Entropy and Ergodicity of Boole-Type Transformations

We review some analytic, measure-theoretic and topological techniques for studying ergodicity and entropy of discrete dynamical systems, with a focus on Boole-type transformations and their generalizations. In particular, we present a new proof of the ergodicity of the 1-dimensional Boole map and prove that a certain 2-dimensional generalization is also ergodic. Moreover, we compute and demonstrate the equivalence of metric and topological entropies of the 1-dimensional Boole map employing “compactified”representations and well-known formulas. Several examples are included to illustrate the results. We also introduce new multidimensional Boole-type transformations invariant with respect to higher dimensional Lebesgue measures and investigate their ergodicity and metric and topological entropies.

The measurable dynamical systems oriented concept of metric entropy (K-S entropy) was introduced by A. N. Kolmogorov [34] as a useful invariant that could be used, for example, to solve the problem of showing the nonequivalence of the Bernoulli 2-and 3-shifts. It was also observed that the metric entropy is related to the exponential growth of distinguishable orbits, which in turn has a certain communication interpretation following from the fact that information theory models can be reformulated as Bernoulli schemes (cf. [35]). As a matter of fact, in that same year, C. Shannon independently introduced his theory of transmission along noisy channels and defined his entropy as a measure of information content. Interestingly, but not really surprising in view of the communication interpretation, Shannon entropy shares many features with metric entropy.
Topological entropy (AKM entropy) was introduced in 1965 by Adler, Konheim and McAndrew [36], and can be viewed as a means of providing an analog of K-S entropy for dynamical systems on topological spaces, independent of any measure-theoretic structure. All three versions of entropy are connected in a variety of aspects, as indicated, for example, in [35]. One of the more profound and rigorous connections is the variational Then one can naturally construct an induced infinite sequence of refinements of the partitions of the form . . ∩ f −n+1 A k n−1 : A k j ∈ ξ, k j ∈ N, j = 0, n − 1} (2) for arbitrary n ∈ N and, whereupon (1), the metric entropy h µ ( f ) ∈ R + is defined as There is an analogous definition of the topological entropy h T ( f ) for a continuous map on a topological space (X; T) described in sources such as [35,36] for compact spaces and in [44,45] for noncompact spaces. Computing metric entropy tends to be quite a bit easier than topological entropy. However, calculation h µ ( f ) via definition (3) is generally very difficult, which naturally led researchers to find simpler methods for determining the metric entropy. Ergodicity, which is an important property in its own right, turns out to be a key to substantial simplifications in computing K-S entropy, such as in the the following result of Shannon-McMillan-Breiman [17,48,49]: Let a measurable mapping f : X → X on the probability space (X; B, µ) be ergodic and let ξ be a countably infinite generating partition of X, for which H µ (ξ) < ∞. Then for µ-almost every x ∈ X, where sets A n ( f ; x) ∈ ξ n ( f ), n ∈ N, are chosen so that x ∈ A n ( f ; x), n ∈ N. The expression (4) simplifies the calculation of metric entropy for ergodic systems, but a fairly high level of difficulty remains. A further, very substantial simplification under the added assumptions of differentiability and dilation, was obtained by Krengel [50] and Rokhlin [51,52]; namely, if the measurable dynamical system (X; B, µ, f ) is ergodic, X is a metric space and f is a differentiable dilation in the sense that the Radon-Nikodym derivative f µ (x) := dµ • f (x)/dµ(x) at all points x ∈ X of the shifted measure µ • f with respect to the probability measure µ is measurable and satisfies the dilation condition inf x∈X f µ (x) > 1, then we have the simple formula strongly based on the Lyapunov exponents [48] of the mapping f : X → X on the probability space (X; B, µ). We note that the dilation condition plays an important role in the existence of an invariant measure µ on X for f : X → X as shown in [47][48][49][53][54][55][56][57] for a wide class of measurable spaces.
