IN AUBRY-MATHER THEORY

GENERIC UNIQUENESS OF MINIMAL CONFIGURATIONS WITH RATIONAL ROTATION NUMBERS

IN AUBRY-MATHER THEORY

ALEXANDER J. ZASLAVSKI Received 18 November 2002

We study (h)-minimal configurations in Aubry-Mather theory, wherehbelongs to a com- plete metric space of functions. Such minimal configurations have definite rotation num- ber. We establish the existence of a set of functions, which is a countable intersection of open everywhere dense subsets of the space and such that for each elementhof this set and each rational numberα, the following properties hold: (i) there exist three different (h)-minimal configurations with rotation numberα; (ii) any (h)-minimal configuration with rotation numberαis a translation of one of these configurations.

1. Introduction

Let Z be a set of all integers. A configuration is a bi-infinite sequence x=(xi)i∈Z

∈R^Z. The setR^Zwill be endowed with the product topology and the partial order de- fined byx < yif and only ifxi< yifor alli∈Z.

We have an order-preserving actionT:Z²×R^Z→R^Zdefined by T(k,x)=T_kx=y⇐⇒k=

k1,k2

∈Z²,

x,y∈R^Z, yi=xi−k1+k2 ∀i∈Z. (1.1) Letx,y∈R^Z. We say that yis a translation ofx if there isn=(n1,n2)∈Z² such that y=T_nx.

Leth:R²→R¹be a continuous function. We extend hto arbitrary finite segments (xj,. . .,xk),j < k, of configurationsx∈R^Zby

hxj,. . .,xk

:=

k−1 i=j

hxi,xi+1

. (1.2)

A segment (xj,. . .,xk) is called (h)-minimal ifh(xj,. . .,xk)≤h(yj,. . .,yk) whenever x_j=y_jandx_k=y_k.

Copyright©2004 Hindawi Publishing Corporation Abstract and Applied Analysis 2004:8 (2004) 691–721 2000 Mathematics Subject Classification: 37J45, 37E45, 70K75 URL:http://dx.doi.org/10.1155/S1085337504310067

We assume thathhas the following properties [3,4]:

(H1) for all (ξ,η)∈R²,h(ξ+ 1,η+ 1)=h(ξ,η);

(H2) lim_|η|→∞h(ξ,ξ+η)= ∞uniformly inξ;

(H3) ifξ1< ξ2andη1< η2, then hξ1,η1

+hξ2,η2

< hξ1,η2

+hξ2,η1

; (1.3)

(H4) if (x₋1,x0,x1)=(y₋1,y0,y1) are (h)-minimal segments andx0=y0, then x₋1−y₋1

x1−y1

<0. (1.4)

A configurationx∈R^Zis (h)-minimal if, for each pair of integers jandksatisfying j < kand each finite segment{y_i}^k_i=j⊂R¹satisfying y_j=x_jandy_k=x_k, the inequality h(x_j,. . .,x_k)≤h(y_j,. . .,y_k) holds. Denote byᏹ(h) the set of all (h)-minimal configura- tions. It is known that the setᏹ(h) is closed [2,3].

The notion of global minimizers ((h)-minimal configurations in the present paper) is crucial to the Aubry-Mather theory. The works by Aubry and Mather were begun inde- pendently and with different motivations but led to similar results by different methods.

While Mather [12] studied area-preserving annulus mappings as they occur as section mappings for Hamiltonian systems of two degrees of freedom, Aubry [1] investigated cer- tain models of solid state physics related to dislocations in one-dimensional crystals. For more details on Aubry-Mather theory, see [1,2,3,4,12,13,14,15,18,19]. For the usage of the notion of global minimizers in the related topics of calculus of variations, partial differential equations, and geometry, see also [3,4,5,6,7,8,10,11,16,17,20,21].

We briefly review the definitions, notions, and some basic results from Aubry-Mather theory [2,3].

