Z d -TOEPLITZ ARRAYS

Transcription

1 Z d -TOEPLITZ ARRAYS MARIA ISABEL CORTEZ Abstract In ths paper we gve a defnton of Toepltz sequences and odometers for Z d actons for d whch generalzes that n dmenson one For these new concepts we study propertes of the nduced Toepltz dynamcal systems and odometers classcal for d = In partcular, we characterze the Z d -regularly recurrent systems as the mnmal almost - extensons of odometers and the Z d -Toepltz systems as the famly of subshfts whch are regularly recurrent Introducton Toepltz sequences have been ntroduced n dynamcal systems by Jacobs and Keane n [] Snce then, they have been extensvely studed n dfferent contexts and they have been used to provde a seres of examples wth nterestng dynamcal propertes (see for example [6], [0], [7], [6]) Toepltz flows are characterzed as mnmal almost - symbolc extensons of odometers systems by Markley and Paul In [8] Downarowcz and Lacrox publsh a proof of ths theorem In addton, as t was shown by Gjerde and Johansen n [0], Toepltz systems also correspond, up to conjugacy, to the famly of expansve Brattel-Vershk systems assocated to Brattel dagrams wth the equal path number property The am of ths paper s to extend the defnton of both odometers and Toepltz flows to Z d -actons and to settle down a characterzaton result, n ths general context, n the sense of Markley and Paul A frst approach to ths problem was made by Downarowcz n [5], where he ntroduces the Z 2 -Toepltz arrays Snce any element of a Z d -subshft may be seen as a tlng of R d, Z d -Toepltz arrays are a class of nterestng examples of perfect tlngs In Secton 2, we gve some basc defntons relevant for the study of Z d -actons and n Secton 3 we ntroduce the generalzed noton of an odometer In Secton 4, we ntroduce Z d -regularly recurrent systems and we characterze them as the mnmal almost - extensons of odometers In Secton 5, we dentfy the set of egenvalues of odometers and the set of contnuous egenvalues of regularly recurrent systems and we characterze those whch are measure-theoretcally conjugate to ther maxmal equcontnuous factors In Secton 6, we defne Z d -Toepltz arrays and we show that they are the famly of regularly recurrent Z d -subshfts We prove that every Topeltz array has, as n the case d =, a perodc structure that allows to dentfy the maxmal equcontnuous factor of the assocated Toepltz system We generalze the noton of a regular Toepltz sequence to hgher dmensons In Secton 7, we ntroduce the concept 99 Mathematcs Subject Classfcaton Prmary: 54H20; Secondary: 37B50 Key words and phrases Almost - extensons, Toepltz, tlng, Odometer

2 2 MI Cortez of a semcocycle The last secton contans an example of a Z 2 -Toepltz system wth a determned fnte number of ergodc measures and another example of a unquely ergodc Z 2 -Toepltz system wth postve entropy 2 Basc Defntons and Background Let d be an nteger In ths artcle, by a topologcal dynamcal system we mean a par (X, Z d ), where Z d acts, by homeomorphsm, on a compact metrc space X Gven v Z d and x X we wll dentfy v wth the assocated homeomorphsm and we denote by v(x) the acton of v on x The dynamcal system (X, Z d ) s free f v(x) = x for some x X mples v = 0 For a subgroup Z Z d somorphc to Z d, the Z-orbt of x X s O Z (x) = {v(x) : v Z} and the Z-system assocated to x s (Ω Z (x), Z), where Ω Z (x) s the closure of O Z (x) and the acton of Z on Ω Z (x) s the restrcton to Z and Ω Z (x) of the acton of Z d on X When Z = Z d we wrte orbt and assocated system nstead of Z d -orbt and Z d -assocated system, respectvely The set of return tmes of x X to A X s T A (x) = {v Z d : v(x) A} The topologcal dynamcal system (X, Z d ) s mnmal f the orbt of any x X s dense n X, and t s sad to be equcontnuous f for every ε > 0 there exsts δ > 0 such that f x, y X satsfy d(x, y) < δ then d(v(x), v(y)) < ε for all v Z d We say that (X, Z d ) s an extenson of (Y, Z d ), or that (Y, Z d ) s a factor of (X, Z d ), f there exsts a contnuous surjecton π : X Y such that π preserves the acton We call π a factor map When the factor map s bjectve, we say that (X, Z d ) and (Y, Z d ) are conjugate The factor map π s an almost - factor map and (X, Z d ) s an almost - extenson of (Y, Z d ) by π f the set of ponts havng one pre-mage s resdual (contans a dense G δ set) n Y In the mnmal case t s equvalent to the exstence of a pont wth one pre-mage The set M(X) of nvarant probablty measures of X s the set of probablty measures µ defned on B(X), the Borel σ-algebra of X, such that µ(v(b)) = µ(b) for all v Z d and B B(X) We say that (X, µ, Z d ), the topologcal dynamcal system (X, Z d ) equpped wth µ M(X), s a measure-theoretc dynamcal system A measuretheoretc factor map φ from (X, µ, Z d ) to (Y, ν, Z d ), s a measurable functon preservng the acton and such that µ(φ (B)) = ν(b) for all B B(Y ) If φ s bjectve we say that (X, µ, Z d ) and (Y, ν, Z d ) are measure-theoretcally conjugate Consder a fnte alphabet Σ endowed wth the dscrete topology and ΣZ d wth the product topology The elements x = {x(z)} z Z d of Σ Z d are called Z d -arrays The shft acton on ΣZ d s defned by v(x) = {x(z + v)} z Z d, for all v Zd and x = {x(z)} z Z d ΣZ d In ths context, we wll wrte x + v nstead of v(x) If Z s a subset of Z d, x(z) denotes {x(z) : z Z} Σ Z When X ΣZ d s closed and nvarant by the shft acton, we say that (X, Z d ) s a subshft 3 d-dmensonal odometers Let {Z } 0 Z d be a decreasng sequence of subgroups somorphc to Z d (or of rank d) and let π : Z d /Z + Z d /Z the functon nduced by the ncluson Z + Z, 0 Consder the nverse lmt G = lm (Z d /Z, π )