In particular, let a smooth (C 1 ) measure preserving mapping f : X → X, defined on a metric space X be dilating [47,48,55,56], so that inf x∈X f µ (x) > 1. Furthermore, let the finite generating partition ξ := {A j ∈ X : j = 1, k}, k ∈ N, be such that sets f (A j ) ⊂ X, j = 1, k, are measurable and the reduced (restriction) mappings f j : A j → f (A j ), j = 1, k, are invertible and measurable. Then extend the inverse mappings f −1 j : f (A j ) → A j , j = 1, k, arbitrarily, but measurably, to the whole space X. Then, the entropy of the probabilistic measure invariant mapping f : X → X can be calculated, owing to the Gibbs Formula (1), as where mappings E(χ A j | f −1 B) : X → R, j = 1, k, denote the corresponding conditional expectations of the characteristic functions χ A j (•) : X → R, j = 1, k, with respect to the σ-algebra f −1 B. Taking into account that for any function F ∈ L 1 (X; R) one has for any x ∈ X the expansion with respect to the Borel σ-algebra B, from the corresponding definition of the integrated conditional expectation value it follows from (6) and (8) that or equivalently, for all x ∈ X and j = 1, k. Now, having substituted the conditional expectation functions (10) into (6), one finally obtains with respect to the covering finite generating partition ξ, which coincides with (5). If the measurable metric space (X; B, µ) is finite dimensional and the probabilistic measure dµ(x) = dλ(x), x ∈ X (the normalized Lebesgue measure on X), then the Radon-Nikodym derivative f µ (x) = |J f (x)| at point x ∈ X is the absolute value of the usual Jacobian of the differentiable mapping f : X → X, and then the Krengel-Rokhlin entropy expression (5) becomes Using the Shannon-McMillan-Breiman expression (4), Formula (5) was also proved by Yuri [57] for multidimensional mappings with finite range structure subject to the corresponding partitions ξ n (ϕ), n ∈ N, induced by a fixed generating partition ξ ∈ 2 X , X ⊂ R n . Among this class of mappings there are so called "fibered" mappings f : X → X satisfying the following conditions [5,14,26,[57][58][59][60][61][62]: (a) there is an invariant Lebesgue equivalent probability measure µ : B → R + , for which there exist positive constants c 1 , c 2 ∈ R + , such that c 1 λ(E) ≤ µ(E) ≤ c 2 λ(E) for every Borel set E ⊂ X; (b) there is a finite or countably infinite digit set D j , j = 1, N; (c) there is a mapping k : X → D, D := × j=1,N D j , such that the sets X i := k −1 {i} = {x ∈ X : k(x) = i}, i ∈ D, are measurable and form a partition ξ(X) of the space X, that is, (d) the restrictions f | X i : X i → X, i ∈ D, are injective and smooth.
Then one can see [5,51,57] that under the additional conditions imposed on the map f : X → X : for some point τ ∈ X, it appears to be ergodic and equivalent to the weak Bernoulli shift mapping T f : with respect to the isomorphism ψ : determined for the admissible rank-n cylinder sets X n (k 1 , k 2 , k 3 , . . . , k n ) ⊂ X, n ∈ N, for which X n (k 1 , k 2 , k 3 , . . . , k n ) := ∩ j=1,n X k j (x), and on which we shall not dwell. In many concrete cases the ergodicity of a mapping f : X → X can be proved more effectively using standard measure theoretical calculations. In particular, using the construction above one can employ a slightly modified approach to [14,48,49] for proving ergodicity, making use of the following two classical measure theory [63,64] lemmas. Owing to the weak equivalence of the above "fibered" mappings f : X → X, X ⊂ R n , to the Bernoulli shifts [47,48,57] (14), one can state the following important result.

Theorem 2.
Let the cylinder sets of a smooth "fibered" mappings f : X → X with finite range structure satisfy the conditions of Lemma 2 for the case of the Lebesgue measure λ on X. Then, if the invariant measure µ is absolutely equivalent to the Lebesgue measure λ on X, the mapping f : X → X is ergodic.
