Z^d-actions with prescribed topological and ergodic properties
aa r X i v : . [ m a t h . D S ] N ov Z d -ACTIONS WITH PRESCRIBED TOPOLOGICAL ANDERGODIC PROPERTIES YURI LIMA
Abstract.
We extend constructions of Hahn-Katznelson [6] and Pavlov [8]to Z d -actions on symbolic dynamical spaces with prescribed topological andergodic properties. More specifically, we describe a method to build Z d -actionswhich are (totally) minimal, (totally) strictly ergodic and have positive topo-logical entropy. Introduction
Ergodic theory studies statistical and recurrence properties of measurable trans-formations T acting in a probability space ( X, B , µ ), where µ is a measure invariantby T , that is, µ ( T − A ) = µ ( A ), for all A ∈ B . It investigates a wide class ofnotions, such as ergodicity, mixing and entropy. These properties, in some way,give qualitative and quantitative aspects of the randomness of T . For example,ergodicity means that T is indecomposable in the metric sense with respect to µ and entropy is a concept that counts the exponential growth rate for the numberof statistically significant distinguishable orbit segments.In most cases, the object of study has topological structures: X is a compactmetric space, B is the Borel σ -algebra of X , µ is a Borel measure probability and T is a homeomorphism of X . In this case, concepts such as minimality and topologicalmixing give topological aspects of the randomness of T . For example, minimalitymeans that T is indecomposable in the topological sense, that is, the orbit of everypoint is dense in X .A natural question arises: how do ergodic and topological concepts relate to eachother? How do ergodic properties forbid topological phenomena and vice-versa?Are metric and topological indecomposability equivalent? This last question wasanswered negatively in [4] via the construction of a minimal diffeomorphism of thetorus T which preserves area but is not ergodic.Another question was raised by W. Parry: suppose T has a unique Borel proba-bility invariant measure and that ( X, T ) is a minimal transformation. Can (
X, T )have positive entropy? The difficulty in answering this at the time was the scarcityof a wide class of minimal and uniquely ergodic transformations. This was solved af-firmatively in [6], where F. Hahn and Y. Katznelson developed an inductive methodof constructing symbolic dynamical systems with the required properties. The prin-cipal idea of the paper was the weak law of large numbers.
Date : October, 15, 2010.1991
Mathematics Subject Classification.
Key words and phrases. Z d -actions, minimality, unique ergodicity, positive topological entropy,symbolic dynamics. Later, works of Jewett and Krieger (see [9]) proved that every ergodic measure-preserving system ( X, B , µ, T ) is metrically isomorphic to a minimal and uniquelyergodic homeomorphism on a Cantor set and this gives many examples to Parry’squestion: if an ergodic system ( X, B , µ, T ) has positive metric entropy and Φ :( X, B , µ, T ) → ( Y, C , ν, S ) is the metric isomorphism obtained by Jewett-Krieger’stheorem, then ( Y, S ) has positive topological entropy, by the variational principle.It is worth mentioning that the situation is quite different in smooth ergodictheory, once some regularity on the transformation is assumed. A. Katok showedin [7] that every C α diffeomorphism of a compact surface can not be minimaland simultaneously have positive topological entropy. More specifically, he provedthat the topological entropy can be written in terms of the exponential growth ofperiodic points of a fixed order.Suppose that T is a mesure-preserving transformation on the probability space( X, B , µ ) and f : X → R is a measurable function. A successful area in ergodictheory deals with the convergence of averages n − · P nk =1 f (cid:0) T k x (cid:1) , x ∈ X , when n converges to infinity. The well known Birkhoff ’s Theorem states that such limitexists for almost every x ∈ X whenever f is an L -function. Several results havebeen (and still are being) proved when, instead of { , , . . . , n } , average is madealong other sequences of natural numbers. A remarkable result on this directionwas given by J. Bourgain [3], where he proved that if p ( x ) is a polynomial withinteger coefficients and f is an L p -function, for some p >
1, then the averages n − · P nk =1 f (cid:0) T p ( k ) x (cid:1) converge for almost every x ∈ X . In other words, convergencefails to hold for a negligible set with respect to the measure µ . In [1], V. Bergelsonasked if this set is also negligible from the topological point of view. It turned out,by a result of R. Pavlov [8], that this is not true. He proved that, for every sequence( p n ) n ≥ ⊂ Z of zero upper-Banach density, there exist a totally minimal, totallyuniquely ergodic and topologically mixing transformation ( X, T ) and a continuousfunction f : X → R such that n − · P nk =1 f ( T p k x ) fails to converge for a residualset of x ∈ X .Suppose now that ( X, T ) is totally minimal, that is, (
X, T n ) is minimal forevery positive integer n . Pavlov also proved that, for every sequence ( p n ) n ≥ ⊂ Z of zero upper-Banach density, there exists a totally minimal, totally uniquelyergodic and topologically mixing continuous transformation ( X, T ) such that x T p n x ; n ≥ } for an uncountable number of x ∈ X .In this work, we extend the results of Hahn-Katznelson and Pavlov, giving amethod of constructing (totally) minimal and (totally) uniquely ergodic Z d -actionswith positive topological entropy. We carry out our program by constructing closedshift invariant subsets of a sequence space. More specifically, we build a sequence offinite configurations ( C k ) k ≥ of { , } Z d , C k +1 being essentially formed by the con-catenation of elements in C k such that each of them occurs statistically well-behavedin each element of C k +1 , and consider the set of limits of shifted C k -configurationsas k → + ∞ . The main results are Theorem 1.1.
There exist totally strictly ergodic Z d -actions ( X, B , µ, T ) with ar-bitrarily large positive topological entropy. We should mention that this result is not new, because Jewett-Krieger’s Theoremis true for Z d -actions [10]. This formulation emphasizes to the reader that theconstructions, which may be used in other settings, have the additional advantageof controlling the topological entropy. d -ACTIONS WITH PRESCRIBED TOPOLOGICAL AND ERGODIC PROPERTIES 3 Theorem 1.2.
Given a set P ⊂ Z d of zero upper-Banach density, there exist atotally strictly ergodic Z d -action ( X, B , µ, T ) and a continuous function f : X → R such that the ergodic averages | P ∩ ( − n, n ) d | X g ∈ P ∩ ( − n,n ) d f ( T g x ) fail to converge for a residual set of x ∈ X . In addition, ( X, B , µ, T ) can havearbitrarily large topological entropy. The above theorem has a special interest when P is an arithmetic set for whichclassical ergodic theory and Fourier analysis have established almost-sure conver-gence. This is the case (also proved in [3]) when P = { ( p ( n ) , . . . , p d ( n )) ; n ∈ Z } , where p , . . . , p d are polynomials with integer coefficients: for any f ∈ L p , p > n → + ∞ n n X k =1 f (cid:16) T ( p ( k ) ,...,p d ( k )) x (cid:17) exists almost-surely. Note that P has zero upper-Banach density whenever one ofthe polynomials has degree greater than 1. Theorem 1.3.