3 Z d -Toepltz arrays 3 More precsely, G s defned as the subset of the product Π 0 Z d /Z consstng of the elements g = (g ) 0 such that π (g + ) = g for all 0 The set G s a group equpped wth the addton defned by g + h = (g + h ) 0, where + s the operaton nduced on Z d /Z by the addton n Z d Every Z d /Z s endowed wth the dscrete topology and Π 0 Z d /Z wth the product topology Thus G s a compact topologcal group whose topology s spanned by the cylnder sets [; a] = {g G : g = a}, wth a Z d /Z and 0 If H s a subgroup of G then t acts by homeomorphsms on G by h(g) = h+g, h H, g G Snce for all h H and for all cylnders [; a] we have h([; a]) [; a + h ], the topologcal dynamcal system (G, H) s equcontnuous Moreover, f H s dense n G then (G, H) s a mnmal equcontnuous system Consder the homomorphsm τ : Z d Π 0 Z d /Z defned for v Z d by τ(v) = {τ (v)} 0, where τ : Z d Z d /Z s the canoncal projecton The mage of Z d by τ s dense n G, whch mples that the Z d -acton v(g) = τ(v) + g, v Z d, g G, s well defned and (G, Z d ) s a mnmal equcontnuous system We call (G, Z d ) an odometer system or smply an odometer It s straghtforward that an odometer (G, Z d ) s a free dynamcal system f and only f τ : Z d G s one to one, whch s equvalent to 0 Z = {0} Notce that for all g n a cylnder set [; a] of an odometer G = lm (Z d /Z, π ), the set of return tmes of g to [; a] s Z, 0 Through ths paper we wll use these propertes and we wll dentfy G wth (G, Z d ) Lemma Let G j = lm (Z d /Z (j), π ) be two odometers (j =, 2) There s a factor map π : (G, Z d ) (G 2, Z d ) f and only f for every Z (2) there exsts some Z () k such that Z () k Z (2) Proof If π : G G 2 s a factor map then by contnuty, gven 0 and a Z d /Z (2) there exst k 0 and b Z d /Z () k such that [k; b] π [; a] Let v Z () k, we have that v(g) [k; b] for all g [k; b], whch mples that π(v(g)) = v(π(g)) [; a] Snce π(g) [; a] t holds that T [;a] (π(g)) = Z (2), whch proves that v Z (2) Suppose that for every 0 there exsts Z n () Z (2) Snce the sequences {Z (j) } 0 (j =, 2) are decreasng, we can take n n + for all 0 The functon π : G G 2 defned by π((g ) 0 ) = (j n (g n )) 0 where j n : Z d /Z n () Z d /Z (2) s the functon nduced by the ncluson Z n () Z (2), s a factor map, A scale s a sequence {A } 0 GL(d, Z) such that for every 0 there exsts Q GL(d, Z) satsfyng A + = A Q Let G = lm (Z d /Z, π ) be an odometer Any sequence {A } 0 of nteger matrces such that for all 0 the columns of A represent a base of Z s a scale We say that {A } 0 s a scale assocated to G f the odometer lm (Z d /A Z d, π ) s conjugate to G

4 4 MI Cortez It s drect that the scale {A } 0 s assocated to the odometer G = lm (Z d /A Z d, π ), but an odometer can be assocated to several scales We can formulate Lemma n terms of scales: Lemma 2 Let G j = lm (Z d /Z (j), π ) be two odometers (j =, 2) There s a factor map π : G G 2 f and only f gven {A (j) } 0 a scale assocated to G j (j =, 2) for all A (2) there exsts A () k such that A () k = A (2) Q for some Q GL(d, Z) We could thnk that all d-dmensonal odometers correspond to a product (up to conjugaton) of d one-dmensonal odometers It can be proved that a product of d onedmensonal odometers concdes wth an odometer havng an assocated scale consstng of dagonal matrces However, t s not true that all d-dmensonal odometers (d 2) admt a scale formed by dagonal matrces Examples can be constructed n any dmenson d 2: Example 3 Examples of d-dmensonal odometers whch are not conjugate to a product of d one-dmensonal odometers If d = 2, consder the sequence {A } 0 GL(2, Z) gven by [ A = ] If d > 2 consder {A } 0 GL(d, Z) defned by 3 + f k = mod Z 3 A (k, k) = + f k = 2 mod Z f k = 0 mod Z 3 7 f k = mod Z 3 A (k, k + ) = 3 7 f k = 2 mod Z 3 3 f k = 0 mod Z 3 A (k, j) = 0 f j {,, d} \ {k, k + }, k =,, d In both cases {A } 0 s a scale and 0 A Z d = {0} Ths means that G = lm (Z d /A Z d, π ) contans a copy of Z d and therefore G {0} Suppose there exsts a factor map π : G G wth G an odometer havng an assocated scale formed by dagonal matrces {D } 0 wth D (k, k) = d (k) for k {,, d} By Lemma 2, we have that d (k) dvdes every element n the k-th row of some A j Snce mcd{a j (k, l) : l =,, d} =, we have that D = d, d (k) = and then G = {0} Ths proves that G s not conjugate to a product of d one-dmensonal odometers 4 Characterzaton of mnmal almost - extensons of odometers Let (X, Z d ) and (Y, Z d ) be two topologcal dynamcal systems (Y, Z d ) s sad to be the maxmal equcontnuos factor of (X, Z d ) f t s an equcontnuos factor of (X, Z d ) such that for any other equcontnuous factor (Y, Z d ) of (X, Z d ) there exsts a factor map π : Y Y that satsfes π f = f, wth f : X Y and f : X Y factor maps

5 Z d -Toepltz arrays 5 It s well known that every topologcal dynamcal system has a maxmal equcontnuous factor and f (X, Z d ) s a mnmal almost - extenson of a mnmal equcontnuous system (Y, Z d ), then (Y, Z d ) s the maxmal equcontnuous factor of (X, Z d ) (for more detals see []) 4 Regularly recurrent systems A subset S of Z d s sad to be syndetc f there exsts a fnte subset K of Z d such that Z d = S + K = {s + k : s S, k K} Let (X, Z d ) be a topologcal dynamcal system and let x X The pont x s unformly recurrent f for every open neghborhood V of x the set T V (x) s syndetc It s well known that (Ω Z d(x), Z d ) s mnmal f and only f x s unformly recurrent A pont x X s regularly recurrent f for every open neghborhood V of x there s a subgroup Z of Z d somorphc to Z d such that Z T V (x) We say that a system s regularly recurrent f t s the orbt closure of a regularly recurrent pont Snce every subgroup Z of Z d somorphc to Z d s syndetc, regularly recurrent systems are mnmal In ths secton we wll show that regularly recurrent systems are exactly the mnmal almost - extensons of the odometers Lemma 4 Let (X, Z d ) be a mnmal topologcal dynamcal system and let x X If Z Z d s a group somorphc to Z d then (Ω Z (x), Z) s mnmal Proof Let V X be a neghborhood of x Pck a mnmal set M n X Z d /Z (wth the natural product acton) Ths set projects onto a mnmal subset of X, hence onto X Thus for every x X there exsts a pont (x, a) M and ths pont s unformly recurrent Addng a on the second axs s a conjugacy, hence (x, 0) s also unformly recurrent Ths mples that {z : z(x) V, z Z} s syndetc Lemma 5 Let (X, Z d ) be a topologcal dynamcal system and let x X be a regularly recurrent pont For all closed neghborhood V of x there exsts a subgroup Z of Z d somorphc to Z d such that Z T V (x) and {w(ω Z (x))} w Z d /Z s a clopen partton of X Proof Let Z Z d be a subgroup somorphc to Z d If u and w are two elements of Z d n the same class of Z d /Z then u(ω Z (x)) = w(ω Z (x)) So, t makes sense to speak about w(ω Z (x)) for w Z d /Z By mnmalty of (X, Z d ) we have that X = w Z d /Z w(ω Z(x)) From Lemma 4, for every w Z d /Z the system (w(ω Z (x)), Z) s mnmal Thus f u, w Z d /Z satsfy w(ω Z (x)) u(ω Z (x)) then u(ω Z (x)) = w(ω Z (x)) Ths mples that {w(ω Z (x))} w Z d /Z s a clopen coverng of X Let V X be a closed neghborhood of x and let Z Z d be a subgroup somorphc to Z d such that Z T V (x) Consder the subgroup Z of Z d spanned by the set {w Z d : Ω Z (x) = w(ω Z (x))} Snce Z Z, we have that Z s somorphc to Z d, and, because Ω Z (x) = Ω Z (x), the group Z s contaned n T V (x) Fnally, for w Z d due to Ω Z (x) = w(ω Z (x)) f and only f Ω Z (x) = w(ω Z (x)), t holds that {w(ω Z (x))} w Z d /Z s a clopen partton of X Corollary 6 Let (X, Z d ) be a topologcal dynamcal system and let x X The pont x s regularly recurrent f and only f there exsts {C } 0, a fundamental system of