We are now in a position to apply Lemmas 1 and 2. Let a measurable set B ⊂ [0, 1) be f -invariant and calculate the Lebesgue measure

Example 2. A very interesting example is given by the classical continued fraction expansion via the Gauss ergodic mapping
f whose fibering is defined by the mapping k The probabilistic invariant measure is the well-known Gauss measure dµ( coinciding with the well-known [47,48,65] result. The ergodicity of the map (28) can be easily proved by reducing it [49] via the continued fraction expansion to a Bernoulli shift and applying Lemmas 1 and 2. Namely, take a number x ∈ [0, 1) and denote by [x 0 , x 1 , . . . , x n , . . . ] its continuous fraction expansion: where x i ∈ Z + for all indices i ∈ Z + . Observe here that the induced continuous fraction mapping acts by left shifting as T ϕ [x 0 , x 1 , . . . , x n , . . . ] = [x 1 , . . . , x n , . . . ] for any expansion (30). This expansion [x 0 , x 1 , . . . , x n , . . . ] can be reduced to n-th by defining for every t ∈ [0, 1) the rational t-fraction where P n (x 0 , x 1 , . . . , x n−1 ; t) and Q n (x 0 , x 1 , . . . , x n−1 ; t) are coprime polynomials in the variables x 0 , x 1 , . . . , x n−1 ∈ Z + and t ∈ [0, 1) for all n ∈ N. If we define the n-th order polynomials P n = P n (x 0 , x 1 , . . . , x n−1 ) := P n (x 0 , x 1 , . . . , x n−1 ; 0) and Q n = Q n (x 0 , x 1 , . . . , x n−1 ) := Q n (x 0 , x 1 , . . . , x n−1 ; 0), it is easy to see that the following iterative expressions hold: P n+1 (x 0 , x 1 , . . . , x n−1 , x n ; t) = x n P n + P n−1 + tP n , Q n+1 (x 0 , x 1 , . . . , x n−1 , x n ; t) = x n Q n + Q n−1 + tQ n for any t ∈ [0, 1) and arbitrary n ∈ N. By setting the parameter t = 0 in (32), one readily obtains derives the following iterative relationships for all n ∈ N : P n+1 = x n P n + P n−1 , Q n+1 = x n Q n + Q n−1 (33) with initial conditions P 0 = 0, P 1 = 1 and Q 0 = 1, Q 1 = x 0 ∈ Z + . In particular, the following invariant condition Q n P n−1 − P n Q n−1 = (−1) n and inequality Q n−1 ≤ Q n readily follow from (33) for all n ∈ N. Let k 0 , k 1 , . . . , k n−1 ∈ N for every n ∈ N and define the cylindrical intervals I n ⊂ [0, 1) as the corresponding collection of rational tfractions Defining the inverse and estimate the Lebesgue measure of the interval (34) to be where we took into account that 0 < Q n−1 ≤ Q n for all n ∈ N. At this stage, we are in a position to estimate the Lebesgue measure λ(B ∩ I n ) for the intersection B ∩ I n of an invariant set B = f −1 B = f −n B ⊂ [0, 1) and arbitrary cylindrical interval I n ⊂ [0, 1), n ∈ Nas which is consistent with Lemma 2 with C = 4. Consequently, from the estimate (38) one deduces that the measure λ(B) = 1 or λ(B) = 0, thus proving the ergodicity both of the Lebesgue measure dλ(x), x ∈ [0, 1), and the invariant Gauss measure dµ(

One-Dimensional Boole-Type Mappings, Invariant Ergodic Measures and Their Entropies
The classical one-dimensional Boole [4] mapping is defined as Adler and Weiss in [46] proved that (39) is ergodic with respect to the infinite invariant σ-finite Lebesgue measure dλ(x) := dx, x ∈ R := X. Their proof of the ergodicity relied heavily on the measure theoretic reduction of β to the corresponding induced [47][48][49] The β-invariance of the Lebesgue measure dλ(x) := dx, x ∈ R, is easily checked making use of the Perron-Frobenius theory; namely, for preimages coinciding exactly with the Lebesgue measure on R.

Remark 1.
