Minimal Cantor Sets: The Combinatorial Construction of Ergodic Families and Semi-Conjugations

Combinatorially obtained minimal Cantor sets are acquired as the inverse limit of certain directed topological graphs where specific nonnegative integer matrices, called winding matrices, are used to describe the projection between each graph. Examples of non-uniquely ergodic combinatorially obtained minimal Cantor sets first appeared in the 2006 article Algebraic topology for minimal Cantor sets of Gambaudo and Martens and are constructed using winding matrices whose entries grow “fast enough.” In this work, we will introduce families of minimal Cantor sets which may be combinatorially obtained in such a way that the corresponding winding matrices possess unbounded entries given by explicit sequences of nonnegative integers. For each of these families, the growth rate needed to achieve either unique or non-unique ergodicity will be specifically addressed and the result will be applied to the case of minimal Cantor sets corresponding to Lorenz maps. We will explore the construction of topological semi-conjugations between combinatorially obtained minimal Cantor sets. Theorems guaranteeing the existence of a topological semi-conjugation between specific families of these sets will be proved and utilized to introduce examples possessing additional intriguing properties. We will also show that there exist both finitely and infinitely nonuniquely ergodic minimal Cantor sets semi-conjugated to an adding machine in such a way that the semi-conjugation map is almost-everywhere injective with respect to the unique ergodic invariant probability measure on the adding machine. Furthermore, we will prove that this construction can be realized by the dynamical system (ω(c), q) where q is a logistic unimodal map with critical point c and omega-limit set ω(c).


CHAPTER 1
Introduction and Preliminaries

Introduction
In 1883, Cantor introduced the now well-known middle-thirds Cantor set within a footnote to his paperÜber unendliche, lineare Punktmannichfaltigkeiten (see [1,Part 5]). One method of constructing this set begins by taking A 0 = [0, 1] and then deleting the open middle third of this interval to obtain A 1 = [0, 1 3 ]∪[ 2 3 , 1]. The set A 2 is formed by deleting the open middle third from each of the disjoint intervals that make up A 1 . In general, the set A n (n ≥ 1) is formed by deleting the open middle third from each of the disjoint intervals that make up A n−1 . The sets A 0 , A 1 , A 2 , and A 3 can been seen in Figure 1. The middle-thirds Cantor set C is defined as the intersection of the sets A n ; that is, C = ∞ n=0 A n .
The middle-thirds Cantor set possesses a number of interesting properties. It is clearly nonempty (for instance, it contains 0). It is bounded and, as the intersection of closed sets, it is closed (hence compact). It also contains no intervals since the complement [0, 1] \ C has Lebesgue measure one. Furthermore, it is a perfect set since it is a closed set with no isolated points. This can be verified by observing that any x ∈ C lies in exactly one of the maximal disjoint intervals of length 1  1 that make up A n , say A x n . If x is not an endpoint of the interval A x n , take x n to be the lefthand endpoint of A x n . Otherwise, take x n to be the endpoint of A x n that is not x. Since endpoints are never deleted, x n ∈ C for each n. Moreover, x n = x and |x − x n | ≤ 1 3 n for each n. It follows that x is not isolated.
A generalization of the properties of the middle-thirds Cantor set allows for the definition of a Cantor set. Because of this, the set consisting of those points that do not escape to −∞ under iteration of T 3 must be a subset of [0,1]. It will be shown that this set is precisely the middle-thirds Cantor set.
For a given unimodal map u : [0, 1] → [0, 1] with critical point c, the omega limit set ω(c) is compact and satisfies that u(ω(c)) = ω(c). The set ω(c) may be a cycle of intervals, a Cantor set, a finite set, a countable set, or the union of a Cantor set and a countable set. Unimodal maps under which the omega-limit set of the critical point is a minimal Cantor set are known to exist and have been studied widely (see, for instance, [3,4,5,6,7]). The most well-known examples include the Feigenbaum map (see [8]) and the Fibonacci map (see [5]), which will be described in more detail in Chapter 4.

Combinatorially Obtained Minimal Cantor Sets
In 2006, Gambaudo and Martens gave conditions under which a minimal Cantor set can be obtained via the inverse limit of particular directed topological graphs (see [4]). In their article, the authors introduced the concept of using specific nonnegative integer matrices, called winding matrices, to describe the projection between each of these graphs. This section provides a summary of these and related results.
Definition 1.6. Let Y 1 , Y 2 , . . . be a countable collection of topological spaces and suppose that for each n ∈ N there exists a continuous mapping g n : Y n+1 → Y n .
The sequence of spaces and mappings {Y n , g n } ∞ n=1 is called an inverse limit sequence and may be represented by the diagram For n < m, the continuous mapping g mn : Y m → Y n is given by Definition 1.7. The inverse limit of an inverse limit sequence {Y n , g n } ∞ n=1 is given by lim ←− gn Y n = (y 1 , . . . , y n . . .) ∈ ∞ n=1 Y n : g mn (y m ) = y n for all n ≤ m .
Note that as a subset of the product space ∞ n=1 Y n carrying the product topology, the inverse limit Y = lim ←− gn Y n may be equipped with the corresponding subspace topology.
Definition 1.8. A graph is an ordered pair G = (V, E) comprised of a set V of vertices together with a set E of edges which are unordered, 2-element subsets of V.
We will consider only those graphs for which each edge {U, V } satisfies that where all edges are oriented in the same direction.
Gambaudo and Martens introduced an inverse limit sequence in which each topological space consists of a specific type of directed topological graph. Definition 1.10. A directed topological graph G is called a combinatorial cover if the following three properties are satisfied: 1. The graph G is finite and the set of vertices carries the discrete topology.
2. The graph G is irreducible (any pair of distinct vertices is connected by a directed path).
3. Except for one vertex 0 G ∈ G, each vertex of G has exactly one outgoing edge. The vertex 0 G is called the splitting vertex.
Definition 1.11. In a combinatorial cover G, a vertex U ∈ G is called an image of a vertex V ∈ G if there is a single directed edge from V to U . The shortest directed path from a vertex V ∈ G to 0 G is called the path of V and is denoted by λ(V ). Furthermore, if U is an image of 0 G , then λ(U ) together with the directed edge from 0 G to U is called a loop of G. When the meaning is clear, the notation λ(U ) will also be used to denote such a loop.
An illustration of a combinatorial cover and these concepts can be seen in Figure 3.
In order to take the inverse limit involving a sequence of combinatorial covers, it is necessary to define a particular type of mapping between two combinatorial covers.
Definition 1.12. Let G and H be combinatorial covers. A map π : G → H is called a combinatorial refinement if the following three properties are satisfied: 1. The map π preserves the graph structure.
2. The map π satisfies that π(0 G ) = 0 H ; that is, the splitting vertex of the graph G is mapped to the splitting vertex of the graph H.  . An illustration of a combinatorial cover. Note that U 1 , U 2 , and U 3 are the images of the splitting vertex 0 G . The path of U 2 , λ(U 2 ), is indicated by the dashed edges.
exists an image V ∈ G of the splitting vertex 0 G ∈ G such that 1 H = π(V ). Lemma 1.13. (Gambaudo & Martens) Consider an inverse limit sequence consisting of combinatorial covers {G n } ∞ n=1 and corresponding combinatorial refinements It is only natural to ask what restriction(s), if any, must be placed upon an inverse limit sequence of combinatorial covers and combinatorial refinements in order to guarantee that a combinatorially obtained minimal Cantor set will result.
The answer to this question requires the definition of a winding matrix.
Definition 1.15. For G n and G n+1 combinatorial covers and π n : G n+1 → G n a combinatorial refinement, let {U 1 , . . . , U r } be the images of the splitting vertex 0 Gn and {V 1 , . . . , V s } be the images of the splitting vertex 0 G n+1 . The r × s matrix W n with entries is called the winding matrix corresponding to π n .
Note that w ij gives the number of distinct vertices in the loop λ(V j ) ∈ G n+1 that are projected to the vertex U i ∈ G n under the combinatorial refinement π n .
Equivalently, w ij gives the number of times the loop λ(V j ) ∈ G n+1 is wrapped Example 1.16. Suppose that for some n the projection between the combinatorial covers G n+1 and G n as pictured in Figure 4 has corresponding winding matrix This matrix describes the projection in the following manner: The outermost loop of G n+1 is wrapped twice around the outer loop of G n and once around the inner loop of G n . The middle loop of G n+1 is wrapped once around the outer loop of G n and thrice around the inner loop of G n . The innermost loop of G n+1 is wrapped thrice around the outer loop of G n and zero times around the inner loop of G n .
Theorem 1.17. (Gambaudo & Martens) Consider an inverse limit sequence consisting of combinatorial covers {G n } ∞ n=1 and corresponding combinatorial re-  guarantees that the map f : G → G obtained by taking the inverse limit G = lim ←− πn G n is a minimal Cantor set. Now suppose that in the above scenario the winding matrix W is given by 1 1 0 1 instead. It can be shown that for any n, W n is of the form * * 0 * .
Because of the zero in the bottom left corner, Theorem 1.17 guarantees that the map f : G → G obtained by taking the inverse limit G = lim ←− πn G n is not a minimal Cantor set.