6 6 MI Cortez clopen neghborhoods of x, such that there s a subgroup Z Z d somorphc to Z d such that for all y C the set of return tmes of y to C s Z, for every 0 Proof If x X has a fundamental system of neghborhoods as s wrtten above, t s a regularly recurrent pont If x s a regularly recurrent pont we take V an open neghborhood of x and we apply Lemma 5 to V We obtan a group Z T V (x), somorphc to Z d, such that {w(ω Z (x))} w Z d /Z s a clopen partton of X We set C = Ω Z (x) whch s a clopen set wth T C (y) = Z for all y C So, gven C n and Z n, we take an open neghborhood V n+ C n of x and we apply Lemma 5 to V n+ As n the case n =, we obtan C n+ and Z n+ If we take lm dam(v n ) = 0, we obtan that {C } 0 s a fundamental system of clopen neghborhoods of x Theorem 7 A mnmal topologcal dynamcal system (X, Z d ) s an almost - extenson of an odometer G by π f and only f (X, Z d ) s a regularly recurrent system Moreover, the set of regularly recurrent ponts of X s exactly the pre-mage of the set of ponts n G whch have only one pre-mage by π Proof Let (X, Z d ) be a mnmal - extenson of an odometer G = lm (Z d /Z, π ) Let π : X G be the almost - factor map and let x X be such that {x} = π {π(x)} Snce π s contnuous, f π(x) = (a ) 0 G then {π ([; a ])} 0 s a decreasng sequence of clopen neghborhoods of x that satsfes 0 π ([; a ]) = {x} We know that for every g [; a ] t holds T [;a ](g) = Z, therefore for all g π ([; a ]), T π ([;a ])(g) = Z So, by Corollary 6 we conclude that x s a regularly recurrent pont of X Let X be a regularly recurrent system and let x X be a regularly recurrent pont By Corollary 6 there exsts a decreasng sequence {C } 0 of clopen neghborhoods of x such that 0 C = {x}, and there s a subgroup Z somorphc to Z d such that T C (y) = Z for all y C, 0 Snce C + C, we have that Z + Z, 0 So, we can defne the odometer G = lm (Z d /Z, π ) We defne π : X G by π = (f ) 0 where f s the contnuous map f : X Z d /Z gven by f (y) = z f and only f y z(c ) for y X, z Z and 0 The functon π s a factor map, and, snce 0 C = {x}, we have that f {0} = {x} So, π s an almost - extenson If π : X G s another almost - factor map and G an odometer, G and G are the maxmal equcontnuous factor of (X, Z d ) (therefore, they are conjugate) Thus there exsts a factor map π : G G such that π π = π, whch mples that {x} = π {π(x)} We conclude that the set of regularly recurrent ponts s exactly the pre-mage of the ponts n G whch have only one pre-mage 5 Egenvalues of odometers, measure-theoretc factor maps 5 Egenvalues Let (X, µ, Z d ) be a measure-theoretc dynamcal system A vector α R d s an egenvalue of X f there exsts f L 2 µ(x) \ {0} such that f(v(x)) = exp(2πα T v)f(x) for all x X and v Z d We call f an egenfuncton assocated to α We say that an egenvalue s a contnuous egenvalue f t has an

7 Z d -Toepltz arrays 7 assocated contnuous egenfuncton Snce an odometer G s a compact group, the normalzed Haar measure λ of G s the only nvarant probablty measure of G Thus when we speak about G as a measuretheoretc dynamcal system, we mean G equpped wth the measure λ and on G we consder the acton of Z d vewed as a subset of G Proposton 8 Let G = lm n (Z d /Z n, π n ) be an odometer The set of egenvalues of G s gven by E G = n 0 {α Rd : α T z Z, z Z n } Q d Moreover, every egenvalue of G s a contnuous egenvalue Proof It s clear that E G Q d because α T z Z for all z Z n f and only f α = v T A n for some v Z d and A n GL(d, Z) such that Z n = A n Z d For n 0 we call C n = [n; 0] Snce v, w Z d satsfy C n + v = C n + w f and only f w and v belong to the same class n Z d /Z n, t makes sense to wrte C n + v for v Z d /Z n Notce that the collecton P n = {C n + v : v Z d /Z n } s a clopen partton of G Let α E G and let n 0 be such that α T z Z for all z Z n Ths means that exp(2πα T v) = exp(2πα T w) for all v Z d and w v + Z n, whch mples that f = v Z d /Z n exp(2πα T v) Cn +v s a well defned contnuous functon that verfes f(g + w) = exp(2πα T w)f(g) for all g G and w Z d Let α R d be an egenvalue of G and let f L 2 λ (G)\{0} be an assocated egenfuncton For v Z d we have that ( ) exp(2πα T v) fdλ = fdλ C n C n +v Snce C n + v = C n + v + z for all z Z n, t holds that ( ) (5) exp(2πα T z) fdλ = fdλ for all z Z n C n C n Observe that E(f P n ) = ( ) exp(2παt v) fdλ Cn +v λ(c v Z n ) d C /Z n n Snce B(P n ) B(G), by the ncreasng Martngale theorem, we have that E(f P n ) converges to f n L 2 λ (G) Because f 0, ths mples there exsts m 0 such that C m fdλ 0 and, by (5), we conclude that α T z Z for all z Z m, whch means that α E G Corollary 9 Let (X, Z d ) be a regularly recurrent system and let G be ts maxmal equcontnuous factor The set of contnuous egenvalues of X s E G Proof It s clear that E G s contaned n the set of contnuous egenvalues of X Conversely, f α s a contnuous egenvalue of X we can take f : X S an assocated contnuous egenfuncton whch s a factor map between (X, Z d ) and the dynamcal system (f(x), Z d ), where the acton of v Z d on exp(2πx) f(x) s gven by v(exp(2πx)) = exp(2π(α T v + x)), whch s an sometry Thus the system (f(x), Z d ) s equcontnuous and therefore there exsts a factor map π : G f(x) Snce π s an egenfuncton assocated to α we conclude that α E G