Here we remark that the Boole transformation (39) is σ-finite and nonsingular on R; that is, for Moreover, for nonsingular transformations f : X → X the following properties [1] are equivalent: (i) f is conservative and ergodic; (ii) for We now consider a limiting version of the corresponding distributed Krengel type entropy of the Boole mapping (39) with respect to the σ-finite set of probabilistic measures dµ (r) := dx/(2r) on compact intervals [−r, r] ⊂ R as r → ∞, which unfortunately can readily be shown to yield since I(r) is bounded on (0, ∞). Not only is this entropy result counterintuitive, it is not actually valid since we cannot use the Krengel-Rokhlin formula (5) directly to calculate the entropy of maps of σ-finite measure preserving dynamical systems on spaces of infinite measure. Consequently, we propose to use a result from [11] for approximate Boole transformations that satisfy the necessary conditions for the validity of (5), which enables the calculation of the desired result as a limit. To this end, we define where β α (x) := αx − (1/x) and 0 < α < 1. The following (differential) probability measure on R, is absolutely continuous with respect to the (differential) Lebesgue measure dx and β α -invariant for all α ∈ (0, 1): It is worth mentioning that the measure (43), suitably de-regularized, weakly tends to the Lebesgue measure dλ = dx on R; that is, w-lim α↑1 = dλ subject to the space C 0 (R; R). Now we can use the invariant probability measure (43) in the Krengel-Rokhlin formula (5) to obtain the following result in the limit: and Therefore, inasmuch as isomorphic measurable dynamical systems have the same entropy, (46) implies, for example, that β is not isomorphic to the doubling map (18). Additionally, it should be mentioned that the metric entropy of (42) can also be obtained using Pesin's formula whenever α is positive and α = 1 (cf. [66]).
Admittedly, one might well question the limit-based computation (44) of the entropy. In particular, it is not rigorous because the very first equality, due to the fact entropy is only upper semi-continuous, is really just a lower bound. Therefore, we now use a compactification procedure to both confirm the result and also determine the topological entropy, which, as is well known [37], bounds the measure-theoretic entropy. We start by using what is essentially 1-dimensional stereographic projection and smooth extension to obtain a representation of the Boole map on the unit circle S 1 : where ϕ(x) := 2x Observe thatβ is a smooth (= C ∞ ) surjection of the unit circle onto itself, having among others the properties that there is a unique fixed point at the north pole, (0, 1) corresponding to θ = s = π/2 andβ(1, 0) =β(−1, 0) = (0, −1), soβ 2 (1, 0) =β 2 (−1, 0) = β(0, −1) = (0, 1). It is also worth noting that the differential of arclength ds for which dμ := ds/2π is a natural (differential) probability measure on S 1 satisfies It is straightforward to show from its definition that probability measureμ on S 1 is absolutely continuous with respect to the Lebesgue measure associated to the representation of the unit circle as R/Z and that it isβ-invariant. Moreover, a simple calculation reveals thatβ is a dilationμ-almost everywhere on S 1 ; in fact, d dsβ (s) > 1 except at the fixed point (0,1). Consequently, the Krengel-Rokhlin formula can be used to compute the entropy, so that owing to the change of variables formula for integrals and (44), we find that Serendipitously, this formula for the K-S entropy of the measurable dynamical system S 1 , M,μ,β can also be used to obtain the formula for the topological entropy of the topological dynamical system S 1 , T,β , where M is the (Borel)μ-measurable subsets of the unit circle and T is the Euclidean subspace topology on S 1 . In fact, it is not difficult to verify that the hypotheses of Theorem 3 of [35] are satisfied for S 1 , M,μ,β and S 1 , T,β , so it follows that Next, we present a new proof of the ergodicity of the Boole transformation (39)-a variant of the approach in [59]-that is more concise than the original due to Adler and Weiss [46]. Theorem 3. The one-dimensional Boole transformation (39) is ergodic with respect to the invariant Lebesgue measure λ on R.