Definition 1.1. The configurationsx∈R^Zandx^∗∈R^Zcross (a) ati∈Zifxi=x^∗_i and (xi−1−x^∗_i−1)(xi+1−x_i+1^∗ )<0, (b) betweeniandi+ 1 if (xi−x^∗_i )(xi+1−x^∗_i+1)<0.

Definition 1.2. The configurationx∈R^Zis periodic with period (q,p)∈(Z\ {0})×Zif T_(q,p)x=x.

Remark 1.3. Assume thath=h(ξ1,ξ2)∈C²(R²) and (∂²h/∂ξ1∂ξ2)(u,v)<0 for all (u,v)∈ R². It is not difficult to show that (H3) and (H4) hold. Moreover, we can show that if h∈C²(R²), then (H3) holds if and only if

(u,v)∈R²: ∂²h

∂ξ1∂ξ2

(u,v)<0

(1.5) is an everywhere dense subset ofR².

We have the following result [3, Corollary 3.16, Theorem 3.17].

Proposition1.4. There exists a continuous functionα^(h):ᏹ(h)→R¹with the following properties:

(i)for allx∈ᏹ(h),i∈Z,

xi−x0−iα^(h)(x)<1; (1.6) (ii)ifx∈ᏹ(h)is periodic with period(q,p)∈Z², thenα^(h)(x)=p/q;

(iii)for allα∈R¹, the set{x∈ᏹ(h) :α^(h)(x)=α} = ∅. Remark 1.5. We callα^(h)(x) the rotation number ofx∈ᏹ(h).

For eachα∈R¹, define

ᏹ(h,α)= x∈ᏹ(h) :α^(h)(x)=α. (1.7) We studyᏹ(h,α) with rationalα∈R¹.

Let a rationalα=p/qbe an irreducible fraction, whereq≥1 andpare integers. De- note byᏹ^per(h,α) the set of all periodic (h)-minimal configurationsx∈ᏹ(h,α) which satisfyT_(q,p)x=x, equivalently,x_i−q+p=x_i, for alli∈Z.

For the proof of the following result, see [2,3].

Proposition 1.6. ᏹ^per(h,α)is a nonempty closed totally ordered set. Moreover, if x∈ ᏹ^per(h,α), thenxis a minimizer ofh_qp:P_qp→R¹, where

hqp(x)=hx0,. . .,xq

, Pqp= x∈R^Z:T(q,p)x=x. (1.8)

Two elements ofᏹ^per(h,α) are called (h)-neighboring if there does not exist an ele- ment ofᏹ^per(h,α) between them. The following two propositions describe the structure of the setᏹ(h,α). For their proofs, see [3].

Proposition1.7. Suppose thatx⁻< x⁺are(h)-neighboring elements of the setᏹ^per(h,α).

Then there existy⁽¹⁾,y⁽²⁾∈ᏹ(h,α)such that

x⁻< y⁽¹⁾< x⁺, x⁻< y⁽²⁾< x⁺,

i→−∞lim y⁽¹⁾_i −x⁻_i =0, lim

i→∞y_i⁽¹⁾−x⁺_i =0,

i→−∞lim y⁽²⁾_i −x⁺_i =0, lim

i→∞y_i⁽²⁾−x⁻_i =0.

(1.9)

Suppose thatx⁻< x⁺are (h)-neighboring elements ofᏹ^per(h,α). Define ᏹ⁺h,α,x⁻,x⁺=

x∈ᏹ(h,α) : lim

i→−∞x_i−x⁻_i =0, lim

i→∞x_i−x⁺_i =0, ᏹ⁻h,α,x⁻,x⁺=

x∈ᏹ(h,α) : lim

i→−∞xi−x_i⁺=0, lim

i→∞xi−x_i⁻=0. (1.10) We denote byᏹ⁺(h,α) (resp.,ᏹ⁻(h,α)) the union of the setsᏹ⁺(h,α,x⁻,x⁺) (resp., ᏹ⁻(h,α,x⁻,x⁺)) extended over all pairs of (h)-neighboring elementsx⁻< x⁺ofᏹ^per(h,α).