Given a set P ⊂ Z d of zero upper-Banach density, there exists atotally strictly ergodic Z d -action ( X, B , µ, T ) and an uncountable set X ⊂ X forwhich x
6∈ { T p n x ; n ≥ } , for every x ∈ X . In addition, ( X, B , µ, T ) can havearbitrarily large topological entropy. Yet in the arithmetic setup, Theorem 1.3 is the best topological result one canexpect. Indeed, Bergelson and Leibman proved in [2] that if T is a minimal Z d -action, then there is a residual set Y ⊂ X for which x ∈ { T ( p ( n ) ,...,p d ( n )) x ; n ∈ Z } ,for every x ∈ Y . 2. Preliminaries
We begin with some notation. Consider a metric space X , B its Borel σ -algebraand a G group with identity e . Throughout this work, G will denote Z d , d >
1, orone of its subgroups.2.1.
Group actions.Definition 2.1. A G -action on X is a measurable transformation T : G × X → X ,denoted by ( X, T ), such that(i) T ( g , T ( g , x )) = T ( g g , x ), for every g , g ∈ G and x ∈ X .(ii) T ( e, x ) = x , for every x ∈ X .In other words, for each g ∈ G , the restriction T g : X −→ Xx T ( g, x )is a bimeasurable transformation on X such that T g g = T g T g , for every g , g ∈ G . When G is abelian, ( T g ) g ∈ G forms a commutative group of bimeasurable trans-formations on X . For each x ∈ X , the orbit of X with respect to T is the set O T ( x ) . = { T g x ; g ∈ G } . YURI LIMA If F is a subgroup of G , the restriction T | F : F × X → X is clearly a F -action on X . Definition 2.2.
We say that (
X, T ) is minimal if O T ( x ) is dense in X , for every x ∈ X , and totally minimal if O T | F ( x ) is dense in X , for every x ∈ X and everysubgroup F < G of finite index.Remind that the index of a subgroup F , denoted by ( G : F ), is the number ofcosets of F in G . The above definition extends the notion of total minimality of Z -actions. In fact, a Z -action ( X, T ) is totally minimal if and only if T n : X → X is a minimal transformation, for every n ∈ Z .Consider the set M ( X ) of all Borel probability measures in X . A probability µ ∈ M ( X ) is invariant under T or simply T - invariant if µ ( T g A ) = µ ( A ) , ∀ g ∈ G, ∀ A ∈ B . Let M T ( X ) ⊂ M ( X ) denote the set of all T -invariant probability measures. Suchset is non-empty whenever G is amenable, by a Krylov-Bogolubov argument appliedto any F φ lner sequence of G . Definition 2.3. A G measure-preserving system or simply G - mps is a quadruple( X, B , µ, T ), where T is a G -action on X and µ ∈ M T ( X ).We say that A ∈ B is T -invariant if T g A = A , for all g ∈ G . Definition 2.4.
The G -mps ( X, B , µ, T ) is ergodic if it has only trivial invariantsets, that is, if µ ( A ) = 0 or 1 whenever A is a measurable set invariant under T . Definition 2.5.
The G -action ( X, T ) is uniquely ergodic if M T ( X ) is unitary, and totally uniquely ergodic if, for every subgroup F < G of finite index, the restricted F -action ( X, T | F ) is uniquely ergodic. Definition 2.6.
We say that (
X, T ) is strictly ergodic if it is minimal and uniquelyergodic, and totally strictly ergodic if, for every subgroup
F < G of finite index, therestricted F -action ( X, T | F ) is strictly ergodic.The result below was proved in [11] and states the pointwise ergodic theorem for Z d -actions. Theorem 2.7.
Let ( X, B , µ, T ) be a Z d -mps. Then, for every f ∈ L ( µ ) , there isa T -invariant function ˜ f ∈ L ( µ ) such that lim n → + ∞ n d X g ∈ [0 ,n ) d f ( T g x ) = ˜ f ( x ) for µ -almost every x ∈ X . In particular, if the action is ergodic, ˜ f is constant andequal to R f dµ . Above, [0 , n ) denotes the set { , , . . . , n − } , [0 , n ) d the d -dimensional cube[0 , n ) × · · · × [0 , n ) of Z d and by a T -invariant function we mean that f ◦ T g = f , forevery g ∈ G . These averages allow the characterization of unique ergodicity. Let C ( X ) denote the space of continuous functions from X to R . Proposition 2.8.
Let ( X, T ) be a Z d -action on the compact metric space X . Thefollowing items are equivalent.(a) ( X, T ) is uniquely ergodic. d -ACTIONS WITH PRESCRIBED TOPOLOGICAL AND ERGODIC PROPERTIES 5 (b) For every f ∈ C ( X ) and x ∈ X , the limit lim n → + ∞ n d X g ∈ [0 ,n ) d f ( T g x ) exists and is independent of x .(c) For every f ∈ C ( X ) , the sequence of functions f n = 1 n d X g ∈ [0 ,n ) d f ◦ T g converges uniformly in X to a constant function.Proof. The implications (c) ⇒ (b) ⇒ (a) are obvious. It remains to prove (a) ⇒ (c).Let M T ( X ) = { µ } . We’ll show that f n converges uniformly to ˜ f = R f dµ . Bycontradiction, suppose this is not the case for some f ∈ C ( X ). This means thatthere exist ε > n i → ∞ and x i ∈ X such that (cid:12)(cid:12)(cid:12)(cid:12) f n i ( x i ) − Z f dµ (cid:12)(cid:12)(cid:12)(cid:12) ≥ ε. For each i , let ν i ∈ M ( X ) be the probability measure associated to the linearfunctional Θ i : C ( X ) → R defined byΘ i ( ϕ ) = 1 n id X g ∈ [0 ,n i ) d ϕ ( T g x i ) , ϕ ∈ C ( X ) . Restricting to a subsequence, if necessary, we assume that ν i → ν in the weak-star topology. Because the cubes A i = [0 , n i ) d form a F φ lner sequence in Z d , ν ∈ M T ( X ). In fact, for each h ∈ Z d , (cid:12)(cid:12)(cid:12)(cid:12)Z (cid:0) ϕ ◦ T h (cid:1) dν − Z ϕdν (cid:12)(cid:12)(cid:12)(cid:12) = lim i →∞ n id (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X g ∈ A i + h ϕ ( T g x i ) − X g ∈ A i ϕ ( T g x i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max x ∈ X | ϕ ( x ) | · lim i →∞ A i ∆( A i + h ) A i = 0 . But (cid:12)(cid:12)(cid:12)(cid:12)Z f dν − Z f dµ (cid:12)(cid:12)(cid:12)(cid:12) = lim i →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z f dν i − Z f dµ (cid:12)(cid:12)(cid:12)(cid:12) = lim i →∞ (cid:12)(cid:12)(cid:12)(cid:12) f n i ( x i ) − Z f dµ (cid:12)(cid:12)(cid:12)(cid:12) ≥ ε and so ν = µ , contradicting the unique ergodicity of ( X, T ). (cid:3) Subgroups of Z d . Let F be the set of all subgroups of Z d of finite index. Thisset is countable, because each element of F is generated by d linearly independentvectors of Z d . Consider, then, a subset ( F k ) k ∈ N of F such that, for each F ∈ F ,there exists k > F k < F , for every k ≥ k . For this, just consider anenumeration of F and define F k as the intersection of the first k elements. Suchintersections belong to F because( Z d : F ∩ F ′ ) ≤ ( Z d : F ) · ( Z d : F ′ ) , ∀ F, F ′ < Z d . Restricting them, if necessary, we assume that F k = m k · Z d , where ( m k ) k ≥ isan increasing sequence of positive integers. Observe that ( Z d : F k ) = m kd . Suchsequence will be fixed throughout the rest of the paper. YURI LIMA
Definition 2.9.
