Just as for the alternating Lüroth mapLαH and the classical (non-alternating) Lüroth mapLαfH, we could consider a non-alternating version of the map Lα. To do this, define the partition
e
α :={[t2,t1],[tn+1,tn):n≥2}, where thetns are the same as for the partitionα, and then define
the mapLαe :U →U by setting Lαe(x):= (x−tn+1)/an forx∈[tn+1,tn), forn≥2; (x−t2)/a1 forx∈[t2,t1]; 0 forx=0.
In other words, the mapLαe has all positive slopes instead of all negative slopes.
The non-alternatingαe-Lüroth mapLαe also generates a series expansion of the numbers inU. In this case, the expansion is given by
x= ∞
∑
n=1 (∏
i<n aℓi ) tℓn+1=tℓ1+1+aℓ1tℓ2+1+aℓ1aℓ2tℓ3+1+. . . .We write [ℓ1, ℓ2, ℓ3, . . .]αe. This expansion can be finite or infinite, but just as for the classical Lüroth map, every finite expansion can alternatively be written as an eventually periodic expan- sion with periodic part consisting of infinitely many 1’s. The convergents and cylinder sets in
this non-alternating case are defined in the obvious way. It is clear that the Lebesgue measure of the cylinder setCαe(ℓ1, . . . , ℓn)is equal to itsLα counterpart.
It is also possible to consider a non-alternating version of the α-Farey map. The map Fαe would again be defined on the partitionαe, the difference being that the right-hand branch would have a positive slope. The definition of the left-hand remains basically the same. That is,
Fαe(x):= {
(x−t2)/a1 forx∈[t2,1]; an−1(x−tn+1)/an+tn forx∈[tn+1,tn).
Concerning the topological properties of Fαe, if we replace the tent map FαD by the binary expansion mapFαfD:x7→2x( mod 1), then the mapsFαe andFαfDare topologically conjugate, via the conjugating homeomorphismθαe which is given by
θαe(x):= ∞
∑
k=1
2−∑ki=1ℓei,
where the ℓei are now the entries of the expansion ofx with respect to Lαe. The function θαe is
Chapter 3
Ergodic theoretic properties of
F
α
and
L
α
In this chapter we will investigate various measure theoretic and ergodic theoretic properties of the maps Lα and Fα. It turns out that the Lebesgue measure is invariant for every map Lα and that there exists a unique Lebesgue-absolutely continuous invariant measure forFα. We will give an exact expression for the density of this measure. Also, we prove that bothLα andFα are exact, and thus ergodic. First, we give some background material useful for the results of the following sections.
3.1
Measure and ergodic theoretic preliminaries
In this section, we give an outline of the main results we need in this and the following chapters. Whilst there are many available references dealing with various aspects of ergodic theory, the main reference used here is Walters [81]. Throughout, let(X,B)be a measurable space.
Definition 3.1.1. A measure µ is said to be invariant for a map T :X →X provided that for everyµ-measurable setB⊂X, we haveµ◦T−1(B):=µ(T−1(B)) =µ(B). We also say that the mapT preservesthe measureµ.
In practice, it could be difficult to check that a map preserves a given measure using only this definition, as it is often the case that no specific information is known about a general measurable set. However, it is enough to have knowledge of a particular semi-algebra that generatesB, as the following theorem shows.
Theorem 3.1.2. Suppose that(X,B,µ)is a measure space, T:X→X is a map and also suppose thatS is a generating semi-algebra forB. Then, ifµ◦T−1(B) =µ(B)for every set B∈S, we have that the map T preserves the measureµ.
Proof. See Theorem 1.1 in [81].
Note that if the given measurable space is the unit interval U, then the collection S of all sets of the form [0,b] and(a,b]with 0 ≤a<b≤1 is a generating semi-algebra for the Borel σ-algebra of subsets ofU.
Definition 3.1.3. LetT :X→X be a transformation and letµ be aT-invariant Borel probability measure. ThenT is said to beergodic with respect toµ provided that wheneverA∈B is such thatT−1(A) =Awe have thatµ(A) =1 or µ(A) =0. In other words, an ergodic transformation only has trivial invariant subsets.
The first major result in ergodic theory was proved in 1931 by G.D. Birkhoff. There are now various proofs available, the reader is referred to either [81], [16] or, for the proof usually known as the “non-standard” one, the paper of Kamae and Keane [41]. We shall state it here in the simplest case of an ergodic finite measure-preserving system.