Proposition1.8. (1)Ifx∈ᏹ⁻(h,α,x⁻,x⁺)∪ᏹ⁺(h,α,x⁻,x⁺), wherex⁻,x⁺∈ᏹ^per(h,α) are(h)-neighboring andx⁻< x⁺, thenx⁻< x < x⁺.

(2)ᏹ(h,α)=ᏹ^per(h,α)∪ᏹ⁺(h,α)∪ᏹ⁻(h,α).

(3)The setsᏹ^per(h,α)∪ᏹ⁺(h,α)andᏹ^per(h,α)∪ᏹ⁻(h,α)are totally ordered.

(4)ᏹ⁺(h,α)= {x∈ᏹ(h,α) : x > T(q,p)x}andᏹ⁻(h,α)= {x∈ᏹ(h,α) :x < T(q,p)x}. Letk≥2 be an integer. In this paper, we consider a complete metric space of functions h:R²→R¹which belong toC^k(R²). This space is defined inSection 2and is denoted by Mk. We establish the existence of a setᏲ⊂Mkwhich is a countable intersection of open everywhere dense subsets ofMkand such that for eachh∈Ᏺand each rationalα=p/q withpandqrelatively prime, the following properties hold:

(i) there exist (h)-minimal configurationsx⁽⁺⁾,x⁽⁻⁾, andx⁽⁰⁾with rotation number αsuch thatx⁽⁺⁾_i−q+p > x⁽⁺⁾_i ,x⁽_i−⁻_q⁾+p < x_i⁽⁻⁾, andx⁽⁰⁾_i−q+p=x_i⁽⁰⁾for all integersi;

(ii) any (h)-minimal configuration with rotation numberαis a translation of one of the configurationsx⁽⁺⁾,x⁽⁻⁾, andx⁽⁰⁾.

2. Spaces of functions

Letk≥2 be an integer. For f = f(x1,x2)∈C^k(R²) andq=(q1,q2)∈ {0,. . .,k}², satisfy- ingq1+q2≤k, we set

|q| =q1+q2, D^qf = ∂^|q|f

∂x1^q1∂x^q2²

. (2.1)

Denote byMkthe set of allh∈C^k(R²) which have the property (H1), satisfying ∂²h

∂x1∂x2

ξ1,ξ2

≤0 ∀ ξ1,ξ2

∈R², (2.2)

and have the following property:

(H5) there existδh∈(0, 1) andch>0 such that hx1,x2

≥δ_hx1−x2

2

−c_h ∀ x1,x2

∈R². (2.3)

Clearly (H5) implies (H2).

Denote byMk0the set of allh∈Mksuch that ∂²h

∂x1∂x2

ξ1,ξ2

<0 ∀ ξ1,ξ2

∈R². (2.4)

For eachN,>0, we set E_k(N,)= h1,h2

∈Mk×Mk:D^qh1

x1,x2

−D^qh2

x1,x2≤ for eachq_{∈ {}0,. . .,k_}²satisfying|q_{| ≤}k

and eachx1,x2

∈R²satisfyingx1,x2≤N

∩ h1,h2

∈Mk×Mk:h1

x1,x2

−h2

x1,x2<

+max h1

x1,x2,h2

x1,x2∀ x1,x2

∈R².

(2.5)

Using the following simple lemma, we can easily show that for the setMkthere exists the uniformity which is determined by the baseEk(N,),N,>0.

Lemma2.1. Leta,b∈R¹,∈(0, 1), and|a−b|<+max{|a|,|b|}. Then

|a−b|<+²(1−)⁻¹+(1−)⁻¹min |a|,|b|

. (2.6)

It is not difficult to see that the uniformity determined by the baseE_k(N,),N,>0, is metrizable (by a metricdk) and complete [9]. For the setMk, we consider the topology induced by the metricd2, which is called the weak topology, and the topology induced by the metricd_k, which is called the strong topology.