Given a subgroup
F < Z d , we say that two elements g , g ∈ Z d are congruent modulo F if g − g ∈ F and denote it by g ≡ F g . The set ¯ F ⊂ Z d is a complete residue set modulo F if, for every g ∈ Z d , there exists a unique h ∈ ¯ F such that g ≡ F h .Every complete residue set modulo F is canonically identified to the quocient Z d /F and has exactly ( Z d : F ) elements.2.3. Symbolic spaces.
Let C be a finite alphabet and consider the set Ω( C ) = C Z d of all functions x : Z d → C . We endow C with the discrete topology and Ω( C ) withthe product topology. By Tychonoff’s theorem, Ω( C ) is a compact metric space.We are not interested in a particular metric in Ω( C ). Instead, we consider a basisof topology B to be defined below.Consider the family R of all finite d -dimensional cubes A = [ r , r + n ) × · · · × [ r d , r d + n ) of Z d , n ≥
0. We say that A has length n and is centered at g =( r , . . . , r d ) ∈ Z d . Definition 2.10. A configuration or pattern is a pair b A = ( A, b ), where A ∈ R and b is a function from A to C . We say that b A is supported in A with encodingfunction b .Let Ω A ( C ) denote the space of configurations supported in A and Ω ∗ ( C ) the spaceof all configurations in Z d :Ω ∗ ( C ) . = { b A ; b A is a configuration } . Given A ∈ R , consider the map Π A : Ω( C ) → Ω A ( C ) defined by the restrictionΠ A ( x ) : A −→ C g x ( g )In particular, Π { g } ( x ) = x ( g ). We use the simpler notation x | A to denote Π A ( x ). Definition 2.11. If A ∈ R is centered at g , we say that x | A is a configuration of x centered at g or that x | A occurs in x centered at g .For A , A ∈ R such that A ⊂ A , let π A A : Ω A → Ω A be the restriction π A A ( b ) : A −→ C g b ( g )As above, when there is no ambiguity, we denote π A A ( b ) simply by b | A . It is clearthat the diagram below commutes.Ω( C ) Π A / / Π A $ $ HHHHHHHHH Ω A ( C ) π A A (cid:15) (cid:15) Ω A ( C )These maps will help us to control the patterns to appear in the constructions ofSection 3.By a cylinder in Ω( C ) we mean the set of elements of Ω( C ) with some fixedconfiguration. More specifically, given b A ∈ Ω ∗ ( C ), the cylinder generated by b A isthe set Cyl( b A ) . = { x ∈ Ω( C ) ; x | A = b A } . d -ACTIONS WITH PRESCRIBED TOPOLOGICAL AND ERGODIC PROPERTIES 7 The family B := { Cyl( b A ) | b A ∈ Ω ∗ ( C ) } forms a clopen set of cylinders generating B . Hence the set C = { χ B ; B ∈ B } of cylinder characteristic functions generatesa dense subspace in C (Ω( C )). Let µ be the probability measure defined by µ (Cyl( b A )) = |C| −| A | , ∀ b A ∈ Ω ∗ ( C ) , and extended to B by Carathe´odory’s Theorem. Above, | · | denotes the number ofelements of a set.Consider the Z d -action T : Z d × Ω( C ) → Ω( C ) defined by T g ( x ) = ( x ( g + h )) h ∈ Z d , also called the shift action . Given B = Cyl( b A ) and g ∈ Z d , let B + g denote thecylinder associated to b A + g = (˜ b, A + g ), where ˜ b : A + g → { , } is defined by˜ b ( h ) = b ( h − g ), ∀ h ∈ A + g . With this notation, χ B ◦ T g = χ B + g . (2.1)In fact, χ B ( T g x ) = 1 ⇐⇒ T g x ∈ B ⇐⇒ x ( g + h ) = b ( h ) , ∀ h ∈ A ⇐⇒ x ( h ) = ˜ b ( h ) , ∀ h ∈ A + g ⇐⇒ x ∈ B + g. Definition 2.12. A subshift of (Ω( C ) , T ) is a Z d -action ( X, T ), where X is a closedsubset of Ω( C ) invariant under T .2.4. Topological entropy.
For each subset X of Ω( C ) and A ∈ R , letΩ A ( C , X ) = { x | A ; x ∈ X } denote the set of configurations supported in A which occur in elements of X andΩ ∗ ( C , X ) the space of all configurations in Z d occuring in elements of X ,Ω ∗ ( C , X ) = [ A ∈R Ω A ( C , X ) . Definition 2.13.
The topological entropy of the subshift (
X, T ) is the limit h ( X, T ) = lim n → + ∞ log | Ω [0 ,n ) d ( C , X ) | n d , (2.2)which always exists and is equal to inf n ∈ N n d · log | Ω [0 ,n ) d ( C , X ) | .2.5. Frequencies and unique ergodicity.Definition 2.14.
Given configurations b A ∈ Ω A ( C ) and b A ∈ Ω A ( C ), the set ofocurrences of b A in b A is S ( b A , b A ) . = { g ∈ Z d ; A + g ⊂ A and π A A + g ( b A ) = b A + g } . The frequency of b A in b A is defined asfr( b A , b A ) . = | S ( b A , b A ) || A | · YURI LIMA
Given F ∈ F and h ∈ Z d , the set of ocurrences of b A in b A centered at h modulo F is S ( b A , b A , h, F ) . = { g ∈ S ( b A , b A ) ; A + g is centered at a vertex ≡ F h } and the frequency of b A in b A centered at h modulo F is the quocientfr( b A , b A , h, F ) . = | S ( b A , b A , h, F ) || A | · Observe that if ¯ F ⊂ Z d is a complete residue set modulo F , thenfr( b A , b A ) = X g ∈ ¯ F fr( b A , b A , g, F ) . To our purposes, we rewrite Proposition 2.8 in a different manner.