Summary of Results
The study of a given dynamical system (X, f ) is usually conducted from a perspective that "makes sense" for the space X and map f under consideration. If X is a C r -differentiable manifold and f is a C r -diffeomorphism, the system can be approached from the point of view of differentiable dynamics. If X is a topological space and f is continuous, topological dynamics may be utilized. If X is a measure space and f measurable, the methods of measurable dynamics may be used.
In the case of a combinatorially obtained minimal Cantor set (X, f ), the space X is both a topological space and a measure space, and the map f is both continuous and measurable. The theory of topological dynamics and the theory of measurable dynamics both apply, and for this reason the topological and measuretheoretical properties of such a system are of great interest.
The remaining chapters of this work are concerned with exploring the measuretheoretical property of ergodicity and the construction of topological semi-conjugations in regards to combinatorially obtained minimal Cantor sets.
for all A ∈ Σ. Furthermore, f is said to be ergodic if for every A ∈ Σ with f −1 (A) = A either µ(A) = 0 or µ(A) = 1. In this case the measure µ is called an ergodic measure.
Definition 1.20. Let X and Y be topological spaces and let f : X → X and g : Y → Y be continuous functions. The map f is said to be topologically semiconjugate to the map g if there exists a continuous surjection p : Y → X such that The study of ergodicity is important as ergodic theory allows us to come to a better understanding of how predictable an unpredictable dynamical system can be. Combinatorially obtained minimal Cantor sets are known to carry a set of ergodic invariant probability measures and non-uniquely ergodic minimal Cantor sets are known to exist. Both the paper of Gambaudo and Martens (see [4]) and the 2011 doctoral thesis of Winckler (see [9]) contain examples of combinatorially obtained minimal Cantor sets that are not uniquely ergodic. In each of these examples, the authors rely upon winding matrices whose unbounded entries grow "fast enough" but do not address the necessary growth rate. Chapter 2 will provide details concerning these two examples before going on to establish the following main result on the growth rate needed to guarantee either unique or non-unique ergodicity for a specific family of combinatorially obtained minimal Cantor sets.
Theorem 2.7. Let a, b, m, p ∈ Z + be such that a > b and m ≥ 2. Then the minimal Cantor set corresponding to the sequences of m × m winding matrices is uniquely ergodic when p = 1 and is not uniquely ergodic when p > 1. In the second case, there are in fact m ergodic invariant probability measures.
The remainder of Chapter 2 consists of theorems that establish the ergodicity of families of minimal Cantor sets that are combinatorially obtained from winding matrices with a similar structure to those that appear in Theorem 2.7. These results are then applied in the case of a specific type of map.
Chapters 3 and 4 use the results of Gambaudo and Martens to explore the construction of topological semi-conjugations between minimal Cantor sets, which is a topic of interest since such semi-conjugations are known to preserve a variety of dynamical properties. In Chapter 3, theorems guaranteeing the existence of a topological semi-conjugation between specific families of combinatorially obtained minimal Cantor sets will be established and utilized to prove the following main results.
a n a n a n . . . a n is finitely non-uniquely ergodic for an appropriately chosen sequence of integers For such a sequence {a n } ∞ n=1 , this minimal Cantor set can be semiconjugated to a specific uniquely ergodic minimal Cantor set in such a way that the semi-conjugation map is almost everywhere injective with respect to the unique ergodic probability measure on the latter minimal Cantor set.
Theorem 3.18. The minimal Cantor set that is combinatorially obtained from the sequence of m n × (m n + 1) (m n ≥ 3 and m n = n + 2) winding matrices a n a n . . . a n a n 1 mn−1 na n + 1 na n 2 2 . . .
is infinitely non-uniquely ergodic for an appropriately chosen sequence of integers For such a sequence {a n } ∞ n=1 , this minimal Cantor set can be semiconjugated to a specific uniquely ergodic minimal Cantor set in such a way that the semi-conjugation map is almost everywhere injective with respect to the unique ergodic probability measure on the latter minimal Cantor set. List of References