The following result shows that a generic function in Mk belongs to Mk0 and by Remark 1.3has the properties (H1), (H2), (H3), and (H4).

Theorem2.2. There exists a setᏲ0⊂Mk0which is a countable intersection of open (in the weak topology) everywhere dense (in the strong topology) subsets ofMk.

Proof. Forh∈Mkandγ∈(0, 1), defineh_γ:R²→R¹by hγ

x1,x2

=hx1,x2

+γx1−x22

, x1,x2

∈R². (2.7)

It is easy to see that forh∈Mkandγ∈(0, 1),h_γ∈Mk0and ∂²h_γ

∂x1∂x2

ξ1,ξ2

≤ −2γ, ξ1,ξ2

∈R², (2.8)

andhγ→hasγ→0⁺in the strong topology.

Let f ∈Mk, letγ∈(0, 1), and leti≥1 be an integer. By (2.5) and (2.8), there exists an open neighborhoodᐁ(f,γ,i) of fγinMkwith the weak topology such that the following property holds:

(P1) for eachg∈ᐁ(f,γ,i) and each (ξ1,ξ2)∈R²satisfying|ξ1|,|ξ2| ≤i, the inequality

∂²g/∂x1∂x2(ξ1,ξ2)≤ −γholds.

DefineᏲ0= ∩^∞n=1∪ {ᐁ(f,γ,i) : f ∈Mk, γ∈(0, 1), i≥n}.Clearly, Ᏺ0 is a count- able intersection of open (in the weak topology) everywhere dense (in the strong topol- ogy) subsets ofMk. We will show that Ᏺ0⊂Mk0. Leth∈Ᏺ0, (ξ1,ξ2)∈R². Choose a natural numbernsuch that|ξ1|+|ξ2|< n. There exist f ∈Mk,γ∈(0, 1), and an inte- geri≥nsuch thath∈ᐁ(f,γ,i). It follows from property (P1) and the choice ofnthat (∂²h/∂x1∂x2)(ξ1,ξ2)≤ −γ. Therefore,h∈Mk0. This completes the proof ofTheorem 2.2.

3. The main results

We will prove the following result.

Theorem 3.1. Let k≥2be an integer and αa rational number. Then there exists a set Ᏺα⊂Mk0which is a countable intersection of open (in the weak topology) everywhere dense (in the strong topology) subsets of Mk such that for each f ∈Ᏺα, the following assertions hold:

(1)ifx,y∈ᏹ^per(f,α), then there exist integersm,nsuch thatyi=xi−m+nfor alli∈Z;

(2)ifx,y∈ᏹ⁺(f,α), then there exist integersm,nsuch thaty_i=x_i−m+nfor alli∈Z; (3)ifx,y∈ᏹ⁻(f,α), then there exist integersm,nsuch thatyi=xi−m+nfor alli∈Z. It is not difficult to see thatTheorem 3.1implies the following result.

Theorem3.2. Letk≥2be an integer. Then there exists a setᏲ⊂Mk0which is a countable intersection of open (in the weak topology) everywhere dense (in the strong topology) subsets ofMksuch that for each rational numberαand each f _∈Ᏺ, assertions (1), (2), and (3) of Theorem 3.1hold.

Theorem 3.1follows from the next two propositions.

Proposition3.3. Let k≥2be an integer andα a rational number. Then there exists a setᏲα⁺⊂Mk0which is a countable intersection of open (in the weak topology) everywhere dense (in the strong topology) subsets ofMk such that for each f ∈Ᏺα⁺, assertions (1) and (2) of Theorem 3.1hold.

Proposition3.4. Let k≥2be an integer andα a rational number. Then there exists a setᏲα⁻⊂Mk0which is a countable intersection of open (in the weak topology) everywhere dense (in the strong topology) subsets of Mksuch that for each f ∈Ᏺα⁻, assertions (1) and (3) of Theorem 3.1hold.

Our goal is to proveProposition 3.3.Proposition 3.4is proved analogously.