It is also worth mentioning here the well-known result [1,13,47,48,70,71] that the doubling map (18) is isomorphic to the 1-dimensional Boole-type transformation which is invariant with respect to the probability measure dµ(x) = dx/[π(1 + x 2 )], x ∈ R, and has entropy coinciding with that of (19). The Boole mapping (39) was also generalized [1] in the form where a, b j ∈ R, j = 1, N, α, β j ∈ R + , j = 1, N, and analyzed in [1,3,11,72,73]. For α = 1, a = 0, the ergodicity result was proved in [3,[74][75][76] by making use of a specially devised inner function method. The related spectral aspects of the mapping (66) were in part also studied in [1,3]. In spite of these results, the case α = 1 is still challenging, with the only available related results [1,3] being for the special case of (66) for α = 1/2, and arbitrary a, b ∈ R and β ∈ R + . Invariant measures and ergodicity related to (67) were analyzed in [11,[70][71][72] using their equivalence to following from the commutative diagram for which the condition ϕ • T = ϕ • f # , where ϕ(s) := (2β) 1/2 cot(πs) + 2a, s ∈ [0, 1), holds. The Krengel-Rokhlin formula (5) can be used to calculate the corresponding measuretheoretic entropy of the f # , yielding h µ ( f # ) = ln 2, which is the entropy of the shift map (19). It is also important to mention that the theory of inner functions in [1,[74][75][76] was used to prove the existence an f # -invariant probability measure µ # on R, such that the generalized Boole type transformation (66) is ergodic for any N > 1, α = 1 and a = 0. It appears that the transformation (66) is not ergodic for α = 1 and a = 0 since it is totally dissipative, with wandering set where a ∈ R, α ∈ R + and the measure ν on R (not necessary absolutely continuous with respect Lebesgue measure) has compact support supp ν ⊂ R and satisfies the natural conditions R dν(s) which guarantee the boundedness of its entropy.

Multi-Dimensional Boole Transformations: Their Entropy and Ergodicity
Multi-dimensional endomorphisms of measurable spaces are of great interest [14,47] in many mathematical subdisciplines, including number theory, numerics, dynamical systems theory and diverse physical applications. In this regard, we should mention [14,61,62], which treat many interesting measure preserving and ergodic multi-dimensional mappings. Recently, in [11,70,72,73] a class of new multi-dimensional Boole type transformations β ρ : R n → R n of the following form were introduced and analyzed for any n ∈ N and arbitrary permutations ρ ∈ S n , where the signs " ± " are chosen from the nondegeneracy condition J β ρ (x) = 0, x ∈ R n \{0}. For the case n = 2, (x, y) ∈ R 2 \{0, 0}, one obtains the two-dimensional Boole type mapping: and for the case n = 3, (x, y, z) ∈ R 3 \{0, 0, 0}, the pair of nontrivial three-dimensional Boole type mapping: The infinitesimal σ-finite Lebesgue measure dλ(x, y) := dxdy, (x, y) ∈ R 2 is β (21)invariant, as can be easily checked via the Perron-Frobenius eigenfunction condition as follows: For the corresponding preimages (u ± , v ± ) : (21) (u ± , v ± ) = (x, y) ∈ R 2 , one verifies that the measure satisfies coinciding with the Lebesgue measure dλ(x, y) := dxdy. As for the ergodicity of the Lebesgue measure preserving mapping (73), the approach based on Theorem 2 employing smooth-fibered multi-dimensional mappings does not seem to be viable. Inasmuch as that the ergodicity result of [46] for the one-dimensional Boole mapping (39) is largely based on the induced Kakutani transformation technique, one can expect that it is also applicable to the two-dimensional Boole map (72). Now we recall that the notion of the induced transformation [47][48][49] for an infinite measure preserving mapping f : X → X, which was used by Adler and Weiss [46] to prove the ergodicity of the map (39), is rather closely related to the classical Poincaré recurrence theorem [18,48]. Namely, let (X; B, µ, f ) be a measure preserving discrete dynamical system, where A ⊂ X is a set of positive measure satisfying the covering condition modulo a set of measure zero.

Remark 2.