Introduction
The number of ergodic invariant probability measures that a given combinatorially obtained minimal Cantor set carries can be investigated by using the winding matrices associated with it. A brief overview of the background involved in this process is outlined below; all details are provided by Gambaudo and Martens in [1].
Let (X, f ) be a combinatorially obtained minimal Cantor set with inverse limit representation X = lim ←− πn X n where each combinatorial cover X n possesses d n loops.
Let M(X) be the space of signed invariant measures on (X, f ) where Σ is the σ-algebra on X. The space of signed measures on X n has σ-algebra Σ n generated by the elements of X n . Let U 1 , . . . , U dn be the d n images of the splitting vertex 0 Xn so that the loops of X n are λ(U 1 ), . . . , λ(U dn ). For j = 1, . . . , d n , the loop λ(U j ) carries an invariant measure ν n j : Σ n → R with λ(U j ) as support and ν n j (A) = 1 if and only if A ∈ λ(U j ).
Let H 1 (X n ) be the vector space generated by ν n 1 , . . . , ν n dn . For a given measure µ ∈ M(X), the inclusion p n : X → X n induces a map (p n ) * : M(X) → H 1 (X n ) where (p n ) * (µ) is the measure obtained when Σ is restricted to Σ n . Similarly, the maps π n : X n+1 → X n induce a linear map (π n ) * : H 1 (X n+1 ) → H 1 (X n ). When represented using the bases {ν n+1 1 , . . . , ν n+1 d n+1 } and {ν n 1 , . . . , ν n dn }, it can be shown that (p n ) * = W n , where W n is the winding matrix corresponding to π n . This result and the fact that X = lim ←− πn X n leads to the conclusion that lim ←− Wn H 1 (X n ) is isomorphic to M(X).
Let I(X) ⊂ M(X) be the set of invariant measures of f and let P(X) ⊂ I(X) consist of the corresponding set of probability measures. Define is the set of invariant measures associated with X n and the corresponding set of probability measures is denoted by P (X n ).
Note that each I(X n ) is a cone in H 1 (X n ), W n (I(X n+1 )) = I(X n ), and lim ←− is well-defined. Hence I(X) is isomorphic to lim ←− Wn I(X n ) and P(X) is isomorphic to for each n is of bounded combinatorics since the largest entry that appears in any matrix is 16 and each matrix is 2 × 2. However, the minimal Cantor set obtained from the sequence of winding matrices {W n } ∞ n=1 where W n = 1 1 1 2 n for each n is not of bounded combinatorics since matrix entries are unbounded (2 n approaches infinity as n approaches infinity). Proof. Since the number of loops d n in X n is uniformly bounded by d for each n, we may assume without loss of generality that d n = d for each n. For j = 1, . . . , d, let t n j be the number of vertices in the loop λ(U j ). The basis measures ν n 1 , . . . , ν n d of H 1 (X n ) may be normalized to probability measures µ n 1 , . . . , µ n d by setting µ n j = 1 t n j ν n j for j = 1, . . . d.
Let P m (X m ) ⊂ H + 1 (X m ) be the set of probability measures and P m n = W nm (P m ). Since P m is the convex hull of the {µ n j }, P m n is the convex hull of the measures µ nm j = W nm (µ m j ). By taking a subsequence it can be assumed that the measures µ nm j converge to measures µ j ∈ P n for j = 1, . . . d. But P (X n ) = P m n , so P (X n ) is the convex hull of the measures {µ j : j = 1, . . . , d} and therefore it is the convex hull of at most d points (and hence has at most d extreme points and, consequently, at most d ergodic measures).
For the sake of contradiction, suppose (X, f ) carries more than d ergodic measures. For n sufficiently large, the projection of these ergodic measures would be distinct extremal points of P (X n ), which has at most d extremal points. This contradiction yields the desired result. Proof. It suffices to show that I(X) is one-dimensional. To this end, note that the hyperbolic distance between two points x, y ∈ H + 1 (X n ) is Here, m is the length of the line segment [x, y] and , r are the length of the con- containing [x, y]. Positive matrices contract the hyperbolic distance on positive cones. Since X is of bounded combinatorics, the winding matrices W n are uniformly bounded in size and entries. This implies that the contraction is uniform and is one-dimensional for each n. Since I(X) is isomorphic to lim Wn I(X n ), it is also one dimensional. As P(X) is a subset of I(X), it is too one-dimensional and hence the map f has only one ergodic invariant probability measure.