4. Preliminary results for assertion (1) ofTheorem 3.1

Letm≥1 be an integer. Consider the manifold (R¹/Z)^mand the canonical mappingPm: R^m→(R¹/Z)^m. We have the following result [21, Proposition 6.2].

Proposition 4.1. LetΩbe a closed subset of (R¹/Z)². Then there exists a nonnegative functionφ∈C^∞((R¹/Z)²)such thatΩ= {x∈(R¹/Z)²: φ(x)=0}.

Corollary4.2. LetΩbe a closed subset ofR¹/Z. Then there exists a nonnegative function φ∈C^∞(R¹/Z)such thatΩ= {x∈R¹/Z:φ(x)=0}.

In this section, we assume thatk≥2 is an integer andα=p/qis an irreducible frac- tion, whereq_≥1 andpare integers.

For each f ∈Mk0, define E_α(f)=

q−1 i=0

fx_i,x_i+1, x∈ᏹ^per(f,α), (4.1) (seeProposition 1.6).

Proposition4.3. Let f ∈Mk, letQ be a natural number, and letD,>0. Then there exists a neighborhoodᐁof f inMkwith the weak topology such that for eachg∈ᐁ, each pair of integersn1,n2∈[n1+ 1,n1+Q], and each sequence{xi}ⁿi=²n1⊂R¹which satisfies

min _n2−1

i=n1

fxi,xi+1

,

n2−1 i=n1

gxi,xi+1

≤D, (4.2)

the inequality

n2−1 i=n1

fx_i,x_i+1−

n2−1 i=n1

gx_i,x_i+1≤ (4.3)

holds.

Proof. By (H5), there existδ0∈(0, 1) andc0>0 such that fx1,x2

≥δ0

x1−x22

−c0 ∀ x1,x2

∈R². (4.4)

Choose a positive number1for which 1

Q+c0Q+D<4⁻¹min{1,} (4.5) and a positive number0<1 which satisfies

0+²0

1−0

₋1

+0

1−0

₋1

<4⁻¹1. (4.6) Define

ᐁ= g∈Mk: (f,g)∈Ek 1,0

(4.7)

(see (2.5)).

Assume that g∈ᐁ, n1,n2∈Z,n2∈[n1+ 1,n1+Q], {xi}ⁿ_i=²_n1⊂R¹, and that (4.2) holds. By (2.5) and (4.7) for every (z1,z2)∈R²,

fz1,z2

−gz1,z2<0+0max fz1,z2,gz1,z2. (4.8) It follows from (4.6), (4.8), andLemma 2.1that for every (z1,z2)∈R²,

fz1,z2

−gz1,z2

<0+0²

1−0

₋1

+0

1−0

₋1

min fz1,z2,gz1,z2

<4⁻¹1+ 4⁻¹1min fz1,z2,gz1,z2.

(4.9)

Formulas (4.4) and (4.9) imply that for every (z1,z2)∈R², gz1,z2

≥ fz1,z2

−4⁻¹1−4⁻¹1fz1,z2≥ −4⁻¹1−2c0. (4.10) Set

λ_i=min fx_i,x_i+1,gx_i,x_i+1, i=n1,. . .,n2−1. (4.11) It follows from (4.4), (4.9), (4.10), and (4.11) that fori=n1,. . .,n2−1,

fx_i,x_i+1−gx_i,x_i+1

<4⁻¹1+ 4⁻¹1min fxi,xi+1

+ 2c0,gxi,xi+1

+ 4c0+ 2

≤4⁻¹1+ 4⁻¹1λ_i+c01+¹ 2.

(4.12)

By these inequalities, (4.2), (4.5), and (4.11),

n2−1 i=n1

fx_i,x_i+1−gx_i,x_i+1

≤ n2−n1

4⁻¹1+ 2⁻¹1+1c0

+ 4⁻¹1 n2−1 i=n1

λi

≤ n2−n1

1+1c0

+ 4⁻¹1D

≤Q1+1c0

+ 4⁻¹1D <.