It is worth mentioning here [18,[47][48][49] that if a measurable dynamical system (X; B, µ, f ) satisfies for arbitrarily chosen measurable A ⊂ X, µ(A) > 0, the covering condition (76), then f : X → X is ergodic. In fact, if f is ergodic, then for an arbitrary measurable set B ⊂ X, In general, for a probability measure space (X; B, µ) the following properties are equivalent: (v) for any measurable and invariant function g : X → C, g = g • f µ-almost everywhere implies that g : X → C is constant almost everywhere.
Owing to the condition (76), the first return time τ A ∈ N can now be defined as which exists almost everywhere and is finite.

Definition 1.
Let a measurable dynamical system (X; B, µ, f ) satisfy the condition (76). Then a mapping f A : for almost all x ∈ A is called the transformation induced by f : X → X on the subset A ⊂ X.
The induced transformation is characterized by the following [47][48][49] important theorems: . Let a mapping f : X → X be ergodic and a measurable set A ⊂ X be such that 0 < µ(A) < ∞. Then the average return time is proportional to the measure µ(A); that is, Theorem 5. The induced transformation (78) is measure preserving on the space (A, Moreover, if f : X → X is ergodic, with respect to the measure µ, the induced transformation f A : A → A is ergodic with respect to the measure µ A := µ/µ(A) induced on the set A.
An instructive proof of this theorem is given below in Supplement to the article. As was already mentioned above, this theorem was used in [46] to prove the ergodicity of the Boole mapping (39). In addition, it was shown above that there is a second effective essentially analytical approach to proving the ergodicity, and so it would be useful to present two proofs, if any, of the ergodicity of the two-dimensional Boole type mapping (73).
Concerning the approach based on Theorem 5, its main technical ingredients are intimately related to the construction of a special generating partition of the measure space X, suggested by Kakutani and Rokhlin [51,77] for the corresponding induced mapping Now, if one tries to apply the measure-theoretic construction devised in [46] for proving ergodicity of the two-dimensional Boole mapping (73), some very technical difficulties arise that appear to be too difficult to overcome. Thus, the analytical approach based on Theorem 2 that relies on the relationship between the two-dimensional Boole mapping (73) and the two-dimensional doubling map appears to be the only feasible choice. More specifically (73), which for convenience in what follows we define as F := β (21) , is related to the following two-dimensional transformation T F : [0, 1) 2 (s, t) → ({2s}, {2t}) ∈ [0, 1) 2 on the square Y = [0, 1) 2 ⊂ R 2 , according to the commutative diagram where α −1 : [0, 1) 2 → R 2 is the map owing to changing the variables x = cot(πs), y = cot(πt), (s, t) ∈ [0, 1) 2 , (x, y) ∈ R 2 , subject to the new coordinates (s, t) ∈ [0, 1) 2 and the transformation S −1 : That this approach could be used to prove of the ergodicity theorem of the two-dimensional Boole transformation (73), was announced in [70][71][72] and is now confirmed by the following result.
Theorem 6. The two-dimensional Boole transformation F = β (21) defined in (73) is ergodic with respect to the invariant Lebesgue measure λ on R 2 .
This can be readily estimated as Thus, based on the estimates (86), we readily obtain the following inequalities for the measure (84): for any n ∈ N, where the constants are Having the estimate (88), we are in a position allowing to apply Lemmas 1 and 2. Whence, if a measurable set B ⊂ [0, 1) 2 is F-invariant so that B = F −1 B = F −n B, n ∈ N, we compute that where we made use of the property that the composition F • (S −1 • σ k j ,l j • α) = Id for any j = 0, n − 1. Now, it follows from (88) that so the Lebesgue measure satisfies λ(I n )λ(B) ≤ Cλ(B ∩ I n ) for all n ∈ Z + , where C := C 2 C −1 1 . Therefore, owing to Lemma 2, either λ(B) = 1 or λ(B) = 0, thus confirming the ergodicity of the two-dimensional Boole mapping (73) with respect to the same invariant Lebesgue measure λ on R 2 , and completing the proof.
As mentioned above, the Lebesgue measure on R 3 is invariant with respect to the three-dimensional Boole-type transformations (74), which are also likely to be ergodic, but the search for a proof is still ongoing.