Non-Uniquely Ergodic Minimal Cantor Sets
We are interested in constructing combinatorially obtained minimal Cantor sets that are either uniquely ergodic but not of bounded combinatorics or nonuniquely ergodic. In either case, Theorem 2.4 stipulates that this can only be done if the minimal Cantor set does not have bounded combinatorics; that is, the winding matrices W n must be unbounded in size, possess unbounded entries, or both. To simplify computations, we will focus solely on those minimal Cantor sets that can be combinatorially obtained from square winding matrices of a fixed size.
The next proposition, which is a consequence of Theorem 2.3, specifies the maximum number of ergodic invariant probability measures that such a combinatorially obtained minimal Cantor set can possess.
Proposition 2.5. Let m ∈ N be fixed and let (X, f ) be a minimal Cantor set that is combinatorially obtained from a sequence of winding matrices {W n } ∞ n=1 where each matrix is of size m × m. Then (X, f ) has at most m ergodic invariant probability measures.
Proof. Consider the corresponding inverse limit X = lim ←− πn X n of f . For each n, the winding matrix W n describes the projection from the combinatorial cover X n+1 to the combinatorial cover X n . That is, for the images {V j } of the splitting vertex 0 X n+1 in X n+1 and the images {U i } of the splitting vertex 0 Xn in X n , an entry It is now clear that the number of loops in each X n is uniformly bounded by m, and the result follows from Theorem 2.3.
An illustration of a typical Lorenz map can be seen in Figure 5.
In [2], Winckler showed that there exist minimal Cantor sets under Lorenz maps which can obtained from the sequence of winding matrices {W n } ∞ n=1 , where and Martens provide insight into exactly how quickly the sequences of unbounded matrix entries must grow in order for non-unique ergodicity to be achieved. The main results in this section will introduce combinatorially obtained minimal Cantor sets whose associated winding matrices possess unbounded entries given by explicit sequences of integers. The specific structure of these winding matrices will allow us to address the precise growth rate needed to guarantee either unique or non-unique ergodicity given a specific additional restriction on the winding matrix entries of Winckler's Lorenz minimal Cantor set example.
Theorem 2.7. Let a, b, m, p ∈ Z + be such that a > b and m ≥ 2. Then the minimal Cantor set corresponding to the sequences of m × m winding matrices is uniquely ergodic when p = 1 and is not uniquely ergodic when p > 1. In the second case, there are in fact m ergodic invariant probability measures.
The proof of this result requires some knowledge regarding the convergence of infinite products. (1 + a n ) converges if and only if the series ∞ n=1 a n converges.
Proof of Theorem 2.7. Consider a sequence of winding matrices {W n } ∞ n=1 . As in [2], every invariant measure can be represented by an inverse limit of the sets and hence we should look on the sets We observe that if the limit as n approaches infinity of W k . . . W n K is of dimension m for each k, it will follow that the minimal Cantor set corresponding to {W n } ∞ n=1 possesses m ergodic invariant probability measures. Similarly, if the limit as n approaches infinity of W k · · · W n K is a one-dimensional space for each k, it will follow that the minimal Cantor set corresponding to {W n } ∞ n=1 is uniquely ergodic.
Let the main diagonal entries of the matrices W n be given by an p and all other entries are given by b. In seeking a convenient representation of the matrix product W k . . . W n for n > k, we observe that each W n is symmetric and exhibits two distinct eigenvalues. The first, λ = an p + b(m − 1), is of multiplicity one and has corresponding eigenvector consisting of all 1's. Hence each W n has spectral decomposition P D n P −1 , where and D n is the diagonal matrix with entries d Now for n > k, It can be shown that Let e i be the vector consisting of all zeros except for a single 1 in the i-th position. The angle θ n between W kn e i and W kn e j satisfies that and θ n = arccos W kn e i · W kn e j ||W kn e i || ||W kn e j || .
Thus the minimal Cantor set corresponding to {W n } ∞ n=1 is not uniquely ergodic if for each k there exist some e i , e j with i = j where {θ n } ∞ n=1 does not converge to zero as n approaches infinity. By equation (6), this is the same as showing that the righthand side of equation (5) does not converge to 1 as n approaches infinity.
If no such e i , e j exist for each k, then the minimal Cantor set corresponding to We observe that for any i = j, equation (5) becomes Since the only term in the righthand side of equation (7) dependent upon n is the behavior of the righthand side of (7) as n approaches infinity depends upon the behavior of the product (8) as n approaches infinity. Note that bm a p −b is positive for each and hence by Lemma 2.8 the behavior of the product (8) as n approaches infinity can be determined by the behavior of the series If p = 1, then series (9) diverges by comparison with the harmonic series ∞ =k 1 . Hence the product (8) approaches infinity as n approaches infinity and it can be shown that the righthand side of equation (7) approaches 1 as n approaches infinity. Hence {θ n } ∞ n=1 converges to 0 as n approaches infinity and it follows that the minimal Cantor set corresponding to the sequence of winding matrices {W n } ∞ n=1 is uniquely ergodic.
If p > 1, then series (9) converges by comparison with the p-series ∞ =k 1 p .
Hence the product (8) converges to a nonnegative number, say T , as n approaches infinity and it can be shown that the righthand side of equation (7) converges to T 2 −1 T 2 −1+m as n approaches infinity. This ratio is clearly never equal to 1 for any value of T , and so the righthand side of equation (7) will not be uniquely ergodic. We will further explore this idea by introducing two additional families of non-uniquely ergodic minimal Cantor sets. As will be demonstrated, the ergodicity of the first is ultimately determined by a comparison test involving a geometric series.
Theorem 2.9. Let a, b, m ∈ Z + be such that a > b and m ≥ 2. Then the minimal Cantor set corresponding to the sequences of m × m winding matrices is not uniquely ergodic. In fact, there are m ergodic invariant probability measures.
Note that the requirement that a, b ∈ Z + with a > b ensures that a = 1. If a = 1, then each W n would be identical and the corresponding minimal Cantor set would be of bounded combinatorics, and hence uniquely ergodic by Theorem 2.4.
Proof of Theorem 2.9. We will utilize the method that was employed in the proof of Theorem 2.7.
Let the main diagonal entries of the matrices W n be given by a n and all other entries are given by b. Each W n is symmetric and exhibits two distinct eigenvalues.
The first, λ = a n + b(m − 1), is of multiplicity one and has corresponding eigenvector consisting of all 1's. Hence each W n has spectral decomposition P D n P −1 , where P is the matrix given in (3) and D n is the diagonal matrix with entries d The matrix product W kn is given in equation (4).
Let e i be the vector consisting of all zeros except for a single 1 in the i-th position. For i = j, the angle θ n between W kn e i and W kn e j satisfies equations (5), (6), and (7) above, and it will follow that the minimal Cantor set corresponding to is not uniquely ergodic if it can be shown that there exist e i , e j with i = j such that the sequence of angles {θ n } ∞ n=1 does not converge to zero as n approaches infinity. This is equivalent to showing that the righthand side of equation (7) does not converge to 1 as n approaches infinity.
As above, the only term in the righthand side of equation (7) dependent upon n is and bm a −b is positive for each . Hence by Lemma 2.8, the convergence of the product (10) depends on the convergence of the series ∞ =k bm a −b . This series converges by comparison with the geometric series ∞ =k 1 a (to see this, recall that a, b ∈ Z + with a > b, which implies that a > 2). Hence the righthand side of equation (10) converges to a nonnegative number, say T , as n approaches infinity and, as in the proof of Case 1 of Theorem 2.7, it can be shown that the righthand side of is not uniquely ergodic. In fact, there are m ergodic invariant probability measures.
Proof. The proof is similar to the proofs of Theorems 2.7 and 2.9.
We wish to apply Theorems 2.7, 2.9, and 2.10 in the case of Lorenz minimal Cantor sets. However, each winding matrix that we have considered has unbounded entries located along the main diagonal while Lorenz minimal Cantor sets have winding matrices of form (2) with unbounded entries located along the minor diagonal. The subsequent theorem will allow us to overcome this obstacle.
Recall that a permutation matrix of size m × m can be obtained by permuting the rows of the m × m identity matrix. Left multiplication of a square matrix A by a permutation matrix of the same size will permute the rows of A while right multiplication will permute the columns of A. To prove Theorem 2.11, we will make use of the following lemma from the theory of permutation matrices. We now proceed as in the proofs of Theorems 2.7 and 2.9. As before, let W kn = W k . . . W n . Since W n = P W n for each n, we have that W kn = P W k . . . P W n = P n−k W k . . . W n by Lemma 2.12. The product W k . . . W n is given in (4) and therefore The As a power of a permutation matrix, P n−k is itself a permutation matrix.
Hence the product d 11 m P n−k D will be a scalar multiple of the matrix D with its rows permuted in some manner. Hence for each n, the angle θ n between W kn e i and W kn e j for i = j will satisfy equations (5), (6), and (7). As a consequence,  (2). If we let a n = b n , the winding matrices satisfy that W n = 1 a n a n 1 .
Let P be the 2 × 2 permutation matrix with 1's along the minor diagonal so that W n = P a n 1 1 a n . W n = 1 an p an p 1 , W n = 1 a n a n 1 , or W n = 1 a n n p a n n p 1 are not uniquely ergodic.

Future Work
The structure of each of the winding matrices considered in this chapter allowed for a convenient diagonal representation of the matrix product W kn . This is not the case in general, as can be seen by considering the winding matrices introduced in the two conjectures below. These conjectures generalize the example of Gambaudo and Martens described below Proposition 2.5 and are supported by preliminary numerical trials. In each case, however, the eigenvectors corresponding to the winding matrix W n depend upon n. This makes obtaining a closed form formula for the matrix product W kn challenging, and it is likely that a different technique will need to be employed to establish the ergodicity of these and other minimal Cantor sets that are combinatorially obtained from winding matrices that exhibit less symmetry than those that were considered in Theorems 2.7, 2.9, 2.10, and 2.11. Conjecture 2.14. Let a, b, m, p ∈ Z + be such that a > b and m ≥ 2. Then the minimal Cantor set corresponding to the sequence of m × m winding matrices where are k ≥ 2 copies of an p and m − k copies of b along the main or minor diagonal, respectively, is uniquely ergodic when p = 1 and is not uniquely ergodic where there are k ≥ 2 copies of a n and m − k copies of b along the main or minor diagonal, respectively, is not uniquely ergodic and there are in fact k ergodic invariant probability measures.

Introduction
For f : X → X and g : Y → Y continuous functions, f is said to be topologically conjugate to g if there exists a homeomorphism h : X → Y such that Topological conjugacy plays an important role in studying the dynamics of continuous maps since two topologically conjugate maps are known to have the same dynamical properties.
Recall that for f and g defined as above, f is said to be topologically semiconjugate to g if the function h above is a continuous, surjective function rather than a homeomorphism and the diagram is still commutative. In this case, the map h is called a topological semi-conjugation. Such semi-conjugations also preserve a number of dynamical properties, and if f is semi-conjugate to g, it can be said that the dynamics of (X, f ) are at least as complicated as the dynamics of (Y, g).
It is well-known that topological semi-conjugations do not preserve measuretheoretical properties. For this reason, it is not necessarily a surprising result that a non-uniquely ergodic minimal Cantor set may be semi-conjugated to a uniquely ergodic minimal Cantor set. In this chapter, it will be shown that for specific combinatorially obtained minimal Cantor sets, the above statement is true even when the semi-conjugation map is almost everywhere injective with respect to the ergodic measure associated with the uniquely ergodic minimal Cantor set.