(4.13)

This completes the proof ofProposition 4.3.

Corollary4.4. Let f ∈Mk0and>0. Then there exists a neighborhoodᐁof f inMk

with the weak topology such that for eachg∈ᐁ∩Mk0,E_α(g)≤E_α(f) +.

Proposition4.5. Assume that f ∈Mk0,f_n∈Mk0,n=1, 2,. . .,limn→∞f_n= f in the weak topology,

x⁽ⁿ⁾∈ᏹfn

, n=1, 2,. . .,x∈R^Z,

nlim→∞x⁽ⁿ⁾_i =xi ∀i∈Z. (4.14) Thenx∈ᏹ(f).

Proof. We assume the converse. Then, there exist integersi1< i2and a sequence{yi}ⁱi²=i1⊂ R¹such that

yi1=xi1, yi2=xi2,

i2−1 i=i1

fyi,yi+1

<

i2−1 i=i1

fxi,xi+1

. (4.15)

Set

∆=

i2−1 i=i1

fxi,xi+1

−fyi,yi+1

. (4.16)

For each integern≥1, define a finite sequence{y_i⁽ⁿ⁾}ⁱi²=i1⊂R¹as follows:

y_i⁽ⁿ⁾₁ =x_i⁽ⁿ⁾₁ , y⁽ⁿ⁾_i2 =x⁽ⁿ⁾_i2 , y_i⁽ⁿ⁾=yi, i∈ i1,. . .,i2

\ i1,i2

. (4.17)

It follows from (4.14), (4.15), (4.16), (4.17), and the continuity of f that

nlim→∞

_i2−1

i=i1

fx⁽ⁿ⁾_i ,x_i+1⁽ⁿ⁾−

i2−1 i=i1

fy_i⁽ⁿ⁾,y_i+1⁽ⁿ⁾

=

i2−1 i=i1

fx_i,x_i+1−

i2−1 i=i1

fy_i,y_i+1=∆>0.

(4.18)

Formulas (4.14) and (4.18) imply that the sequences _i2−1

i=i1

fx⁽ⁿ⁾_i ,x_i+1⁽ⁿ⁾

∞ n=1

,

_i2−1

i=i1

fy_i⁽ⁿ⁾,y_i+1⁽ⁿ⁾

∞ n=1

(4.19) are bounded. It follows from this fact,Proposition 4.3, and the equality f =limn→∞fnin the weak topology that

nlim→∞

_i2−1

i=i1

fx⁽ⁿ⁾_i ,x_i+1⁽ⁿ⁾−

i2−1 i=i1

fn

x⁽ⁿ⁾_i ,x⁽ⁿ⁾_i+1=0,

nlim→∞

_i2−1

i=i1

fy⁽ⁿ⁾_i ,y⁽ⁿ⁾_i+1−

i2−1 i=i1

fn

y_i⁽ⁿ⁾,y_i+1⁽ⁿ⁾=0.

(4.20)

Formulas (4.18) and (4.20) imply that

nlim→∞

_i2−1

i=i1

fn

x⁽ⁿ⁾_i ,x_i+1⁽ⁿ⁾−

i2−1 i=i1

fn

y_i⁽ⁿ⁾,y_i+1⁽ⁿ⁾=∆>0. (4.21)

There is an integern0≥1 such that for each integern≥n0,

i2−1 i=i1

fn

x⁽ⁿ⁾_i ,x_i+1⁽ⁿ⁾−

i2−1 i=i1

fn

y_i⁽ⁿ⁾,y_i+1⁽ⁿ⁾>∆

2. (4.22)

This fact contradicts the (f_n)-minimality ofx⁽ⁿ⁾for alln≥n0. The contradiction we have

reached provesProposition 4.5.