Proposition 2.15.
A subshift ( X, T ) is uniquely ergodic if and only if, for every b A ∈ Ω ∗ ( C ) and x ∈ X , fr( b A , x ) . = lim n → + ∞ fr (cid:0) b A , x | [0 ,n ) d (cid:1) exists and is independent of x .Proof. By approximation, condition (b) of Proposition 2.8 holds for C ( X ) if andonly if it holds for C = { χ B ; B ∈ B } . If f = χ Cyl( b A ) , (2.1) implies thatlim n → + ∞ f n ( x ) = lim n → + ∞ n d X g ∈ [0 ,n ) d f ( T g x )= lim n → + ∞ n d X g ∈ [0 ,n ) d χ Cyl( b A + g ) ( x )= lim n → + ∞ n d X g ∈ [0 ,n ) dA + g ⊂ [0 ,n ) d χ Cyl( b A + g ) ( x )= lim n → + ∞ fr (cid:0) b A , x | [0 ,n ) d (cid:1) = fr( b A , x ) , where in the third equality we used that, for a fixed A ∈ R ,lim n → + ∞ |{ g ∈ [0 , n ) d ; A + g [0 , n ) d }| n d = 0 . (cid:3) Corollary 2.16.
A subshift ( X, T ) is totally uniquely ergodic if and only if, forevery b A ∈ Ω ∗ ( C ) , x ∈ X and F ∈ F , fr( b A , x, F ) . = lim n → + ∞ fr (cid:0) b A , x | [0 ,n ) d , , F (cid:1) exists and is independent of x . So, unique ergodicity is all about constant frequencies. We’ll obtain this via theLaw of Large Numbers, equidistributing ocurrences of configurations along residueclasses of subgroups. d -ACTIONS WITH PRESCRIBED TOPOLOGICAL AND ERGODIC PROPERTIES 9 Law of Large Numbers.
Intuitively, if A is a subset of Z d , each letter of C appears in x | A with frequency approximately 1 / |C| , for almost every x ∈ Ω( C ).This is what the Law of Large Number says. For our purposes, we state this resultin a slightly different way. Let ( X, B , µ ) be a probability space and A ⊂ Z d infinite.For each g ∈ A , let X g : X → R be a random variable. Theorem 2.17. (Law of Large Numbers) If ( X g ) g ∈ A is a family of independentand identically distributed random variables such that E [ X g ] = m , for every g ∈ A ,then the sequence (cid:0) X n (cid:1) n ≥ defined by X n = P g ∈ A ∩ [0 ,n ) d X g | A ∩ [0 , n ) d | converges in probability to m , that is, for any ε > , lim n → + ∞ µ (cid:0)(cid:12)(cid:12) X n − m (cid:12)(cid:12) < ε (cid:1) = 1 . Consider the probability measure space ( X, B , µ ) defined in Subsection 2.3. Fixed w ∈ C , let X g : Ω( C ) → R be defined as X g ( x ) = 1 , if x ( g ) = w = 0 , if x ( g ) = w. (2.3)It is clear that ( X g ) g ∈ Z d are independent, identically distributed and satisfy E [ X g ] = Z X X g ( x ) dµ ( x ) = 1 |C| , ∀ g ∈ Z d . In addition, X n ( x ) = P g ∈ [0 ,n ) d X g ( x ) n d = (cid:12)(cid:12) S (cid:0) w, x | [0 ,n ) d (cid:1)(cid:12)(cid:12) n d = fr (cid:0) w, x | [0 ,n ) d (cid:1) , which implies the Corollary 2.18.
Let w ∈ C , g ∈ Z d , F ∈ F and ε > .(a) The number of elements b ∈ Ω [0 ,n ) d ( C ) such that (cid:12)(cid:12)(cid:12)(cid:12) fr( w, b ) − |C| (cid:12)(cid:12)(cid:12)(cid:12) < ε is assymptotic to |C| n d as n → + ∞ .(b) The number of elements b ∈ Ω [0 ,n ) d ( C ) such that (cid:12)(cid:12)(cid:12)(cid:12) fr( w, b, g, F ) − |C| · ( Z d : F ) (cid:12)(cid:12)(cid:12)(cid:12) < ε is assymptotic to |C| n d as n → + ∞ .(c) The number of elements b ∈ Ω [0 ,n ) d ( C ) such that (cid:12)(cid:12)(cid:12)(cid:12) fr( w, b, g, F ) − |C| · ( Z d : F ) (cid:12)(cid:12)(cid:12)(cid:12) < ε for every w ∈ C and g ∈ Z d is assymptotic to |C| n d as n → + ∞ . Proof. (a) The required number is equal to |C| n d · µ (cid:0) { x ∈ Ω( C ) ; (cid:12)(cid:12) fr (cid:0) w, x | [0 ,n ) d (cid:1) − |C| − (cid:12)(cid:12) < ε } (cid:1) and is asymptotic to |C| n d , as the above µ -measure converges to 1.(b) Take A = F + g and ( X h ) h ∈ A as in (2.3). For any x ∈ Ω( C ), X n ( x ) = (cid:12)(cid:12) S (cid:0) w, x | [0 ,n ) d , g, F (cid:1)(cid:12)(cid:12) | A ∩ [0 , n ) d | = fr (cid:0) w, x | [0 ,n ) d , g, F (cid:1) · ( Z d : F ) + o (1) , because (cid:12)(cid:12) A ∩ [0 , n ) d (cid:12)(cid:12) is asymptotic to n d / ( Z d : F ). This implies that for n large (cid:12)(cid:12)(cid:12)(cid:12) fr (cid:0) w, x | [0 ,n ) d , g, F (cid:1) − |C| · ( Z d : F ) (cid:12)(cid:12)(cid:12)(cid:12) < ε ⇐⇒ (cid:12)(cid:12)(cid:12)(cid:12) X n ( x ) − |C| (cid:12)(cid:12)(cid:12)(cid:12) < ε · ( Z d : F )and then Theorem 2.17 guarantees the conclusion.(c) As the events are independent, this follows from (b). (cid:3) Main Constructions
Let C = { , } . In this section, we construct subshifts ( X, T ) with topologicaland ergodic prescribed properties. To this matter, we build a sequence of finitenon-empty sets of configurations C k ⊂ Ω A k ( C ), k ≥
1, such that:(i) A k = [0 , n k ) d , where ( n k ) k ≥ is an increasing sequence of positive integers.(ii) n = 1 and C = Ω A ( C ) ∼ = { , } .(iii) C k is the concatenation of elements of C k − , possibly with the insertion of fewadditional blocks of zeroes and ones.Given such sequence ( C k ) k ≥ , we consider X ⊂ Ω( C ) as the set of limits of shifted C k -patterns as k → + ∞ , that is, x ∈ X if there exist sequences ( w k ) k ≥ , w k ∈ C k ,and ( g k ) k ≥ ⊂ Z d such that x = lim k → + ∞ T g k w k . The above limit has an abuse of notation, because T acts in Ω( C ) and w k Ω( C ).Formally speaking, this means that, for each g ∈ Z d , there exists k ≥ x ( g ) = w k ( g + g k ) , ∀ k ≥ k . By definition, X is invariant under T and, for any k , every x ∈ X is an infiniteconcatenation of elements of C k and additional blocks of zeroes and ones.If C k ⊂ Ω A k ( { , } ) and A ∈ R , Ω A ( C k ) is identified in a natural way to asubset of Ω n k A ( { , } ). In some situations, to distinguish this association, we usesmall letters for Ω A ( C k ) and capital letters for Ω n k A ( { , } ) . In this situation, if w ∈ Ω A ( C k ) and g ∈ A , the pattern w ( g ) ∈ C k occurs in W ∈ Ω n k A ( { , } ) centeredat n k g . In other words, if w k ∈ C k , then S ( w k , W, n k g, F ) = n k · S ( w k , w, g, F ) . (3.1)In each of the next subsections, ( C k ) k ≥ is constructed with specific combinatorialand statistical properties. For example, w ∈ Ω A ( C k ) and W ∈ Ω n k A ( { , } ) denote the “same” element. d -ACTIONS WITH PRESCRIBED TOPOLOGICAL AND ERGODIC PROPERTIES 11 Minimality.