Theorem 3.1.4. Birkhoff’s Ergodic Theorem. Suppose that T :(X,B,µ)→(X,B,µ)is an ergodic measure-preserving transformation and also suppose that µ(X)<∞. Let f be a µ- integrable function. Then, forµ-a.e. x∈X , we have
lim n→∞ 1 n n−1
∑
j=0 f ◦Tj(x) = ∫ f dµ.Definition 3.1.5. A subsetW of X is said to be a wandering set for a map T :X →X if the collection{T−n(W):n≥0}is pairwise disjoint. In particular, this implies that
∞
∑
n=0
1W◦Tn≤1.
Definition 3.1.6. A mapT :X →X is said to be conservativeif every wandering set forT has measure zero.
The following theorem is a useful way of determining that a given transformation is con- servative. The proof is not too long or complicated, so we include it here for completeness. Throughout, we will use the notation “mod µ” to indicate that two sets are equal up to a set of µ-measure zero.
Theorem 3.1.7. Maharam’s Recurrence Theorem. ([1], p.19): Let T be a measure-preserving transformation on a measure space(X,B,µ)and suppose that there exists a set A∈Bof finite measure such that∪∞n=0T−n(A) =X modµ. Then T is conservative.
Proof. First observe thatT−N(X) =X for every positive integerN and so, for eachN, we have that
∞ ∪ n=N
T−n(A) =X (modµ).
Hence, almost everyx∈X must belong to infinitely many setsT−k(A), or, in other words,
∞
∑
n=0
Now, letW be a wandering set forT, so∑∞n=01W ◦Tn≤1. Then, for all positive integersn, the
T-invariance ofµ implies that
∞ > µ(A) =µ(T−n(A))≥ ∫ T−n(A) n
∑
k=0 1W ◦Tk dµ = n∑
k=0 ∫ T−n(A)1W◦T k dµ = n∑
k=0 ∫ X1T −n(A)·1W◦Tkdµ = n∑
k=0 ∫ X (1Tk−n(A)◦Tk)(1W◦Tk)dµ = n∑
k=0 ∫ X (1Tk−n(A)·1W)◦Tk dµ = n∑
k=0 ∫ X1T k−n(A)·1W dµ = ∫ W n∑
k=0 1A◦Tkdµ.Thus, µ(W) =0 andT is conservative.
Definition 3.1.8. A transformation T :X →X of a measure space (X,B,µ)is said to be non- singular if whenever µ(B) = 0, then µ(T−1(B)) =0. That is, the map T preserves sets of measure zero.
Note here that if we have an invariant measure for a mapT, then the mapT is automatically non-singular with respect to this invariant measure.
Definition 3.1.9. A non-singular transformationT of aσ–finite measure space(U,B,µ)is said to beexactif for eachBin the tailσ-algebra∩n∈NT−n(B)we have that eitherµ(B)orµ(U \B) vanishes.
Remark 3.1.10.
1. It is immediately clear that this definition only makes sense for non-invertible transforma- tions. Indeed, ifT :X →X is invertible, it follows thatT−n(B) =Bfor everyn∈N. 2. The tailσ-algebra is not a completely transparent object. It helps to remember that it is
an intersection of sets of sets. In particular, this means that if B∈∩n∈NT−n(B), then B∈T−n(B)for alln∈N. Thus, there exists a sequence of sets(B1,B2,B3, . . .)such that B=T−n(Bn)for everyn∈N.
3. It is easy to see that an exact transformation must be ergodic, for ifT :X →Xis exact and Bis a measurable subset ofXsuch thatT−1(B) =B, thenT−n(B) =Bfor alln∈Nand so the setBbelongs to the tailσ-algebra.
An equivalent formulation of the definition of an invariant measure comes from the transfer operator, which we now define.
Definition 3.1.11. Thetransfer operatorT :L1(µ)→L1(µ)associated with a mapT :X →X on a measure space(X,B,µ)is a positive linear operator given by
∫ BT (f)dµ = ∫ T−1(B) f dµ, for all f ∈L 1(µ) and allB∈B.
Lemma 3.1.12. The measureµ is T -invariant if and only ifT(1X) =1X.
Proof. First notice that from the definition of the transfer operator we obtain
∫ BT (1X)dµ = ∫ T−1(B)1X dµ =µ(T −1(B)). (3.1) Now suppose thatT(1X) =1X. It follows that
∫ BT (1X)dµ = ∫ B1X dµ =µ(B).
Consequently, the condition that T(1X) =1X implies that the measure µ is T-invariant. Con-
versely, ifµ isT-invariant, we have from (3.1) that µ(B) =µ(T−1(B)) =
∫ BT
(1X)dµ ⇒ T (1X) =1X.