Proposition4.6. Let f ∈Mk0, fn∈Mk0,n=1, 2,. . .,limn→∞fn=f in the weak topol- ogy,x⁽ⁿ⁾∈ᏹ^per(fn,α),n=1, 2,. . .,and let the sequence {x0⁽ⁿ⁾}^∞n=1 be bounded. Then the following assertions hold:

(1)there existx∈R^Zand a strictly increasing sequence of natural numbers{n_j}^∞_j=1such that

x_i+q=x_i+p, i∈Z, (4.23)

x⁽ⁿ_i ^j)−→xi asj−→ ∞,∀i∈Z; (4.24) (2)assume thatx∈R^Zand{nj}^∞j=1is a strictly increasing sequence of natural numbers

such that (4.23) and (4.24) hold. Thenx∈ᏹ^per(f,α)and Eα(f)=

q−1 i=0

fxi,xi+1

=lim

j→∞

q−1 i=0

fnj

x⁽ⁿ_i ^j),x⁽ⁿ_i+1^j)=lim

j→∞Eα fnj

. (4.25)

Proof. ByProposition 1.4, the sequence{x⁽ⁿ⁾_i }^∞n=1is bounded for alli∈Z. This fact im- plies that there exist a strictly increasing sequence of natural numbers{n_j}^∞_j=1andx∈R^Z such that (4.23) and (4.24) are valid. Therefore, assertion (1) is true.

We will prove assertion (2). Assume thatx∈R^Z and {n_j}^∞_j=1 is a strictly increas- ing sequence of natural numbers such that (4.23) and (4.24) hold. ByProposition 4.5 and (4.23), x∈ᏹ^per(f,α). Since limn→∞fn= f in the weak topology, it follows from Corollary 4.4that the sequence{E_α(f_n)}^∞_n=1 is bounded from above. Therefore, the se- quence{q−1

i=0 f_n(x⁽ⁿ⁾_i ,x_i+1⁽ⁿ⁾)}^∞_n=1is also bounded from above. It follows from this fact, the equality limn→∞fn= f in the weak topology, andProposition 4.3that

nlim→∞

_q−1 i=0

fn

x⁽ⁿ⁾_i ,x_i+1⁽ⁿ⁾−

q−1 i=0

fx_i⁽ⁿ⁾,x⁽ⁿ⁾_i+1=0. (4.26) By (4.1), (4.23), (4.24), (4.26), andCorollary 4.4,

E_α(f)≤

q−1 i=0

fx_i,x_i+1=lim

j→∞

q−1 i=0

fx⁽ⁿ_i ^j),x_i+1^(nj)

=lim

j→∞

q−1 i=0

f_njx⁽ⁿ_i ^j),x_i+1^(nj)=lim

j→∞E_αf_nj≤E_α(f).

(4.27)

These relations imply (4.25).Proposition 4.6is proved.

Proposition 4.6andCorollary 4.4imply the following result.

Proposition4.7. The function f →Eα(f)is continuous onMk0with the relative weak topology.

Proposition4.8. Assume that f ∈Mk0and that the following property holds:

Ifx⁽¹⁾,x⁽²⁾∈ᏹ^per(f,α), then there existsn=(n1,n2)∈Z²such thatx⁽²⁾=Tnx⁽¹⁾. Then there existsn¯=( ¯n1, ¯n2)∈Z²such that for eachx∈ᏹ^per(f,α),

Tn¯x > x, y∈ᏹ^per(f,α) :x < y < Tn¯x= ∅. (4.28) Proof. Let ¯x∈ᏹ^per(f,α). Then

ᏹ^per(f,α)= Tnx¯:n= n1,n2

∈Z²

= Tnx¯:n= n1,n2

∈Z², 0≤n1≤q−1. (4.29) Formula (4.29) implies that the set

y∈ᏹ^per(f,α) : ¯x < y < T(0,1)x¯ (4.30) is either finite or empty. Therefore, there exists ¯x⁺∈ᏹ^per(f,α) such that

¯

x <x¯⁺, y∈ᏹ^per(f,α) : ¯x < y <x¯⁺= ∅. (4.31) There exists ¯n=( ¯n1, ¯n2)∈Z²such that

Tn¯x¯=x¯⁺. (4.32)