The action (
X, T ) is minimal if and only if, for each x, y ∈ X , every configuration of x is also a configuration of y . For this, suppose C k ⊂ Ω A k ( { , } ) is defined and non-empty.By the Law of Large Numbers, if l k is large, every element of C k occurs inalmost every element of Ω [0 ,l k ) d ( C k ) (in fact, by Corollary 2.18, each of them occursapproximately with frequency 1 / |C k | > C k +1 of Ω [0 ,l k ) d ( C k ) withthis property and consider it as a subset of Ω [0 ,n k +1 ) d ( { , } ), where n k +1 = l k n k .Let us prove that ( X, T ) is minimal. Consider x, y ∈ X and x | A a finite configu-ration of x . For large k , x | A is a subconfiguration of some w k ∈ C k . As y is formedby the concatenation of elements of C k +1 , every element of C k is a configuration of y . In particular, w k (and then x | A ) is a configuration of y .3.2. Total minimality.
The action (
X, T ) is totally minimal if and only if, foreach x, y ∈ X and F ∈ F , every configuration x | A of x centered at 0 also occursin y centered at some g ∈ F . To guarantee this for every F ∈ F , we inductivelycontrol the ocurrence of subconfigurations centered in finitely many subgroups of Z d .Consider the sequence ( F k ) ⊂ F defined in Subsection 2.2. By induction, sup-pose C k ⊂ Ω A k ( { , } ) is non-empty satisfying (i), (ii), (iii) and the additionalassumption(iv) gcd( n k , m k ) = 1 (observe that this holds for k = 1).Take l k large and ˜ C k +1 ⊂ Ω [0 ,l k m k +1 ) d ( C k ) non-empty such that(v) S ( w k , w | [0 ,l k m k +1 − d , g, F k ) = ∅ , for every triple ( w k , w, g ) ∈ C k × ˜ C k +1 × Z d .Considering w | [0 ,l k m k +1 − d as an element of Ω [0 ,l k m k +1 n k − n k ) d ( { , } ), (3.1) impliesthat S ( w k , W | [0 ,l k m k +1 n k − n k ) d , n k g, F k ) = ∅ , ∀ ( w k , w, g ) ∈ C k × ˜ C k +1 × Z d . As gcd( n k , m k ) = 1, the set n k Z d runs over all residue classes modulo F k and so(the restriction to [0 , l k m k +1 n k − n k ) d of) every element of ˜ C k +1 contains everyelement of C k centered at every residue class modulo F k .Obviously, C k +1 must not be equal to ˜ C k +1 , because m k +1 divides l k m k +1 n k .Instead, we take n k +1 = l k m k +1 n k + 1 and insert positions B i , i = 1 , , . . . , d , nextto faces of the cube [0 , l k m k +1 n k ) d . These are given by B i = { ( r , . . . , r d ) ∈ A k +1 ; r i = l k m k +1 n k − n k } . There is a natural surjection Φ : Ω A k +1 ( { , } ) → Ω [0 ,n k +1 − d ( { , } ) obtainedremoving the positions B , . . . , B d . More specifically, if δ ( r ) = 0 , if r < l k m k +1 n k − n k , = 1 , otherwiseand ∆( r , . . . , r d ) = ( δ ( r ) , . . . , δ ( r d )) , (3.2)the map Φ is given byΦ( W )( g ) = W ( g + ∆( g )) , ∀ ( r , . . . , r d ) ∈ [0 , n k +1 − d . Because of the T -invariance of X , we can suppose that x | A is centered in 0 ∈ Z d . In fact,instead of x, y , we consider T g x, T g y . We conclude the induction step taking C k +1 = Φ − ( ˜ C k +1 ).Φ( w k +1 ) ∈ ˜ C k +1 w k +1 ∈ C k +1 Φ1 2 34 5 67 8 9 1 2 34 5 67 8 9
Figure: example of Φ when d = 2 . By definition, w k +1 and Φ( w k +1 ) coincide in [0 , n k +1 − n k − d , for every w k +1 ∈C k +1 . This implies that every element of C k appears in every element of C k +1 centered at every residue class modulo F k .Let us prove that ( X, T ) is totally minimal. Fix elements x, y ∈ X , a subgroup F ∈ F and a pattern x | A of x centered in 0 ∈ Z d . By the definition of X , x | A isa subconfiguration of some w k ∈ C k , for k large enough such that F k < F . As y isbuilt concatenating elements of C k +1 , w k occurs in y centered in every residue classmodulo F and the same happens to x | A . In particular, x | A occurs in y centered insome g ∈ F , which is exactly the required condition.3.3. Total strict ergodicity.