8 8 MI Cortez 52 Measure-theoretc conjugaton Proposton 0 Let (X, Z d ) and (X 2, Z d ) be two mnmal equcontnuous systems If φ : X X 2 s a measure-theoretc factor map then there exsts a topologcal factor map π : X X 2 such that π = φ ae Proof A mnmal equcontnuous system (X, Z d ) s conjugate to a system (G, Z d ), where G s a topologcal compact group wth a contnuous homomorphsm ϕ : Z d G satsfyng ϕ(z d ) = G, and the acton of Z d on G s defned by v(g) = ϕ(v) + g for all v Z d and g G, ([] Theorem 36, [9] Theorem 8) The Haar measure λ s the only nvarant probablty measure of (G, Z d ) ([5] Theorem 620) and every egenfuncton of ths system s contnuous because s a constant multple of a character of G ([5] Theorem 35), whch mples that there exsts an orthonormal bass of L 2 λ (G) consstng of contnuous egenfunctons of (G, Z d ) Let µ be the only nvarant probablty measure of (X, Z d ), for =, 2, and let {f n } n 0 be an orthonormal bass of L 2 µ 2 (X 2 ) consstng of contnuous egenfunctons of (X 2, Z d ) If φ : X X 2 s a measure-theoretc factor map then f n φ s an egenfuncton of (X, Z d ), for all n 0 Thus the ergodcty of the system mples that for every n 0 there exsts a contnuous egenfuncton g n of (X, Z d ), such that f n φ = g n ae Thus t s possble to take a full measure Borel subset A of X such that f n φ = g n on A, for all n 0 Let {x } 0 be a sequence n A whch converges to x A, and let y X 2 an accumulaton pont of {φ(x )} 0 By contnuty of f n on X 2 and by contnuty of f n φ on A, we have f n (y) = f n φ(x) for all n 0 Thus f y and y 2 are two accumulaton ponts of {φ(x )} 0 then g(y ) = g(y 2 ) for all g L 2 µ 2 (X 2 ), whch mples that y = y 2 Ths shows that φ s contnuous on A Snce (X, Z d ) s strctly ergodc, A s dense on X, and snce f n and g n are contnuous on the whole spaces, φ A extends to a contnuous map π on X, whch s a factor map Lemma Let G be an odometer If π : G G s a factor map then π s njectve Proof We set G = lm n (Z d /Z n, π n ) Let g, h G be two elements such that π(g) = π(h) = j For all 0 there exsts v Z d /Z such that [; g ] + v = [; h ] Thus for every 0 there exsts n 0 such that [n; g n ], [n; g n ] + v n π ([; j ]) for all n > n Ths mples that v n Z Thus for n > t holds that [n; g n ], [n; h n ] [; g ] Because ths s true for all 0 we conclude that g = h Snce odometers are unquely ergodc, the nvarant probablty measures of a regularly recurrent system (X, Z d ) concde on the sub σ-algebra π (B(G)), where π s the almost - factor map between X and ts maxmal equcontnuous factor G In partcular, due to the set of regularly recurrent ponts of X s the pre-mage by π of a G δ -set n G, ts measure does not depend on the chosen measure µ M(X) The proof of the next Theorem follows the same deas used n the proof for d = (see [4], [6]) Theorem 2 Let (X, Z d ) be a regularly recurrent system The followng statements are equvalent: () The set of regularly recurrent ponts of X s a full measure set

9 Z d -Toepltz arrays 9 (2) (X, Z d ) s unquely ergodc and t s measure-theoretcally conjugate to ts maxmal equcontnuous factor Proof Let π : X G be the almost - factor map between X and ts maxmal equcontnuous factor Let R X be the set of regularly recurrent ponts Suppose that R s a full measure set Let µ M(X) and let B B(G) We have B = (B R) (B \ R) and µ(b) = µ(b R) Snce π s njectve on R, B R = π (π(b R)) π (B(G)) Thus µ(b) = µ(b R) = λ(π(b R)) Ths mples that (X, Z d ) s unquely ergodc Because π s njectve on R, a full measure set, t s a measure-theoretc conjugaton between X and G Assume (2) Let φ : (X, µ) (G, λ) be the measure-theoretc conjugaton Then π φ s a self-homomorphsm of the odometer By Proposton 0 and Lemma, π s njectve when restrcted to an nvarant set A X wth µ(a) = If the set of regularly recurrent ponts of X s not a full measure set for µ, then by ergodcty, nvarance and Theorem 7, the set of ponts n G wth non-sngleton fbers n X s of full measure λ Let B be the pre-mage of ths set The ntersecton A B supports µ On the other hand, B \ A has the same projecton on G as B, because A removes only one pont from each fber So, B \ A has projecton of full measure λ and t s nvarant, hence the measure λ lfts to an nvarant measure ν supported by ths set Because µ and ν have dsjont supports, ν µ contradctng unque ergodcty Remark 3 Let us ndcate a mstake n the paper [4]: Condton (6) n [4, Theorem 3] clams that for regularty of one-dmensonal Toepltz flows t suffces to fnd one ergodc measure measure-theoretcally conjugate to the odometer Ths statement s false; for example the Oxtoby sequence of [4, Example 03] s not regular and has two ergodc measures, both somorphc to the odometer Clearly, smlar examples exst n hgher dmensons 6 Z d -Toepltz Arrays Let Σ be a fnte alphabet and Z Z d a subgroup somorphc to Z d For x = {x(v)} v Z d Σ Z d we defne: P er(x, Z, σ) = {w Z d : x(w + z) = σ for all z Z}, σ Σ, P er(x, Z) = σ Σ P er(x, Z, σ) When P er(x, Z) we say that Z s a group of perods of x We say that x s a Z d -Toepltz array (or smply a Toepltz array) f for all v Z d there exsts Z Z d subgroup somorphc to Z d such that v P er(x, Z) Proposton 4 The followng statements concernng x ΣZ d are equvalent: () x s Toepltz array (2) There exsts a sequence of postve nteger numbers {p n } n 0 such that p n < p n+, p n dvdes p n+ and { n,, n} d P er(x, p n Z d ) for all n 0 (3) x s regularly recurrent

10 0 MI Cortez Proof Before we prove the equvalence between sentences (),(2) and (3) notce that for every subgroup Z Z d somorphc to Z d, there exsts an nteger p > such that pz d Z In fact, snce Z d /Z s fnte, for all w Z d /Z there exsts k > such that kw = 0 Ths mples that for all v Z d there exsts k > such that kv Z In partcular, there exst p,, p d > wth p e,, p d e d Z, where e,, e d are the canoncal vectors n Z d Thus pz d Z wth p = Π d = p We set D n = { n,, n} d and C n = {y ΣZ d : y(d n ) = x(d n )} for all n 0 Suppose that x s a Toepltz array Let n 0 and v D n We take Z v Z d, subgroup somorphc to Z d, such that v P er(x, Z v ) and p v > such that p v Z d Z v For p = Π v Dn p v we have pz d Z v for all v D n Thus Z n = v D n Z v s a subgroup somorphc to Z d whch satsfes D n P er(x, Z n ) We defne the sequence {p n } n 0 by p 0 = q 0 and for n > 0 we set p n = q n p n, where q n > s an nteger such that q n Z d Z n for all n 0 Thus {p n } n 0 s a sequence of postve nteger numbers such that p n < p n+, p n dvdes p n+ and D n P er(x, Z n ) P er(x, p n Z d ) for all n 0 Suppose there exsts a sequence {p n } n 0 as n statement (2) Snce D n P er(x, p n Z d ) the set of return tmes of x to C n contans p n Z d whch mples that x s regularly recurrent because {C n } n 0 s a fundamental system of clopen neghborhoods of x Suppose that x s regularly recurrent For n 0 we take Z n a subgroup somorphc to Z d such that Z n T Cn (x) It holds that Z d = n 0 P er(x, Z n) whch means that x s a Toepltz array A subshft (X, Z d ) s a Z d -Toepltz system (or smply a Toepltz system) f there exsts a Toepltz array x such that X = Ω Z d(x) From Theorem 7 and Proposton 4 we conclude that the famly of mnmal subshfts whch are almost - extensons of odometers concdes wth the famly of Toepltz systems As t was done for the case d = n [6], n order to know the maxmal equcontnuous factor of a gven Toepltz system, we wll ntroduce the generalzaton, for d, of the concepts of essental perod and perod structure Defnton 5 Let x ΣZ d A group Z Z d of perods of x s called group generated by essental perods of x f P er(x, Z) P er(x, Z ) mples that Z Z Lemma 6 Let x ΣZ d If Z Z d s a group of perods of x then there exsts K Z d a group generated by essental perods of x such that P er(x, Z) P er(x, K) Proof Let Z Z d be a group of perods of x We call Ẑ the set of the groups H Zd somorphc to Z d whch satsfy P er(x, Z) P er(x, H) Let K be the subgroup of Z d generated by H Ẑ H Let H Ẑ and let w be an element n P er(x, Z, σ) for some σ Σ Snce w + z P er(x, Z, σ) for all z Z we have that w + z P er(x, H, σ) for all z Z Ths means that σ = x(w + z) = x(w + z + h) for all z Z and for all h H whch s equvalent to say that w + h P er(x, Z, σ) for all h H Thus, f m s a fnte postve nteger and h s some element n H Ẑ for m then w + k P er(x, Z, σ) where k = m = h So, w + K P er(x, Z, σ) Ths mples that w P er(x, K, σ) It holds that K Ẑ and snce every H whch satsfes P er(x, K) P er(x, H) s also n Ẑ, t follows that K s a group generated by essental perods of x