Semi-Conjugations Involving an Adding Machine
We will take as (Y, g) in the definition of a topological semi-conjugation a dynamical system chosen from a particular family of combinatorially obtained minimal Cantor sets called adding machines. Note that adding machines are also frequently referred to as solenoids or odometers in the literature, where they have been studied by many authors, including Collas and Klein, Gambuado and Martens, and Oversteegen (see [1,2,3]).

Proposition 3.2. An adding machine is uniquely ergodic.
Proof. Since the dimension of each winding matrix is 1, the adding machine possesses at most one and hence exactly one ergodic invariant probability measure by Proposition 2.5.
In sections 3.3 and 3.4, it will be shown that there exist topological semiconjugations between non-uniquely ergodic combinatorially obtained minimal Cantor sets and adding machines that are almost everywhere injective with respect to the unique ergodic measure on the given adding machine. These results require the establishment of a property that will guarantee that certain combinatorially obtained minimal Cantor sets are topologically semi-conjugate to a given adding machine.

Proposition 3.3.
Let (X, f ) with X = lim ←− πn X n be a combinatorially obtained minimal Cantor set and let (Y, g) with Y = lim ←− ψn Y n be an adding machine. If each of the d n loops of X n possess the same number of vertices as the single loop of Y n then for each n ∈ Z + there exist continuous, surjective maps p n : X n → Y n and p n+1 : X n+1 → Y n+1 such that p n • π n = ψ n • p n+1 ; that is, the following diagram commutes.
The proof of this proposition requires assigning "labels" to certain vertices in a given combinatorial cover. Note that if the combinatorially cover X n is made up of multiple loops, there may be multiple vertices in a given position. To illustrate this, let U 1 , . . . , U dn be the d n distinct images of the splitting vertex 0 Xn . The shortest directed path from the splitting vertex to each of these vertices consists of a single directed edge, so each of U 1 , . . . , U dn is said to be in position 1 in X n .
Proof of Proposition 3.3. For each n ∈ N, let t n represent the number of vertices in the single loop of Y n and let s n represent in the number of vertices in each of the d n loops of X n . By hypothesis, t n = s n for each n ∈ N. Hence for each n ∈ N, there exists a continuous, surjective map p n : X n → Y n that projects each loop of X n once around the single loop of Y n by mapping x ∈ X n to the unique vertex y ∈ Y n located in position n (x) in Y n . Specifically, this guarantees the existence of continuous, surjective maps p n : X n → Y n and p n+1 : X n+1 → Y n+1 for each n ∈ N.
It remains to be shown that for any x ∈ X n+1 , the equation holds. To prove this, fix x ∈ X n+1 . The position of x in X n+1 is n+1 (x). Now π n (x) is a vertex x ∈ X n in position n+1 (x) mod s n . By definition, p n (x ) is the unique vertex in Y n that is in position n+1 (x) mod t n since s n = t n for each n ∈ N.
Next consider p n+1 (x). This will be the vertex x ∈ Y n+1 that is in position Since Y n is a single loop, there can only be one vertex in position n+1 (x) mod t n . It follows that (p n • π n )(x) = (ψ n • p n+1 )(x).
Theorem 3.5. Let (X, f ) and (Y, g) be as in Proposition 3.3. Then there exists a continuous, surjective map p : X → Y such that f : X → X and g : Y → Y satisfy that p • f = g • p; that is, the following diagram commutes.
Proof. Since Proposition 3.3 holds for each n ∈ N, there exists a sequence {p n } of continuous, surjective maps p n : X n → Y n satisfying that p n • π n = ψ n • p n+1 .
This sequence extends to a continuous, surjective map p : X → Y satisfying that