In addition to the ocurrence of configurations inevery residue class of subgroups of Z d , we also control their frequency. Consider asequence ( d k ) k ≥ of positive real numbers such that P k ≥ d k < + ∞ . Assume that C , . . . , C k − , C k are non-empty sets satisfying (i), (ii), (iii), (iv) and(vi) For every ( w k − , w k , g ) ∈ C k − × C k × Z d ,fr( w k − , w k , g, F k − ) ∈ (cid:18) − d k − m k − d · |C k − | , d k − m k − d · |C k − | (cid:19) · Before going to the inductive step, let us make an observation. Condition (vi) alsocontrols the frequency on subgroups F such that F k − < F . In fact, if ¯ F k − is acomplete residue set modulo F k − ,fr( w k − , w k , g, F ) = X h ∈ ¯ Fk − h ≡ F g fr( w k − , w k , h, F k − ) (3.3)and, as (cid:12)(cid:12) { h ∈ ¯ F k − ; h ≡ F g } (cid:12)(cid:12) = ( F : F k − ),fr( w k − , w k , g, F ) ∈ (cid:18) − d k − ( Z d : F ) · |C k − | , d k − ( Z d : F ) · |C k − | (cid:19) · (3.4)We proceed the same way as in the previous subsection: take l k large and ˜ C k +1 ⊂ Ω [0 ,l k m k +1 ) d ( C k ) non-empty such thatfr( w k , ˜ w k +1 , g, F k ) ∈ (cid:18) − d k m kd · |C k | , d k m kd · |C k | (cid:19) (3.5) d -ACTIONS WITH PRESCRIBED TOPOLOGICAL AND ERGODIC PROPERTIES 13 for every ( w k , ˜ w k +1 , g ) ∈ C k × ˜ C k +1 × Z d . Note that the non-emptyness of ˜ C k +1 is guaranteed by Corollary 2.18. Also, let n k +1 = l k m k +1 n k + 1 and C k +1 =Φ − ( ˜ C k +1 ).Fix b A ∈ Ω ∗ ( C ). Using the big- O notation, we havefr( b A , W k +1 , g, F ) − fr( b A , W k +1 | [0 ,n k +1 − n k − d , g, F ) = O (1 /l k ) , (3.6)because these two frequencies differ by the frequency of b A in [ n k +1 − n k − , n k +1 ) d and ( n k + 1) d n k +1 d = (cid:18) n k + 1 l k m k +1 n k + 1 (cid:19) d = O (1 /l k ) . The same happens to fr( w k , w k +1 , g, F ) and fr( w k , Φ( w k +1 ) , g, F ), because ∆( g ) = 0for all g ∈ [0 , n k +1 − n k − d . To simplify citation in the future, we write it down:fr( w k , w k +1 , g, F ) − fr( w k , Φ( w k +1 ) , g, F ) = O (1 /l k ) . (3.7)These estimates imply we can assume, taking l k large enough, thatfr( w k , w k +1 , g, F k ) ∈ (cid:18) − d k m kd · |C k | , d k m kd · |C k | (cid:19) , ∀ ( w k , w k +1 , g ) ∈ C k × C k +1 × Z d . We make a calculation to be used in the next proposition. Fix b A ∈ Ω ∗ ( C ) and F ∈ F . The main (and simple) observation is: if b A occurs in W k ∈ C k centeredat g and w k occurs in Φ( w k +1 ) ∈ ˜ C k +1 centered at h ∈ [0 , l k m k +1 − d , then b A occurs in W k +1 ∈ C k +1 centered at g + n k h . This implies that, if ¯ F is a completeresidue set modulo F , the cardinality of S ( b A , W k +1 | [0 ,n k +1 − n k − d , g, F ) is equalto X h ∈ ¯ Fwk ∈C k X wk occurring in wk +1 | [0 ,lkmk +1 − d at a vertex ≡ F h | S ( b A , W k , g − n k h, F ) | + T = X h ∈ ¯ Fwk ∈C k (cid:12)(cid:12) S ( w k , w k +1 | [0 ,l k m k +1 − d , h, F ) (cid:12)(cid:12) · | S ( b A , W k , g − n k h, F ) | + T, where T denotes the number of ocurrences of b A in W k +1 | [0 ,n k +1 − n k − d not entirelycontained in a concatenated element of C k . Observe that ≤ T ≤ d · l k m k +1 · n d − · n k +1 < dn d − · n k +12 n k , where n is the length of b A . Dividing (cid:12)(cid:12) S ( b A , W k +1 | [0 ,n k +1 − n k − d , g, F ) (cid:12)(cid:12) by n k +1 d and using (3.6), (3.7), we getfr( b A , W k +1 , g, F ) = (cid:18) n k +1 − n k +1 (cid:19) d X h ∈ ¯ Fwk ∈C k fr( w k , w k +1 , h, F )fr( b A , W k , g − n k h, F )+ O (1 /l k − ) . (3.8) For each line parallel to a coordinate axis e i Z between two elements of C k in W k +1 or con-taining a line of ones, there is a rectangle of dimensions n × · · · × n × n k +1 × n × · · · × n in which b A is not entirely contained in a concatenated element of C k . We wish to show that fr( b A , x, F ) does not depend on x ∈ X . For this, define α k ( b A , F ) = min (cid:8) fr( b A , W k , g, F ) ; W k ∈ C k , g ∈ Z d (cid:9) β k ( b A , F ) = max (cid:8) fr( b A , W k , g, F ) ; W k ∈ C k , g ∈ Z d (cid:9) . The required property is a direct consequence of the next result. Proposition 3.1. If b A ∈ Ω ∗ ( C ) and F ∈ F , then lim k → + ∞ α k ( b A , F ) = lim k → + ∞ β k ( b A , F ) . Proof.
