of any projected Bernoulli measure for the Gauss map was given by the formula

dimµp = −P∞ n=1pnlogpn −R 2 logxdµp(x) (3.9)

provided that the entropyh(µp) =−P_n∞₌₁pnlogpn <∞. Notice that is not clear

from (3.9) whether or not dimµp is less than 1. (3.9) is now known to be a specific

example of the more general result which says that for an ergodic invariant measure with finite entropy we have the following closed-form formula for the dimension, which links the dimension of the measure with the entropy and Lyapunov exponent of the measure (see for instance Theorem 4.4.2 in [MU1]).

Proposition 3.2.1 (Volume Lemma). If µ is an ergodic T-invariant probability measure on[0,1] and h(µ)<∞ then the Hausdorff dimension of µis given by

dimµ= h(µ)

χ(µ).

It is also a classical result that the dimension of the repeller J is encoded as the zero of the pressure. This result in our setting was proved by Mauldin and Urba´nski, see for instance Theorem 3.15 in [MU3].

Proposition 3.2.2 (Bowen-Manning-McCluskey formula). Let J be the repeller of

T. Then λ= dimJ satisifes P(−λlog|T0|) = 0.

In 1978, Walters [W] developed the thermodynamic formalism of countable branch expanding maps, where he proved a generalised Ruelle-Perron-Frobenius theorem for countable branch maps and potentials with sufficient regularity. He used this to prove the existence of a unique Gibbs state µT for the potential −log|T0|.

Moreoever he proved that µT satisfied a variational principle and was the unique

equilibrium state for −log|T0|, that is, the unique absolutely continuous measure for the system. This means that for all invariant probability measuresµ6=µT with R log|T0|dµ <∞, h(µ)− Z log|T0|dµ < h(µT)− Z log|T0|dµT =P(−log|T0|) = 0

where the last equality follows by Proposition 3.2.2. Notice that this implies that for any invariant measureµfor which R log|T0|dµ <∞, then

dimµ= h(µ)

with equality if and only if µT =µ. Therefore if µT 6=µp for anyp ∈ P, then pro-

videdh(µp)<∞, Walters’ work implies that dimµp <1. No further quantitative

information about the size of dimµp can be obtained via this approach.

The next major breakthrough was in 2001, when Kifer, Peres and Weiss [KPW] showed that under some additional assumptions on the mapT, sup_p∈Pdimµp <

1−ψ for some constant ψ that could be made explicit. In particular, under the assumptions that

1. there exists somes <1 for which

X

n∈N

|In|s <∞,

2. the absolutely continuous measureµT is not a Bernoulli measure,

they proved that there existed adimension gap, that is, sup_p∈Pdimµp<1−ψ for

some constant ψ > 0. Importantly, their formula held even when µp had infinite

entropy.

In particular, they applied their results to the Gauss map, and obtained that sup

p∈P

dimµp<1−10−7.

They also gave a characterisation of when one is in the setting thatµT is not

Bernoulli: they showed thatµT is Bernoulli if and only if

F◦T◦F−1 is linear (3.10) whereF is the diffeomorphism F(t) =µ([0, t]).

Their proof was separated into two cases, dependent on whether the entropy of the measure was finite or infinite. The infinite entropy case was tackled by looking at ‘short expansion intervals’ and the finite entropy case was tackled by looking at sets of large deviations for the frequency of certain digits from the one provided by

µT. We will provide the proof for infinite entropy measures in section 3.4, and for

now we describe their proof for the case where the entropy is finite.

The proof for the finite entropy case follows from looking at the dimension of sets of points whose symbolic coding sees a frequency of a certain digit appearing which differs from the one corresponding to the absolutely continuous measureµT.

Bernoulli measure. For a finite wordw∈Σ∗ and δ >0, fix Γδ_w= ( x∈(0,1) : lim sup n→∞ 1 n n−1 X i=0 1w(Tix)−µT(Iw) > δ ) .

Observe that ifw∈Σ∗ is some word for which |µp(Iw)−µT(Iw)|> δ

then by the Birkhoff Ergodic Theorem,µp(Γδw) = 1 and so dimµp 6dim Γδw.

SinceµT is not Bernoulli, there exists some worda∈Σ∗for whichµT(Iaa)6=

(µT(Ia))2. Let

δT =

|µT(Iaa)−(µT(Ia))2|

3 >0. Fixp and put δ=|µp(Ia)−µT(Ia)|. Then it follows that

|µT(Iaa)−(µp(Ia))2| > |µT(Iaa)−(µT(Ia))2| − |(µT(Ia))2−(µp(Ia))2|

= |µT(Iaa)−(µT(Ia))2| − |µT(Ia)−µp(Ia)||µT(Ia) +µp(Ia)|

> 3δT −2δ.

Therefore, ifδ < δT, dimµp 6dim ΓδaaT and if δ >δT, dimµp 6dim Γδa 6dim ΓδaT.

Therefore

sup

p∈P

dimµp 6max{dim ΓδTaa,dim ΓδTa }. (3.11)

Kifer, Peres and Weiss then proved that for anyδ >0 sup

w

dim Γδ_w<1

which in light of (3.11) implies the existence of a dimension gap.

In [KPW] the authors also present analogous results fork-step Markov mea- sures. More recently, Rapaport [Rap] extended the work of [KPW] to the non- stationary case, to show that there is a uniform dimension gap for all measures with respect to which the digits of thef-expansion are independent but not necessarily i.i.d. Notice that these measures will no longer be invariant.