Semi-Conjugation Between a Finitely Non-Uniquely Ergodic Minimal Cantor Set and an Adding Machine
The main objective of this section is to show that there exists a family of finitely non-uniquely ergodic minimal Cantor sets that can be semi-conjugated to an adding machine in such a way that the semi-conjugation map is almost everywhere injective with respect to the uniquely ergodic probability measure associated with a corresponding adding machine. To do this, we will begin by establishing that the following family of combinatorially obtained minimal Cantor sets possesses members that are non-uniquely ergodic.
Proposition 3.6. The minimal Cantor set that is combinatorially obtained from the sequence of winding matrices {W n } ∞ n=1 , with W n =   a n a n a n na n + 1 2na n 2 na n + 1 2 2na n   is not uniquely ergodic and in fact possesses at least two ergodic invariant probability measures for an appropriately chosen sequence of integers {a n } ∞ n=1 .
Proof. Let and let e 1 , e 2 , e 3 be the standard basis vectors for R 3 . The set of vectors {e 2 , e 3 } forms a basis for V . For any two vectors x, y ∈ V , define d(x, y) = arccos x · y ||x|| ||y|| ; a n a n a n . . . a n na n + 1 (m − 1)na n 2 2 . . . 2 is not uniquely ergodic and in fact possesses at least m − 1 ergodic invariant probability measures for an appropriately chosen sequence of integers {a n } ∞ n=1 .
Proof. The proof is analogous to the proof of Proposition 3.6.
For the remainder of this section, let (X, f ) with X = lim ←− πn X n be the minimal Cantor set that is combinatorially obtained as in Proposition 3.7. We will make the additional presumptions that each of the loops of X n possesses the same number of vertices for each n and that any pair of loops in X n has only the splitting vertex in common. Also, let (Y, g) with Y = lim ←− ψn Y n be an adding machine that is combinatorially obtained from the sequence of 1 × 1 winding matrices { (m − 1)n + 1 a n + 2} ∞ n=1 and also satisfies that the single loop of Y 1 has the same number of vertices as each of the loops of the combinatorial cover X 1 .
Proposition 3.8. For each n ∈ N there exist continuous, surjective maps p n : X n → Y n and p n+1 : X n+1 → Y n+1 satisfying that p n • π n = ψ n • p n+1 . Moreover, there exists a continuous, surjective Proof. Let t n be the number of vertices in Y n and let s n be the number of vertices in each loop of X n for each n ∈ N. Note that for each n ∈ N, the number of vertices t n+1 in the single loop of Y n+1 is given by (m − 1)n + 1 a n + 2 t n and the number of vertices s n+1 in each of the loops of X n+1 is given by the sum of the corresponding column of W n . Since each of the columns of W n sum to (m − 1)n + 1 a n + 2, it follows that the number of vertices in each loop of X n+1 is given by (m − 1)n + 1 a n + 2 s n .
Moreover, since s 1 = t 1 , it follows that s n = t n for each n ∈ N. Hence the hypotheses of Proposition 3.3 and Theorem 3.5 are met and each of the results follows.
It remains to be shown that the semi-conjugation map p of Proposition 3.8 is almost everywhere injective with respect to the uniquely ergodic invariant probability measure associated with the adding machine (Y, g).
For each n ∈ N, recall that t n represents the number of vertices in the single loop of Y n . For each n ∈ N \ {1}, define the set U n = y ∈ Y n : n (y) ∈ {0, 1, . . . , a n−1 t n−1 } .
Note that U n consists of the first a n−1 t n−1 vertices of the single loop of Y n for each n ∈ {2, 3, . . .}.
Proof. Suppose y n+1 ∈ U n+1 . Then n+1 (y n+1 ) ∈ {0, 1, . . . , a n t n }. We observe that #{p −1 n+1 (y n+1 )} is either 1 or m since each of the m loops of X n+1 is projected once around the single loop of Y n+1 . Moreover, each of x n+1 , x n+1 ∈ {p −1 n+1 (y n+1 )} is located in the same position (though possibly in distinct loops) in X n+1 as it is in Y n+1 since each of the loops of X n+1 are the same length as the single loop of Y n+1 .
Inspection of the winding matrix corresponding to π n (see equation (14)) reveals that each loop of X n+1 is wrapped a n times around the first loop of X n .
Since each loop of X n is of length t n , this means that each of the vertices located in a given position chosen from 0, 1, . . . , a n t n in X n+1 will be mapped to a single vertex in one of the positions 0, 1, . . . , t n − 1 in the first loop of X n . In particular, x n+1 , x n+1 ∈ {p −1 n+1 (y n+1 )} are mapped under π n to a single vertex in X n ; that is; x n = x n . Now x k = x k for each k = 1, ..., n by the definition of an inverse limit.
Proof. Let x, x ∈ {p −1 (y)}. Since y n ∈ U n for infinitely many n, Lemma 3.9 yields that there exists an infinite sequence of positive integers {n j } ∞ j=1 satisfying that n 1 < n 2 < . . . and x k−1 = x k−1 for k = 1, . . . , n j . As an immediate consequence, We will now show that the set of points y = (y 1 , . . . , y n , . . .) ∈ Y satisfying that y n ∈ U n for infinitely many n (which, by Proposition 3.10, is exactly the set of points y ∈ Y which have a unique pre-image under p) is of full measure with respect to the unique ergodic measure associated with the adding machine (Y, g).
This will complete the proof that the semi-conjugation map p is almost everywhere injective with respect to this measure.
Proposition 3.11. Let µ be the unique ergodic invariant probability measure associated with the adding machine (g, Y ), let U n be defined as in equation (15) of page 39, and let L 0 = {y ∈ Y : y n ∈ U n for infinitely many values of n}.
Proof. Let We will prove the equivalent statement that µ(L 1 ) = 0. To this end, let where for each , the space {0, . . . , t − 1} is equipped with the discrete topology.
Note that we can associate to each y = (y 1 , . . . , y n , . . .) ∈ Y the corresponding point y = ( 1 (y 1 ), . . . , n (y n ), . . .) ∈ L (and vice versa since Y n consists of a single loop for each n). Hence there is a homeomorphism Now since Y 1 consists of t 1 vertices, Y 1 carries the probability measure which assigns to each vertex of Y 1 the weight 1 t 1 . For n = 2, 3, . . ., the combinatorial cover Y n consists of t n = (m − 1)(n − 1) + 1 a n−1 + 2 t n−1 vertices, so Y n carries the probability measure µ n which assigns to each vertex of Y n the weight 1 (m − 1)(n − 1) + 1 a n−1 + 2 t n−1 .
For each n ∈ {2, 3, . . .}, the number of vertices in Y n that are also in U n is a n−1 t n−1 . The number of vertices in Y n that are not in U n is given by t n − a n−1 t n−1 = (m − 1)(n − 1) + 1 a n−1 + 2 t n−1 − a n−1 t n−1 . Nowμ It can be shown that lim that H : Y → L is a homeomorphism and that Y is uniquely ergodic guarantees that µ(L 1 ) = 0, and we have the desired result.
Together, the results in this section prove the following theorem.
a n a n a n . . . a n na n + 1 (m − 1)na n 2 2 . . . 2 is finitely non-uniquely ergodic for an appropriately chosen sequence of integers For such a sequence {a n } ∞ n=1 , this minimal Cantor set is semi-conjugated to an adding machine in such a way that the semi-conjugation map is almost everywhere injective with respect to the unique ergodic probability measure on the adding machine.