By (3.3), if l is large such that F l < F , then( F : F l ) · α k ( b A , F l ) ≤ α k ( b A , F ) ≤ β k ( b A , F ) ≤ ( F : F l ) · β k ( b A , F l ) . This means that we can assume F = F l . We estimate α k +1 ( b A , F ) and β k +1 ( b A , F )in terms of α k ( b A , F ) and β k ( b A , F ), for k ≥ l . As b A and F are fixed, denote theabove quantities by α k and β k . Take W k +1 ∈ C k +1 . By (3.8),fr( b A , W k +1 , g, F ) ≥ (cid:18) n k +1 − n k +1 (cid:19) d · α k · X h ∈ ¯ Fwk ∈C k fr( w k , w k +1 , h, F ) + O (1 /l k − )= (cid:18) n k +1 − n k +1 (cid:19) d · α k + O (1 /l k − )and, as W k +1 and g are arbitrary, we get α k +1 ≥ (cid:18) n k +1 − n k +1 (cid:19) d · α k + O (1 /l k − ) . (3.9)Equality (3.8) also implies the upper boundfr( b A , W k +1 , g, F ) ≤ (cid:18) n k +1 − n k +1 (cid:19) d · β k · X h ∈ ¯ Fwk ∈C k fr( w k , w k +1 , h, F ) + O (1 /l k − )= (cid:18) n k +1 − n k +1 (cid:19) d · β k + O (1 /l k − )= ⇒ β k +1 ≤ (cid:18) n k +1 − n k +1 (cid:19) d · β k + O (1 /l k − ) . (3.10)Inequalities (3.9) and (3.10) show that α k +1 and β k +1 do not differ very much from α k and β k . The same happens to their difference. Consider w , w ∈ C k +1 and g , g ∈ Z d . Renaming g − n k h by h in (3.8) and considering n − k the inverse of n k modulo m l , the difference fr( b A , W , g , F ) − fr( b A , W , g , F ) is at most X h ∈ ¯ Fwk ∈C k fr( b A , W k , h, F ) | fr( w k , w , n − k ( g − h ) , F ) − fr( w k , w , n − k ( g − h ) , F ) | + O (1 /l k − ) . In fact, just take the limit in the inequality α k ( b A , F ) ≤ fr( b A , x | A k , , F ) ≤ β k ( b A , F ). d -ACTIONS WITH PRESCRIBED TOPOLOGICAL AND ERGODIC PROPERTIES 15 From (3.5),fr( b A , W , g , F ) − fr( b A , W , g , F ) ≤ d k m ld · |C k | X h ∈ ¯ Fwk ∈C k fr( b A , W k , h, F )+ O (1 /l k − ) ≤ d k + O (1 /l k − ) , implying that 0 ≤ β k +1 − α k +1 ≤ d k + O (1 /l k − ) . (3.11)In particular, β k − α k converges to zero as k → + ∞ . The proposition will be provedif β k converges. Let us estimate | β k +1 − β k | . On one side, (3.10) gives β k +1 − β k ≤ O (1 /l k − ) . (3.12)On the other, by (3.9) and (3.11), β k +1 − β k ≥ α k +1 − β k ≥ (cid:18) n k +1 − n k +1 (cid:19) d · α k − β k + O (1 /l k − ) ≥ (cid:18) n k +1 − n k +1 (cid:19) d · [ β k − d k − − O (1 /l k − )] − β k + O (1 /l k − )which, together with (3.12), implies that | β k +1 − β k | ≤ d k − + β k · " − (cid:18) n k +1 − n k +1 (cid:19) d + O (1 /l k − )= 2 d k − + O (1 /l k − ) . As P d k and P /l k both converge, ( β k ) k ≥ is a Cauchy sequence, which concludesthe proof. (cid:3) From now on, we consider (
X, T ) as the dynamical system constructed as above.Note that we have total freedom to choose C k with few or many elements. This iswhat controls the entropy of the system.3.4. Proof of Theorem 1.1.
By (2.2), the topological entropy of the Z d -action( X, T ) satisfies h ( X, T ) ≥ lim k → + ∞ log |C k | n kd · Consider a sequence ( ν k ) k ≥ of positive real numbers. In the construction of C k +1 from C k , take l k large enough such that(vii) ν k · n k +1 ≥ | ˜ C k +1 | ≥ |C k | ( l k m k +1 ) d · (1 − ν k ) . These inequalities implylog |C k +1 | n k +1 d ≥ log | ˜ C k +1 | n k +1 d ≥ ( l k m k +1 ) d · (1 − ν k ) · log |C k | n k +1 d ≥ (1 − ν k ) d +1 · log |C k | n kd and then log |C k | n kd ≥ k − Y i =1 (1 − ν i ) d +1 · log |C | n d = k − Y i =1 (1 − ν i ) d +1 · log 2 . If ν ∈ (0 ,
1) is given and ( ν k ) k ≥ are chosen also satisfyinglim k → + ∞ k Y i =1 (1 − ν i ) d +1 = 1 − ν , we obtain that h ( X, T ) ≥ (1 − ν ) log 2 >
0. If, instead of { , } , we take C with moreelements and apply the construction verifying (i) to (viii), the topological entropyof the Z d -action is at least (1 − ν ) log |C| . We have thus proved Theorem 1.1.4. Proof of Theorems 1.2 and 1.3
Given a finite alphabet C , consider a configuration b A : A → C and any A ⊂ A such that | A | ≤ ε | A | . If b A : A → C , the element w ∈ Ω A ( C ) defined by w ( g ) = b A ( g ) , if g ∈ A \ A , = b A ( g ) , if g ∈ A has frequencies not too different from b A , depending on how small ε is. In fact,for any c ∈ C , | S ( c, b A , g, F ) | − | A | ≤ | S ( c, w, g, F ) | ≤ | S ( c, b A , g, F ) | + | A | and then | fr( c, b A , g, F ) − fr( c, w, g, F ) | ≤ ε . Definition 4.1.
The upper-Banach density of a set P ⊂ Z d is equal to d ∗ ( P ) = lim sup n ,...,n d → + ∞ | P ∩ [ r , r + n ) × · · · × [ r d , r d + n d ) | n · · · n d · Consider a set P ⊂ Z d of zero upper-Banach density. We will make l k growquickly such that any pattern of P ∩ ( A k + g ) appears as a subconfiguration inan element of C k . Let’s explain this better. Consider the d -dimensional cubes( A k ) k ≥ that define ( X, T ). For each k ≥
1, let ˜ A k ⊂ A k be the region containingconcatenated elements of C k − . Inductively, they are defined as ˜ A = { } and˜ A k +1 = [ g ∈ [0 ,l k m k +1 ) d (cid:16) ˜ A k + n k g + ∆( n k g ) (cid:17) , ∀ k ≥ , where ∆ is the function defined in (3.2). d -ACTIONS WITH PRESCRIBED TOPOLOGICAL AND ERGODIC PROPERTIES 17 Lemma 4.2. If P ⊂ Z d has zero upper-Banach density, there exists a totallystrictly ergodic Z d -action ( X, T ) with the following property: for any k ≥ , g ∈ Z d and b : P ∩ ( A k + g ) → { , } , there exists w k ∈ C k such that w k ( h − g ) = b ( h ) , ∀ h ∈ P ∩ ( A k + g ) . Proof.
We proceed by induction on k . The case k = 1 is obvious, since C ∼ = { , } .Suppose the result is true for some k ≥ b : P ∩ ( A k +1 + g ) → { , } .By definition, any 0,1 configuration on A k +1 \ ˜ A k +1 is admissible, so that we onlyhave to worry about positions belonging to ˜ A k +1 . For each g ∈ [0 , l k m k +1 ) d , let b g : P ∩ (cid:16) ˜ A k + n k g + ∆( n k g ) + g (cid:17) → { , } be the restriction of b to P ∩ (cid:16) ˜ A k + n k g + ∆( n k g ) + g (cid:17) . If ε > l k islarge enough, (cid:12)(cid:12)(cid:12) P ∩ (cid:16) ˜ A k +1 + g (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ A k +1 + g (cid:12)(cid:12)(cid:12) < ε (2 n k ) d = ⇒ (cid:12)(cid:12)(cid:12) P ∩ (cid:16) ˜ A k +1 + g (cid:17)(cid:12)(cid:12)(cid:12) < ε · ( l k m k +1 ) d , for any g ∈ Z d . This implies that P ∩ (cid:16) ˜ A k + n k g + ∆( n k g ) + g (cid:17) is non-empty forat most ε · ( l k m k +1 ) d values of g ∈ [0 , l k m k +1 ) d . For each of these, the inductivehypothesis guarantees the existence of w g ∈ C k such that w g ( h − n k g − ∆( n k g ) − g ) = b g ( h ) , ∀ h ∈ P ∩ (cid:16) ˜ A k + n k g + ∆( n k g ) + g (cid:17) . Take any element z ∈ C k +1 and define ˜ z ∈ Ω A k +1 ( { , } ) by˜ z ( h ) = w g ( h − n k g − ∆( n k g )) , if h ∈ ˜ A k + n k g + ∆( n k g )= b ( h ) , if h ∈ A k +1 \ ˜ A k +1 = z ( h ) , otherwise . If ε > z ∈ C k +1 . By its own definition, ˜ z satisfies the requiredconditions. (cid:3) The above lemma is the main property of our construction. It proves the follow-ing stronger statement.