11 Corollary 7 Let x ΣZ d Z d -Toepltz arrays be a Toepltz array There exsts a sequence {Z n } n 0 of groups generated by essental perods of x such that Z n+ Z n and n 0 P er(x, Z n) = Z d Proof From Proposton 4 (2) we conclude there exsts a decreasng sequence {Z n} n 0 of groups of perods of x such that n 0 P er(x, Z n) = Z d We set Z 0 a group generated by essental perods of x such that P er(x, Z 0 ) P er(x, Z 0) For n > 0 we set Z n = Z n Z n whch s a subgroup somorphc to Z d, and snce P er(x, Z n ), P er(x, Z n) P er(x, Z n), Z n s a group of perods of x Thus, by Lemma 6, there exsts a group Z n generated by essental perods of x, such that P er(x, Z n) P er(x, Z n ) Snce Z n s a group generated by essental perods of x, we have Z n Z n Thus {Z n } n 0 s a decreasng sequence of groups generated by essental perods of x such that n 0 P er(x, Z n) = Z d Defnton 8 A sequence of groups as n Corollary 7 s called a perod structure of x In the sequel, we wll show that from a perod structure {Z n } n 0 of a Z d -Toepltz array x t s possble to construct a sequence of nested fnte clopen parttons of Ω Z d(x) From ths sequence of parttons t wll be easy to defne an almost - factor map between the Toepltz system (Ω Z d(x), Z d ) and the odometer G = lm n (Z d /Z n, π n ) Let x ΣZ d be a Toepltz array, let y Ω Z d(x) and let Z Z d be a subgroup somorphc to Z d Snce (Ω Z (y), Z) s mnmal, f Z s a group of perods of y then Ω Z (y) C Z (y), where C Z (y) = {x Ω Z d(x) : P er(x, Z, σ) = P er(y, Z, σ), σ Σ} We wll use the followng conventon: For a Z-perodc subset C of Ω Z d(x), e, such that C + w = C + w whenever w w Z we wll wrte C + v nstead of C + w, where v s the projecton of w to Z d /Z be a Toepltz array and let y Ω Z d(x) If Z Z d s a group generated by essental perods of y then Ω Z (y) = C Z (y) and {C Z (y) + v} v Z d /Z s a clopen partton of Ω Z d(x) Proposton 9 Let x ΣZ d Proof It holds that Ω Z (y)+w C Z (y)+w for all w Z d /Z Snce {Ω Z (y)+w} w Z d /Z s a coverng of Ω Z d(x), so s {C Z (y)+w} w Z d /Z Furthermore, (C Z (y)+w) (C Z (y)+ v) f and only f C Z (y) + w = C Z (y) + v, for w, v Z d /Z, whch mples that {C Z (y) + w} w Z d /Z s a clopen coverng of Ω Z d(x) If C Z + w = C Z + v for some v, w Z d /Z, then k(v w) T CZ (y)(y) for all k Z Ths mples that P er(y, Z) P er(y, Z ), where Z Z d s some subgroup somorphc to Z d generated by v w and d elements of some base of Z Snce Z s a group generated by essental perods of y then Z Z Thus v = w and {C Z (y) + v} v Z d /Z s a clopen partton of Ω Z d(x) Because Ω Z (y) + w s contaned n C Z (y) + w, both sets must be equal because {Ω Z (y) + v} v Z d /Z s a coverng of Ω Z d(x)

12 2 MI Cortez be a Toepltz array If {Z n } n 0 s a perod structure of x then the odometer G = lm n (Z d /Z n, π n ) s the maxmal equcontnuous factor of (Ω Z d(x), Z d ) Proposton 20 Let x ΣZ d Proof By Proposton 9, f {Z n } 0 s perod structure of the Toepltz array x, then {C Zn (x) + w : w Z d /Z n } n 0 s a sequence of nested clopen parttons of Ω Z d(x) Ths mples that the functon f n : Ω Z d(x) Z d /Z n gven by f n (y) = w f and only f y C Zn (x) + w s a well defned contnuous functon, y Ω Z d(x), n 0 The functon π : Ω Z d(x) G gven by π = (f n ) n 0 s a factor map Snce n 0 C Z n(x) = {x}, we have that π {0} = {x} and then π s an almost - factor map Proposton 2 For every odometer G there exsts a Toepltz array x {0, }Z d such that G s the maxmal equcontnuous factor of (Ω Z d(x), Z d ) Proof Let G = lm n (Z d /Z n, π n ) be an odometer We dstngush two cases: Case : There exsts m 0 such that Z n = Z m for all n m In ths case G s the fnte group Z d /Z m and then every mnmal almost - extenson wll be conjugate to G For example, x {0, }Z d defned by x(v) = 0 for all v Z m and x(v) = f not, provdes a Toepltz sequence x such that G s the maxmal equcontnuous factor of the system assocated to x Case 2: For every m 0 there exsts n > m such that Z n Z m In ths case we can take a subsequence {Z n } n 0 such that Z n+ Z n and Z n /Z n+ 3 for all n 0 By Proposton, G s conjugate to the odometer obtaned from ths sequence In order to construct the Toepltz array x we wll defne a sequence {(w n, v n )} n 0 Z d Z d as follows: we set v 0 = 0 and we choose some element w 0 Z d \ Z 0 For n > 0, we take v w n + Z n whch satsfes v = mn{ w : w w n + Z n }, where v = max d v () wth v = (v (),, v (d) ) We set v n = v and we choose w n w n + Z n \ (v n + Z n ) The sequence s well defned because Z n /Z n+ 3 for all n 0 We defne, K 0 = Z d \ (v 0 + Z 0 ) (w 0 + Z 0 ) K n = w + Z n, for n > 0 w (w n +Z n )\(v n +Z n w n +Z n ) The famly of sets {v n + Z n, K n : n 0} s a partton of Z d Thus x {0, }Z d gven by: { 0 f z n 0 (62) x(z) = v n + Z n f z n 0 K n s well defned Snce n j=0 v j + Z j P er(x, Z n, 0) and n j=0 K j P er(x, Z n, ), t holds that Z d = n 0 P er(x, Z n) and then x s a Toepltz array To conclude that G s the maxmal equcontnuous factor of the system assocated to x, by Proposton 20, t suffces to show that {Z n } n 0 s a perod structure of x Let n 0 and Z Z d a subgroup somorphc to Z d such that P er(x, Z n ) P er(x, Z) Gven z Z, ths mples that 0 = x(v n ) = x(v n + z) Thus v n + z n j=0 (v j + Z j ) (w n + Z n ) If v n + z w n + Z n we obtan that x(w) = 0 for all w w n + Z n whch s not possble, and f v n + z v j + Z j for some 0 j < n we get x(w) = 0 for all