Semi-Conjugation Between an Infinitely Non-Uniquely Ergodic Minimal Cantor Set and an Adding Machine
The main result of the previous section shows that there exists a non-uniquely ergodic minimal Cantor set possessing any finite number of ergodic invariant probability measures we like that is semi-conjugated to an adding machine in such a way that the semi-conjugation is almost everywhere injective with respect to the unique ergodic measure associated with the adding machine. It seems natural that if this can be done with a non-uniquely ergodic minimal Cantor set carrying as many ergodic measures as we like, it should also be possible for a minimal Cantor set carrying an infinite number of ergodic measures. The main result of this section will show that this is indeed the case.
Thus far each of the constructions we have considered have consisted of minimal Cantor sets which can be combinatorially obtained from square winding matrices of the same size. Winding matrices are allowed to be rectangular, however, provided that the size of each winding matrix W n leads to the matrix product W kn = W k . . . W n being well-defined for each k < n. In order to introduce an infinitely non-uniquely ergodic minimal Cantor set, we will consider a particular family of minimal Cantor set that can be combinatorially obtained from certain rectangular winding matrices.
Proposition 3.13. For each n, let the W n be given by the m n ×(m n +1) rectangular a n a n . . . a n a n where na n is divisible by m n − 1 and m n = n + 2 for each n. For an appropriately chosen sequence of integers {a n } ∞ n=1 , the minimal Cantor set that is combinatorially obtained from the sequence of winding matrices {W n } ∞ n=1 is not uniquely ergodic and in fact possesses an infinite number of invariant probability measures.
Note that in the proposition above, W 1 is a 3 × 4 matrix, W 2 is a 4 × 5 matrix, W 3 is a 5×6 matrix, and so on. In particular, the matrix product W kn = W k . . . W n is well-defined for each k < n.
Proof of Proposition 3.13. For each n, let and let e mn } forms a basis for V mn . For any two vectors x, y ∈ V mn , define d : V mn → R to be the angle between the vectors x and y as in equation (12). As in Proposition 3.6, if it can be shown that for any i = j with i, j = 1, it will follow that the minimal Cantor set under consideration is not uniquely ergodic. For each n, we may choose a n large enough so that By taking the limit as n approaches infinity of each side of the above inequality, it can be seen that Since this holds for any i = j with i, j = 1, it follows that the set is at least (m k −1)-dimensional for each k. As {m k } is a strictly increasing sequence of positive integers, I = lim ←− Wn I k must be infinite-dimensional. Hence the minimal Cantor set that is combinatorially obtained from the sequence of winding matrices {W n } ∞ n=1 is not only non-uniquely ergodic, but in fact possesses an infinite number of ergodic invariant probability measures.
For the remainder of this section, let (X, f ) with X = lim ←− πn X n be the minimal Cantor set that is combinatorially obtained as in Proposition 3.13 and let (Y, g) with Y = lim ←− ψn Y n be an adding machine where each single loop Y n has the same number of vertices t n as each of the loops in X n ; that is, (Y, g) is obtained combinatorially from the sequence of winding matrices {w n } ∞ n=1 where w n = ((n + 1)a n + 2) for each n, and X 1 and Y 1 are both assumed to have t 1 vertices. We also make the additional presumption that any pair of loops in X n has only the splitting vertex in common.
Lemma 3.14. For each n ∈ N there exist continuous, surjective maps p n : X n → Y n and p n+1 : X n+1 → Y n+1 satisfying that p n •π n = ψ n •p n+1 . Moreover, there exists a continuous, surjective map p : X → Y such that p • f = g • p.
Proof. For each n, each of the loops of X n has the same number of vertices as the single loop of Y n . Since neither Proposition 3.3 nor Theorem 3.5 specify that the winding matrices corresponding to the minimal Cantor set where X = lim ←− πn X n must be square, both findings hold regardless of the size of these winding matrices, and we have the desired result.
Proof. Note that the number of loops in X n+1 is given by the number of columns in W n ; since W n is an m n × (m n + 1) matrix, this number is m n + 1.
Suppose y n+1 ∈ U n+1 . Then n+1 (y n+1 ) ∈ {0, 1, . . . , a n t n }. We observe that #{p −1 n+1 (y n+1 )} is either 1 or m n + 1 since each of the m n + 1 loops of X n+1 is projected once around the single loop of Y n+1 . Moreover, each of is located in the same position (though possibly in distinct loops) in X n+1 as it is in Y n+1 since each of the loops of X n+1 are the same length as the single loop of Y n+1 .
The remainder of the proof is identical to the second paragraph of the proof of Lemma 3.9.
Proof. This is a consequence of Lemma 3.15. The proof is analogous to that of Proposition 3.10.
We will now show that p is almost everywhere injective with respect to the uniquely ergodic invariant probability measure associated with the adding machine (Y, g).
Proposition 3.17. Let µ be the uniquely ergodic invariant probability measure associated with the adding machine (Y, g) and let L 0 = {y ∈ Y : y n ∈ U n for infinitely many values of n}.
Proof. As in the proof of Proposition 3.11, we will let L 1 = Y \ L 0 = {y ∈ Y : y n ∈ U n for all n}.
and prove the equivalent statement that µ(L 1 ) = 0. Again, there exists a homeomorphsim H : Y → L that assigns to each y = (y 1 , . . . , y n , . . .) ∈ Y the corre- Now Y 1 carries the probability measure which assigns to each vertex of Y 1 the weight 1 t 1 . For n = 2, 3, . . ., the combinatorial cover Y n consists of t n = (na n−1 + 2)t n−1 vertices, so Y n carries the probability measure µ n which assigns to each vertex of Y n the weight 1 (na n−1 + 2)t n−1 .
The number of vertices in Y n that are not in U n for each n is given by t n − a n−1 t n−1 = (na n−1 + 2)t n−1 − a n−1 t n−1 . Nowμ It can be shown that lim H : Y → L and the fact that Y is uniquely ergodic guarantees that µ(L 1 ) = 0, and we have the desired result.
We have proved the following theorem.
Theorem 3.18. The minimal Cantor set that is combinatorially obtained from the sequence of m n × (m n + 1) (m n ≥ 3 and m n = n + 2) winding matrices a n a n . . . a n a n is infinitely non-uniquely ergodic for an appropriately chosen sequence of integers {a n } ∞ n=1 . For such a sequence {a n } ∞ n=1 , this minimal Cantor set is semi-conjugated to an adding machine in such a way that the semi-conjugation map is almost everywhere injective with respect to the unique ergodic probability measure on the adding machine.

Kneading Theory and Consequences for Unimodal Maps
Recall that a unimodal map is a map u : [0, 1] → [0, 1] satisfying u(0) = u(1) = 0 and for which there exists a critical point c ∈ (0, 1) such that u is increasing to the left of c and decreasing to the right of c.
One family of unimodal maps that we will consider is a subset of the family of tent maps as defined in Example 1.2. We will consider those maps T a : and a ∈ (0, 2]. Note that each tent map achieves its maximum value at its critical point x = 1 2 . Closely related to the family of tent maps is the family of stunted tent maps, which consist of those functions S a,p : [0, 1] → [0, 1] for which the corresponding tent map T a has been "stunted" by replacing its peak with a plateau at a given Note that if u i (x) = c, we take s i to be 0 without loss of generality. The kneading sequence K of a unimodal map is the given by the itinerary of the critical value u(c) = c 1 ; that is, K = I(c 1 ). Moreover, a binary sequence s ∈ {0, 1} N is said to be admissible for u if there exists x ∈ [0, 1] with itinerary I(x) = s.

Kneading sequences and kneading theory were introduced and explored by
Milnor and Thurston in their paper On Iterated Maps of the Interval (see [1]). We will use the related theory developed by Isola and Politi (see [2]) in order to determine whether or not a given binary sequence s ∈ {0, 1} N is admissible as the kneading sequence of a smooth unimodal map. Once an ordering that reflects the ordering on the real line has been established, it is known that the admissible binary sequences are exactly those which never become greater than the kneading sequence when shifted.
The method of Isola and Politi begins by associating to a given sequence Since c 1 is the maximum value in the image of u and it can be proved that the map τ • I is increasing (see [3]), the next theorem holds.    4.4. For each k ∈ N, the difference between two consecutive cutting times S k and S k−1 is also a cutting time and the kneading map is defined as Example 4.5. The Feigenbaum unimodal map exhibits cutting times given by powers of 2; that is, S 0 = 1 and S k = 2S k−1 for k ∈ N. Hence its kneading map is given by Example 4.6. The Fibonacci unimodal map is named after its cutting times, which are given by the Fibonacci numbers; that is; S 0 = 1, S 1 = 1, and Hence its kneading map is given by Remark 4.7. It is known that two unimodal maps with the same kneading sequence have the same kneading map, and vice versa.
The proof of the following result can be found in [4]. In particular, the above states that the number of ergodic measures of u supported on ω(c) is the same as those ofũ supported on ω(c).