Corollary 4.3.
Let ( X, T ) be the Z d -action obtained by the previous lemma. Forany b : P → { , } , there is x ∈ X such that x | P = b . Also, given x ∈ X , A ∈ R and b : P → { , } , there are ˜ x ∈ X and n ∈ N such that ˜ x | A = x | A and ˜ x ( g ) = b ( g ) for all g ∈ P \ ( − n, n ) d .Proof. The first statement is a direct consequence of Lemma 4.2 and a diagonalargument. For the second, remember that x is the concatenation of elements of C k and lines of zeroes and ones, for every k ≥
1. Consider k ≥ z k ∈ C k such that x | A occurs in z k . For any z ∈ C k +1 , there is g ∈ Z d such that z | A k + g = z k . Constructing ˜ z from z making all substitutions described in Lemma4.2, except in the pattern z | A k + g , we still have that ˜ z ∈ C k +1 . (cid:3) Proof of Theorem 1.2.
Consider f : X → R given by f ( x ) = x (0). Then1 | P ∩ ( − n, n ) d | X g ∈ P ∩ ( − n,n ) d f ( T g x ) = fr (cid:0) , x | P ∩ ( − n,n ) d (cid:1) . For each n ≥
1, consider the setsΛ n = [ k ≥ n (cid:8) x ∈ X ; fr (cid:0) , x | P ∩ ( − k,k ) d (cid:1) < /n (cid:9) Λ n = [ k ≥ n (cid:8) x ∈ X ; fr (cid:0) , x | P ∩ ( − k,k ) d (cid:1) > − /n (cid:9) . Fixed k and n , the sets (cid:8) x ∈ X ; fr(1 , x | P ∩ ( − k,k ) d ) < /n (cid:9) and is clearly open, sothat the same happens to Λ n . It is also dense in X , as we will now prove. Fix x ∈ X and ε >
0. Let k ∈ N be large enough so that d ( x, y ) < ε whenever x | ( − k ,k ) d = y | ( − k ,k ) d . Take y ∈ X such that y | ( − k ,k ) d = x | ( − k ,k ) d and y ( g ) =0 for all g ∈ P \ ( − n, n ) d as in Corollary 4.3. As fr(1 , y | ( − k,k ) d ) approaches to zeroas k approaches to infinity, y ∈ Λ n , proving that Λ n is dense in X . The sameargument show that Λ n is a dense open set. Then X = \ n ≥ (Λ n ∩ Λ n )is a countable intersection of dense open sets, thus residual. For each x ∈ X ,lim inf n → + ∞ | P ∩ ( − n, n ) d | X g ∈ P ∩ ( − n,n ) d f ( T g x ) = 0lim sup n → + ∞ | P ∩ ( − n, n ) d | X g ∈ P ∩ ( − n,n ) d f ( T g x ) = 1 , which concludes the proof of Theorem 1.2.4.2. Proof of Theorem 1.3.
Choose an infinite set G = { g i } i ≥ in Z d disjointfrom P such that P ′ = G ∪ P ∪ { } also has zero upper-Banach density and let( X, T ) be the Z d -action given by Lemma 4.2 with respect to P ′ , that is: for every b : P ′ → { , } , there exists x b ∈ X such that x b | P ′ = b . Consider X = (cid:8) x b ∈ X ; b (0) = 0 and b ( g ) = 1 , ∀ g ∈ P (cid:9) . This is an uncountable set (it has the same cardinality of 2 G = 2 N ) and, for every x b ∈ X and g ∈ P , the elements T g x b and x b differ at 0 ∈ Z d , implying that x b
6∈ { T g x b ; g ∈ P } . This concludes the proof. Acknowledgments
I would like to thank Vitaly Bergelson for suggesting the topic and for his guid-ance/advices and enormous optimism during my visit to The Ohio State University;to The Ohio State University for its great hospitality; to Carlos Gustavo Moreira,Enrique Pujals and Marcelo Viana for their mathematical support at IMPA; toFaperj-Brazil for its financial support; to the referee for pointing out some errors. d -ACTIONS WITH PRESCRIBED TOPOLOGICAL AND ERGODIC PROPERTIES 19 References V. Bergelson , Ergodic Ramsey Theory - an Update , Ergodic Theory of Z d -actions, LondonMath. Soc. Lecture Note Series (1996), 1–61.2. V. Bergelson and A. Leibman , Polynomial extensions of van der Waerden’s and Sze-mer´edi’s theorems , Journal of AMS (1996), 725–753.3. J. Bourgain , Pointwise ergodic theorems for arithmetic sets , Publ. Math. IHES (1989),5–45.4. H. Furstenberg , Strict ergodicity and transformations of the torus , Amer. J. of Math. (1961), 573–601.5. H. Furstenberg , Poincar´e recurrence and number theory , Bull. Amer. Math. Soc. (1981),no. 3, 211–234.6. F. Hahn and Y. Katznelson , On the entropy of uniquely ergodic transformations , Trans.Amer. Math. Soc. (1967), 335–360.7.
A. Katok,
Lyapunov exponents, entropy and periodic orbits for diffeomorphisms , Inst.Hautes tudes Sci. Publ. Math. (1980), 137–173.8. R. Pavlov , Some counterexamples in topological dynamics , Ergodic Theory & DynamicalSystems (2008), 1291–1322.9. K. Petersen , Ergodic Theory , Cambridge University Press (1983).10.
B. Weiss , Strictly ergodic models for dynamical systems , Bull. Amer. Math. Soc. (1985),no. 2, 143–146.11. N. Wiener , The ergodic theorem , Duke Math. J. (1939), no. 1, 1–18. Instituto Nacional de Matem´atica Pura e Aplicada, Estrada Dona Castorina 110,22460-320, Rio de Janeiro, Brasil.
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