Supplement: Proof of Theorem 6
Let f : X → X be ergodic and consider a measurable set A ⊂ X satisfying the condition 0 < µ(A) < ∞ together with its induced mapping f A : A → A. As the condition (76) a priori [18,[46][47][48][49] holds, one can construct the following disjoint measurable first return iteration subsets where n∈N X n = X, X n ∩ X m = ∅, m = n ∈ N, and for which is satisfied. Using the sets (91), one constructs for all n ∈ N the sets A n := X n ∩ A, B n := X n ∩Ā, satisfying the disjoint sum property For any measurable subset E ⊂ A we note that which gives rise to the equality Employing the representation (94) and the measure invariance, one readily deduces that for all n ∈ N. Consequently, we find that which reduces to Whence follows the invariance of the positive quantity η A ∈ R + with respect to n ∈ N and the boundedness of the measure µ(B 1 ) ≤ µ(A), since the measure µ(A) = µ( n∈N A n ) < ∞ is assumed bounded. It follows immediately from the first equality of (97) that η A = µ(A) > 0, that is In light of (96), it follows from (100) and (99) that owing to the convergence condition (99) for the measure µ(A) < ∞, so µ( f −1 A E) = µ(E) for any measurable set E ⊂ A. Therefore, the suitably induced measure on A ⊂ X, µ A = µ/µ(A), is also invariant with respect to the induced map f A : A → A.
If we assume now that the induced mapping f A : A → A is ergodic and take an finvariant set D ⊂ X such that µ(D ∩ A) > 0, then since either µ(D ∩ A) > 0 or µ(D ∩Ā) > 0, it follows from the expansion (95) that since the initial assumption ∪ n∈N f −n A = X guarantees that f −1 A A = A modulo a set of measure zero. As the induced mapping is assumed to be ergodic, from (102) and the condition µ(D ∩ A) > 0, we immediately conclude that D ∩ A = A. Thus, based once more on the initial assumption ∪ n∈N f −n A = X one readily finds that meaning that the mapping f : X → X is also ergodic. Similarly, one can also prove the converse statement. In particular, if the mapping f : X → X is ergodic and a set E ⊂ A of positive measure is f A -invariant , then Taking into account the invariance condition (104), let us construct the set G := E ∪ n∈N (B n ∩ f −n E) and calculate its inverse image under f as follows: Hence, G is f -invariant, so the ergodicity of f implies that G = X modulo a set of measure zero. Having now taken into account that, by construction, the subset n∈N (B n ∩ f −n E) ⊂Ā, it follows that the set A ⊆ E modulo a set of measure zero in X. Consequently, from the assumption that E ⊂ A, one deduces finally that A = E, which means that the induced mapping f A : A → A is also ergodic, proving the theorem.

Conclusions
We delineated several aspects of discrete measurable and differential dynamical systems essential to our investigation of the classical Boole transformation and some of its generalizations and extensions. The aspects of primary interest were metric and topological entropy and ergodicity and their interrelationships. For example, the Rokhlin-Krengel formula for metric entropy of ergodic systems played a key role in our treatment of both topological and Kolmogorov-Sinai entropy. One of the main results obtained was the calculation of the topological and metric entropy of the classical (one-dimensional) Boole map employing a limiting approach as well as a compactification method based on stereographic projection. In addition, we presented a new proof of the ergodicity of the classical Boole map employing the ideas of Li and Schweiger and studied the entropy of some simple generalizations of the Boole map.
An interesting class of multi-dimensional extensions of the classical Boole map was introduced and the members were shown to be invariant under the Lebesgue measures of the corresponding dimension. It was proved, using an approach based on fibered mappings, that a particular two-dimensional member of the class is ergodic and a similar result for the higher dimensional generalizations was conjectured. Moreover, we conjectured that the techniques used to compute both topological and metric entropy for the one-dimensional Boole map could be adapted for these multi-dimensional extensions as well. As for related future research, we plan to investigate both our ergodicity and entropy conjectures in considerable detail.