13 Z d -Toepltz arrays 3 w w j + Z j whch also contradcts the constructon of x So, z Z n and we conclude that Z n s a group of essental perods of x Remark 22 Example 3 and Proposton 2 mply that, for d 2, there are Toepltz systems n {0, }Z d such that ther maxmal equcontnuous factors are not products of d one-dmensonal odometers 6 Aperodc part of a Toepltz array Let x ΣZ d be a Toepltz array, let π : Ω Z d(x) G be the almost - factor map between Ω Z d(x) and ts maxmal equcontnuous factor G, and let {Z n } n 0 be a perod structure of x We defne D G = {g G : y, y 2 π {g} such that y (0) y 2 (0)}, and for y Ω Z d(x) the set Aper(y) = Z d \ n 0 P er(y, Z n ) In analogy of the case d =, n the next proposton we wll show that Aper(y) does not depend on the choce of a perod structure {Z n } n 0 and that t s exactly the aperodc part of y Proposton 23 If y Ω Z d(x) and π(y) = g G then: () w Aper(y) f and only f g + w D G (2) w / Aper(y) f and only f there exsts a subgroup Z of Z d somorphc to Z d such that w P er(y, Z) (3) y s a Toepltz array f and only f Aper(y) = (4) If y π {g} then y (w) = y(w) for all w Z d \ Aper(y) Proof If w Aper(y) then for all n 0 there exsts z n Z n such that (y + w)(0) (y + w + z n )(0) Snce g + w and g + w + z n are n [n; g n ], we have that lm n π(y + w + z n ) = lm n g + w + z n = g + w Thus for every accumulaton pont y of {y + w + z n } n 0 t holds that π(y ) = g + w and y (0) (y + w)(0) So, g + w D G If g + w D G then there s y π (g + w) such that (y + w)(0) y (0) By mnmalty and snce y, y + w are n n 0 π ([n; g n ]) we have that for all n 0 there exsts z n Z n such that (y + w + z n )(0) = y (0) (y + w)(0) whch mples that w Aper(y) To show (2) t s obvous that f w / Aper(y) then w P er(y, Z) for some subgroup Z Z d somorphc to Z d Conversely, suppose that w P er(y, Z) for some Z Z d subgroup somorphc to Z d By Lemma 6, we can suppose that Z s a group generated by essental perods of y From Proposton 9, {C Z (y) + w} w Z d /Z s a clopen partton of Ω Z d(x) Let x C Z (y) be a Toepltz array and let {Z n} n 0 be a perodc structure of x Consder the sequence {Z n} n 0 gven by Z 0 = Z and Z n a group of essental perods of x such that P er(x, Z n Z n) P er(x, Z n) for all n > 0 Snce {C Z n (x ) + w : w Z d /Z n} n 0 s a sequence of nested clopen parttons of Ω Z d(x) such that n 0 C Z n (x ) = {x }, we can prove, as t was done n the proof of Proposton 20, that G = lm n (Z d /Z n, π n ) Snce {Z n} n 0 and {Z n } n 0 defne the

14 4 MI Cortez same odometer, Lemma mples that there exsts n 0 such that Z n Z 0 = Z Thus P er(y, Z) P er(y, Z n ), whch mples that w / Aper(y) Propertes (3) and (4) follow of property () 62 Regular Toepltz arrays Let x ΣZ d be a Toepltz array and π : Ω Z d(x) G the almost - factor map between Ω Z d(x) and ts maxmal equcontnuous factor G = lm n (Z d /Z n, π n ) For all n 0 we defne r n = P er(x, Z n) [Z d /Z n ] Z d, /Z n where [Z d /Z n ] s a subset of Z d whch contans exactly one representatve element of every class n Z d /Z n Snce P er(x, Z n ) P er(x, Z n+ ) and Z d /Z n+ = Z d /Z n Z n /Z n+ we have that r n+ r n Thus lm n r n = r (0, ] exsts The Topeltz array x s sad to be regular f r = Proposton 24 Let x ΣZ d be a Toepltz array The followng statements are equvalent: () x s regular (2) The set of Toepltz arrays of Ω Z d(x) s a full measure set for every µ M(X) (3) λ(d G ) = 0, where λ s the Haar measure on G (4) (Ω Z d(x), Z d ) s unquely ergodc and t s measure-theoretcally conjugate to ts maxmal equcontnuous factor Proof The statements (2) and (4) are equvalent by Theorem 2 As t was done n [6], the set of Toepltz arrays of Ω Z d(x) s gven by v Z d C + v π {B(G)}, where C = {y Ω Z d(x) : 0 / Aper(y)} Thus, for all µ M(Ω Z d(x)) t holds that µ({y Ω Z d(x) : Aper(y) = }) = r, whch shows that () and (2) are equvalent We have G\{g G : π {g} = } = v Z d(d G+v), whch means that v Z d(d G+v) s the complement of {g : π {g} s a Toepltz array} Thus f the set of Toepltz arrays s a full measure set for some µ M(Ω Z d(x)), then the complement of v Z d(d G + v) s a full measure set for λ, whch mples that λ(d G ) = 0 Conversely, f λ(d G ) = 0 then λ( v Z d(d G + v)) = 0, whch mples λ({g : π {g} s a Toepltz array}) = Let µ M(Ω Z d(x)) Snce µ(π A) = λ(a) for all A B(G), the set of Toepltz array s a full measure set for µ Ths shows that (2) s equvalent to (3) 7 Semcocycles The noton of a semcocycle has been extensvely used n the theory of one-dmensonal Toepltz flows (see [4]) In ths paper t s not used but we develop t for hgher dmensonal actons for further utlty Let x ΣZ d be a Toepltz array and let {p n } n 0 be the sequence of nteger numbers of Proposton 4(2) Snce for all n 0 there exsts q n > such that p n+ = q n p n, the odometer G = lm n (Z d /p n Z d, π n ) s a free odometer, that s an odometer whch s a free dynamcal system Thus the functon τ : Z d G defned n Secton 3 s an homomorphsm between the groups Z d and τ(z d ) So, we can dentfy τ(z d ) wth Z d