Obtaining the Desired Unimodal Map
This section will culminate in the main result proving the existence of a logistic unimodal map q for which (ω(c), q) is non-uniquely ergodic and may be semiconjugated to an adding machine in such a way that the semi-conjugation map is almost everywhere injective.
To prove this result, we will begin by showing that there exists a minimal Cantor set under the tent map T of Theorem 4.1 that is homeomorphic to a non-uniquely ergodic combinatorially obtained minimal Cantor set satisfying the hypotheses of Theorem 3.12.
Remark 4.9. By taking m = 3 in Theorem 3.12, it can be said that the minimal Cantor set (X, f ) with X = lim ←− πn X n that is combinatorially obtained from the sequence of winding matrices {W n } ∞ n=1 , where W n =   a n a n a n na n + 1 2na n 2 na n + 1 2 2na n   , is non-uniquely ergodic for an appropriately chosen sequence of integers {a n } ∞ n=1 .
For such a sequence {a n } ∞ n=1 , this minimal Cantor set is semi-conjugated to the adding machine (Y, g) with Y = lim ←− ψn Y n satisfying that, for each n, the single loop Y n has the same number of vertices t n as each of the three loops in X n . Moreover, the semi-conjugation map p is almost everywhere injective with respect to the unique ergodic probability measure µ on (Y, g). For the induction hypothesis, suppose that for each word Proof. For the sake of contradiction, suppose that o 1 (I ω ) ∩ o 2 (I ω ) = ∅ and that o 1 (I ω ) = o 2 (I ω ). Then there are three possibilities: either o 2 (I ω ) ⊂ int(o 1 (I ω )), ∂o 1 (I ω )∩int(o 2 (I ω )) = ∅, or o 1 (I ω ) and o 2 (I ω ) intersect at a common endpoint only.
Case 1: Suppose o 2 (I ω ) ⊂ int(o 1 (I ω )). Then For the remainder of this section, let (X, f ) with X = lim ←− πn X n and (Y, g) with Y = lim ←− ψn Y n be the minimal Cantor sets defined as in Remark 4.9. Then t n is the number of vertices in the single loop of Y n as well as the number of vertices in each loop of X n . Inspection of the sequence of winding matrices associated with the sequence of combinatorial covers {π n } ∞ n=1 given in Remark 4.9 reveals that t n+1 = (2n + 1)a n + 2 t n for each n ∈ N. ⊂ I n 0 satisfying that ) is a proper subset of one of I n 0 , I n 1 , I n 2 for = 1, . . . , t n − 1 and k ∈ {0, 1, 2}. Lemma 4.12 additionally allows us to presume that if T 1 (I n k 1 ) ∩ T 2 (I n k 2 ) = ∅, then T 1 (I n k 1 ) = T 2 (I n k 2 ). Moreover, Lemma 4.11 results in a natural inclusion of the orbit of I n+1 k for each k within the orbits of I n j for j = 0, 1, 2. We can specifically assume that this inclusion reflects the winding matrices {W n } ∞ n=1 as defined in Remark 4.9.
For each n ∈ N and k ∈ {0, 1, 2}, the intervals I n+1 k are pairwise disjoint with orbits under T that come to coincide at some iterate t, where 1 ≤ t ≤ t n − 1.
The orbits of these intervals can thus be thought of in the sense of a combinatorial cover, as will be explained in detail within the proof of the next proposition.
Proposition 4.13. For each n ∈ N and each k ∈ {0, 1, 2} define Then there exists a combinatorial cover X O n and a homeomorphism h n : O n → X O n for each n ∈ N.
Proof. For each k ∈ {0, 1, 2} and n ∈ N, consider the subset of the orbit of I n k under the tent map T given by λ(I n k ) = {I n k , T (I n k ), . . . , T tn−1 (I n k )}.
By definition, λ(I n k ) consists of exactly t n intervals for each k and n.
To form the combinatorial cover X O n , begin by creating a vertex for each interval in λ(I n 0 ). Place a directed edge between two vertices x and y if and only if the interval T (I n 0 ) associated with the vertex x satisfies that T +1 (I n 0 ) is the interval associated with the vertex y. Proceed in the same fashion with λ(I n 1 ) and λ(I n 2 ), but if any of the intervals in either of these is the same as an interval I in I n 0 , associate the interval with the same vertex that I was associated with.
Note that this construction results in the vertex associated with I n−1 As we wish to consider minimal Cantor sets, the next step is to the prove that there also exists a sequence of combinatorial refinements {π O n : Proposition 4.14. For each n, there exists a combinatorially refinement π O n : X O n+1 → X O n that can be described by the sequence of winding matrices associated with the combinatorial refinements π n : X n+1 → X n .
Due to how the sets O n were chosen, a combinatorial refinement π O n : X O n+1 → X O n may be defined in such a way that the corresponding sequence of winding matrices is given by {W n } ∞ n=1 (the same sequence used to obtained the non-uniquely ergodic minimal Cantor set (X, f ) of Remark 4.9). Proof. Lemmas 4.13 and 4.14 yield the following diagram.
This diagram implies the existence of a continuous, surjective map Proof. Each of the three loops of the combinatorial cover X n consists of t n vertices with the only common vertex being the splitting vertex. Each of the three loops of the combinatorial cover X O n also consists of t n vertices, but in this case each loop may have multiple vertices in common.
Let U 1 , U 2 , and U 3 be the images of the splitting vertex in X n and let U O 1 , U O 2 , and U O 3 be the images of the splitting vertex in X O n . Define the p O n : X n → X O n so that the vertex x ∈ λ(U i ) that is located in position n (x) in X n is mapped to the vertex y ∈ λ(U O i ) located in position n in X O n . By definition, p O n is continuous and onto for each n and satisfies that p O n • π n = π O n • p O n+1 .
The sequence of maps {p n : X n → X O n } ∞ n=1 may then be extended to a continuous, surjective map p O : Proposition 4.18. There exists a continuous, surjective map ρ : Proof. Since each of the loops of X O n has the same number of vertices as the single loop of Y n for n = 0, 1, . . ., Theorem 3.5 guarantees the existence of a continuous, surjective map ρ : X O → Y with the desired property.
We have the following commutative diagram.
This theorem brings us one step close to our ultimate goal of proving that the minimal Cantor set of Remark 4.9 can be taken to be (ω(c), q) for some logistic map q with critical point c. However, O is not the omega limit set of the critical point of T . We can make it the omega limit set of the critical point of a particular corresponding stunted tent map, however. To this end, let ν be the maximal element of the Cantor set O and let S be the stunted tent map corresponding to T with plateau at height ν. Now (ω(ν), S) is exactly the same as (O, T ). Moreover, ν is the critical value of S and we can consider the kneading sequence K = I(ν) of S. Proof. Since two unimodal maps with the same kneading invariant also have the same kneading map (see Remark 4.7), it follows immediately from Theorem 4.8 and the previous results in this section that (ω(c), q) is non-uniquely ergodic.
Together, the results of this section prove the subsequent result. isfying that ω(c) is a non-uniquely ergodic minimal Cantor set that can be semiconjugated to a specific adding machine in such a manner that the semi-conjugation map is almost everywhere injective with respect to the single ergodic invariant probability measure associated with the adding machine.
Even though we chose to consider the simplest possible sequence of winding matrices from Theorem 3.12, the lemmas, propositions, and proofs within this section can be easily modified to prove the result when considering any sequence of winding matrices that matches the criteria laid out in the statement of Theorem 3.12. This results in the following more general theorem.

Future Work
In this chapter, we proved that there exist finitely non-uniquely ergodic minimal Cantor sets obtained as the omega-limit set of certain logistic unimodal maps that may be semi-conjugated to a corresponding adding machine in such a way that the semi-conjugation map is almost everywhere injective with respect to the unique ergodic measure on the adding machine. It is likely that the lemmas, propositions, and proofs contained within this chapter can be modified in order to prove the same result where the logistic map is infinitely non-uniquely ergodic instead. can be semi-conjugated to a specific adding machine in such a way that the semiconjugation map is almost everywhere injective with respect to the single ergodic