15 Z d -Toepltz arrays 5 and wrte Z d nstead of τ(z d ) The odometer G nduces on Z d the topology generated by the famly of sets {w + p n Z d : w Z d, n 0} that we call Θ G The functon v x(v) s contnuous wth respect Θ G because { n,, n} d P er(x, p n Z d ) for all n 0 The last one means that x : Z d Σ s a semcocycle on G n the followng sense: Defnton 25 Let G = lm n (Z d /Z n, π n ) be a free odometer and let K be a compact metrc space A functon f : Z d K s a semcocycle on G f t s contnuous wth respect Θ G, where Θ G s the topology on Z d nherted from G The functons f : Z d K may be seen as elements of the topologcal dynamcal system (KZ d, Z d ), where KZ d s endowed wth the product topology and the acton of v Z d on f = {f(z)} z Z Z d K d s the shft acton, t means v(f) = {f(v + z)} n 0 We wll skp the proofs of Theorems 26 and 27 below, because they follow by the same deas as used n [4] for dmenson one s a semcocycle on some odometer G then f s a regularly recurrent pont of (KZ d, Z d ) Theorem 26 If f KZ d Theorem 26 provdes another characterzaton of Z d -Toepltz arrays: we have showed that every Toepltz array x ΣZ d s a semcocycle on some odometer G By Theorem 26, f x ΣZ d s a semcocycle on some odometer G wth values n a fnte set Σ then x s regularly recurrent and therefore a Toepltz array Proposton 7 and Theorem 26 mply that (Ω Z d(f), Z d ) s a mnmal almost - extenson of some odometer, where Ω Z d(f) represents the closure orbt of the semcocycle f n KZ d Notce that G need not be the maxmal equcontnuos factor of (Ω Z d(f), Z d ), for nstance, n the frst part of ths secton t was shown that every Toepltz array s a semcocycle on an odometer whch s a product of d one-dmensonal odometers Whle for d > t s not true that any Toepltz system has a maxmal equcontnuous factor whch s a product of d one-dmensonal odometers Let f KZ d be a semcocycle on an odometer G Snce we have dentfed τ(z d ) wth Z d t makes sense to defne F = {(v, f(v)) : v Z d } G K and F (g) = {k K : (g, k) F } for g G We call C f the set of g G such that F (g) = and D f = G \ C f Snce f s contnuous we have that F (v) = {v} for all v Z d Thus C f s the subset where f can be contnuously extended by f(g) = F (g) The semcocycle f s sad to be nvarant under no rotaton f F (g + g ) = F (g ) for every g G mples that g = 0 Theorem 27 A topologcal dynamcal system (X, Z d ) s a mnmal almost - extenson of (G, Z d ) f and only f t s conjugate to (Ω Z d(f), Z d ), where f s a semcocycle on G, nvarant under no rotaton We say that a Toepltz array x s non perodc f x + v = x mples that v = 0 A semcocycle defned by a non perodc Toepltz array s not extendable to a contnuous functon on the whole odometer For contrast, a constant semcocycle defnes a perodc array Notce that x s non perodc f and only f x s a semcocycle on G, ts maxmal

16 6 MI Cortez equcontnuous factor In fact, f x s a semcocycle on G then t s a free dynamcal system and therefore x s non perodc Conversely, f x s non perodc then G s a free dynamcal system Snce x s contnuous wth respect to Θ G, x s a semcocycle on G In Proposton 28 we mean x as a semcocycle on ts maxmal equcontnuous factor G Proposton 28 If x s non perodc and j G then: () σ F (j) f and only f there exsts y π {j} such that y(0) = σ (2) D x = D G Proof Always we can suppose that π(x) = 0 If σ F (j) then j s the lmt of some sequence {n } 0 Z d such that lm x(n ) = σ Thus every accumulaton pont y of {x + n } 0 satsfes y(0) = σ and π(y) = π(x) + j = j By mnmalty of Ω x and by contnuty of π, f y π {j} satsfes y(0) = σ then σ F (j) Property (2) follows drectly from () 8 Examples In ths secton we wll gve two examples of Z 2 -Toepltz arrays In the frst example we wll construct a Toepltz array x such that M(ΩZ2( x)) has a determned fnte number of ergodc measures and n the second one the Toepltz array x wll be constructed such that (ΩZ 2( x), Z2 ) s unquely ergodc wth postve entropy We set some notaton that we use n both examples Let {q n } n 0 be a sequence of nteger numbers such that q n 3 for all n 0 We set p 0 = and p n = Π n =0 q for n > 0 For n > 0 we put { qn r n = 2 f q n s even q n 2 otherwse and l n = q n r n We defne D 0 = {0} 2 and n n D n = {z Z : l p z r p } 2 Z 2 = Notce that D n s the dsjont unon of the sets D n,v = D n + v, for v S n, where S n = {p n z Z : l n z r n } 2 The boundary of S n s S n = {(t, t 2 ) S n : t or t 2 s n {r n p n, l n p n }} Snce q n 3 for all n 0, then {r n } n 0 and {l n } n 0 are ncreasng sequences and thus Z 2 = n 0 D n Let q > be an nteger and consder the alphabet Σ = Σ 0 = {σ,, σ q } For n > 0 we take Σ n = {B n,,, B n,kn } a set of dfferent blocks n Σ Dn such that for all k k n, () B n,k (D n,0 ) = B n,, (2) B n,k (D n,v ) Σ n for all v S n, where B 0, = σ for all q From property () and snce {D n } n>0 covers Z 2, we have that there s only one element x n Z n 0 {x Σ 2 : x(d n ) = B n, } Property (2) mples that x(d n + v) Σ n for =

17 Z d -Toepltz arrays 7 D n,0 p n+ p ṅ p n+ Fgure The square D n+ for q n = 6 The ponts represent the set S n all v p n Z 2 Thus, property (2) nsures that x(d n + v) = B n, for all v p n Z 2, whch means that D n P er( x, p n Z 2 ) So, x s a Z 2 -Toepltz array For all n 0 and k q we defne C n,k = {x ΩZ 2( x) : x(d n) = B n,k }, C n = k n k= C n,k and P n = {C n,k + w : w D n, k k n } From property (2) we have that P n covers the orbt of x and snce the sets n P n are clopen, P n s a clopen coverng of ΩZ 2( x) Lemma 29 If for all n > 0 the followng statements are satsfed: () If there exsts w D n such that for some k, k k n, B n,k (v+w) = B n,k (v) for all v D n such that v + w D n, then w = 0, (2) B n,k (v) = B n,k (v) for every v S n and k, k k n, then the coverngs P n are parttons spannng the topology of ΩZ 2( x) Proof Let z Z 2 and let w D n Suppose that x + p n z + w C n Let B n,k be the block n Σ n such that ( x+p n z+w)(d n ) = B n,k Snce x+p n z s also n C n, there exsts k k n such that ( x + p n z)(d n ) = B n,k Ths mples that B n,k (w + v) = B n,k (v) for all v D n satsfyng v + w D n From statement () we have w = 0 and thus we conclude that T Cn ( x) = p n Z 2, whch mples T Cn (x) = p n Z 2 for all x C n, because (ΩZ 2( x), Z2 ) s mnmal Thus f (C n,k + v) (C n,k + w) for some v, w D n and k, k k n, then v w p n Z 2, whch mples that w v = 0 If v = w then C n,k C n,k, whch s possble f and only f B n,k = B n,k, e, when k = k Ths proves that P n s a partton Suppose that x and x 2 are two ponts of ΩZ 2( x) whch belong to the same set of P n Namely, x, x 2 C n,jn + v n for some v n D n and j n k n Let y, y 2 C n,jn be such that x = y +v n for =, 2, and let u Z 2 be some vector n D n If v n +u D n then y (v n +u) = y 2 (v n +u) whch mples that x (u) = x 2 (u) If v n +u / D n, consder z Z 2 \ {0} and w D n such that v n + u = p n z + w Snce y + p n z and y 2 + p n z

18 8 MI Cortez are n C n, (y + p n z)(d n ) = B n,l and (y 2 + p n z)(d n ) = B n,l2 for some l, l 2 k n Thus y (v n + u) = B n,l (w) and y 2 (v n + u) = B n,l2 (w) It holds w u = v n p n z, whch mples that w u / D n because z 0 Snce u D n and w D n, ths s possble only f w S n Thus, from statement (2), we have B n,l (w) = B n,l2 (w) and then x (u) = y (v n +u) = y 2 (v n +u) = x 2 (u) Ths proves that x (D n ) = x 2 (D n ) So, f x and x 2 are n the same set of P n, for all n 0, then x = x 2, whch means that {P n } n 0 spans the topology of ΩZ 2( x) For n 0 we defne the set n = {(x,, x kn ) T (R + ) kn : k n = x = p n } and the ncdence matrx A n M kn kn+ (Z) between P n and P n+ by A n (k, j) = {v D n+ : C n+,j + v C n,k }, k k n, j k n+ We denote by lm n ( n, A n ) the nverse lmt A 0 A 0 A 2 2, that s, lm n (, A n ) = {(x n ) n 0 Π n 0 n : A n x n+ = x n, n 0} Lemma 30 If the coverngs P n are parttons spannng the topology of ΩZ2( x) then we can dentfy M(ΩZ 2( x)) wth the nverse lmt lm n( n, A n ) Proof Suppose that the coverngs P n are parttons that span the topology of ΩZ 2( x) By property (2) we have that P n+ s fner that P n Ths mples that C n,k = k n+ j= v J(n,k,j) C n+,j + v, wth J(n, k, j) = {v D n+ : C n+,j + v C n,k } Thus k n k= A n (k, j) = q n and for µ M(ΩZ 2( x)), µ(c n,k) = k n+ j= A n(k, j)µ(c n+,j ) for all n 0 The frst one mples that lm n ( n, A n ) s well defned and the second one that (µ n = (µ(c n, ),, µ(c n,kn ))) n 0 s n ths nverse lmt Conversely, gven (u n = (u n,,, u n,kn )) n 0 lm n ( n, A n ), snce the P n are clopen and span the topology of ΩZ 2( x), there exsts only one µ M(Ω Z 2( x)) satsfyng µ(c n,k) = u n,k for all k k n and n 0 We wll construct the dfferent examples by choosng approprate sequences {Σ n } n 0 and {q n } n 0 8 An example of a Z 2 -Toepltz system wth a determned fnte number of ergodc measures Let n > 0 and let k {,, q} We set q n = s n q + for some s n > and S n,k = {(t, t 2 ) S n : t or t 2 s equal to p n ( l n + )}, {0,,q n } (k+qz) We have that S n s the dsjont unon of the sets S n,k and the cardnalty of every one of these sets s 4s n For n > 0 we set k n = q and we defne B n,k, for k q, as follows: () B n,k (D n,0 ) = B n,, (2) B n,k (D n,v ) = B n, f v = p n ( l n +, ), for {2,, q}, (3) B n,k (D n,v ) = B n,, for v S n,, wth {,, q} (4) B n,k (D n,v ) = B n,k, for all v S n such that B n,k (D n,v ) was not defned n the prevous steps

19 Z d -Toepltz arrays 9 B n,3 B n,3 B n, B n,2 B n,3 B n, B n,2 B n,3 B n,2 B n,2 B n, B n,2 B n,3 B n, B n,3 B n, B n,3 B n,2 B n,2 B n, B n, B n,3 B n, B n,2 B n,3 B n, B n,2 B n,3 Fgure 2 For q = 3, s n = 2 and k {, 2, 3}, the pcture represents the block B n+,k f we consder that every empty square corresponds to the block B n,k Pont (3) from the constructon nsures that statement (2) of Lemma 29 s satsfed The exstence of w D \ {0} such that B,k (v + w) = B,k (v) for some k, k k and for every v D n satsfyng v + w D, contradcts (3) and (4) from the constructon Usng the same argument for n >, t s possble to show by an nducton argument that statement () of Lemma 29 s also satsfed Thus we conclude that {P n } n 0 s a sequence of parttons spannng the topology of ΩZ2( x), and by Lemma 30, the set M(ΩZ 2( x)) s gven by lm n( n, A n ) Let n 0 In ths case, the ncdence matrx A n M q q (N) between P n and P n+ s gven by A n (, j) = { 4sn + f j qn 2 (q )(4s n + ) f j = For {,, q} For m n and j {,, q}, we defne u (j) m = p 2 m e j, where e j s the j-th untary vector n R q Smple computatons yelds A n A m u (j) m+ = qp 2 n + l n,m q

20 20 MI Cortez where l n,m = (q2 n q(4s n +)) (qm q(4s 2 m +)) Notce that {l qn q 2 m 2 n,m } m n s a decreasng sequence, then t converges to some α n [0, ], and so, lm A n A m u (j) m m+ = u(n,j) = qp 2 ( α n ) n + qα n e j The ponts u (n,),, u (n,q) generate the convex m n A n A m m+ By choosng q n > 4q for all n 0, where δ s some pont n (0, ), we have that α δ 2 n > 0 for all n 0 n Ths mples that for all n 0, u (n,),, u (n,q) are lnearly ndependent vectors and then, they are the extreme ponts of m n A n A m m+ Snce A n u (n+,j) = u (n,j) for all j {,, q}, we have that u () = {u (n,) } n 0,, u (q) = {u (n,q) } n 0 are the extreme ponts of lm n ( n, A n ) Thus (ΩZ 2( x), Z2 ) has exactly q ergodc measures If the sequence {q n } n 0 s constant then α n = 0 for all n 0, whch mples that n ths case (ΩZ 2( x), Z2 ) s unquely ergodc 82 An example of a unquely ergodc Z 2 -Toepltz system wth postve entropy We take k 0 = q, q 0 = k 0 + 2, k n = f(k n ) and q n = k n + 2 for n > 0, where f : N N s the functon defned by f(n) = (n2 )!, for all n N (n + )! n Remark 3 Observe that f(n) s the number of parttons P = {A } = n wth A = n 2 such that A = n + for all {,, n } of a set A Gven the alphabet Σ = Σ 0, consder the subset Σ n of Σ D n such that B Σ n f and only f: () B(D n,0 ) = B n,, (2) B(D n,v ) Σ n \ {B n, } for v S n \ ({0} S n ), (3) B(D n,v ) = B n,kn for all v S n, (4) {v S n \ ({0} S n ) : B(D n,v ) = B n,l } = k n +, for all l {2,, k n } From remark 3, we easly see that Σ n = k n The pont (3) from the constructon nsures that statement (2) of Lemma 29 s satsfed We have that B n,k (D n,v ) = B n, f and only f v = 0 Ths and () from the constructon mply that statement () of Lemma 29 s satsfed Thus {P n } n 0 s a sequence of parttons whch spans the topology of ΩZ2( x), and by Lemma 30, the set M(ΩZ 2( x)) s gven by lm n( n, A n )

21 Z d -Toepltz arrays 2 B n,kn B n,kn B n,kn B n,kn A B n, B n,kn B n,kn Fgure 3 The blank regon A s flled by a concatenaton k 2 n blocks from Σ D n The block shown above belongs to Σ n+ f the concatenaton fllng A uses exactly k n + copes of every block from Σ n \ {B n, } Let n 0 The ncdence matrx A n M kn k n+ (N) between P n and P n+ s gven by f = A n (, j) = k n + f = 2,, k n 5k n + 5 f = k n For j {,, k n+ }, snce A n n+ = { (, k p 2 n +,, k n +, 5k n + 5) T }, we n+ have lm( n, A n ) = {( (, k n +,, k n +, 5k n + 5) T ) n 0 } n p 2 n+ Ths mples that (ΩZ 2( x), Z2 ) s unquely ergodc wth unque nvarant probablty measure µ M(ΩZ2( x)) defned by f j = p 2 + k µ(c,j ) = + f j = 2,, k p 2 + f j = k For every 0 5k +5 p 2 + Consder U and V, two open coverngs of Ω x We defne N(U) = mn{ U : U s a subcoverng of U} and U V = {U V : U U, V V} The topologcal entropy of (ΩZ 2( x), Z2 ) s defned by h top (ΩZ 2( x), Z2 ) = sup lm sup ln N( U v), U n L n v L n