Entropy and its variational principle for noncompact metric spaces

En trop y and its v ariational princip le for noncompact metric spa ces Mauro P at r˜ ao Octob er 26, 2018 Abstract In the present paper, we in tro duce a natural extension of AKM- top ological en trop y for noncompact spaces and pro v e a v a riational principle whic h states that the top ological entrop y , the supremum of the measure theoretical en tropies and the minim um of th e metric the- oretical entropies alw a ys coincide. W e apply the v ariat ional pr inciple to sho w that the top ological en trop y of automorphisms of simply con- nected nilpotent Lie groups alw a ys v anishes. This sho ws that the classical form ula for the en trop y of an automorphism of a noncompact Lie group is ju st an upp er b ound for its top ological en trop y . AMS 20 0 0 subje ct classific ation : Primary: 37B40 37A35, Secondary: 22E25. Key wor ds: T op ological entrop y , v ariational principle, non- compact metric spaces, automorphisms of Lie groups. 1 In tro duct ion T op ological en tropy w as in tro duced in [1] by Adler, Konheim and MacAn- drew (AKM) to study the dynamic of a con tin uous map T : X → X defined on a compact space X . They ha ve also conjectured the w ell kno wn v ariational principle whic h states that h ( T ) = sup µ h µ ( T ) , where h ( T ) is the top ological en tropy of T , h µ ( T ) is the µ -en tropy of T and the suprem um is tak en ov er all T - in v ar ian t probabilit ies µ o n X . 1 R. Bow en has extended in [2] the concept of to p ological en tropy for non- compact metric spaces. This approa c h uses the concept of generator sets, whic h are defined b y means of a giv en metric. F or compact spaces this defi- nition of en tro p y is, in fact, indep enden t of the metric and, indeed, coincides with the AKM-top ological en tro py . Ho w ev er, for noncompact space s Bow en’s en tropy dep ends on the metric. In Lemma 1.5 of [4], it w as prov ed the following v a riational principle f o r lo cally compact spaces sup µ h µ ( T ) = inf d h d ( T ) , where h d ( T ) is the d -en tropy of T and the infim um is tak en ov er all metric d on X . In the presen t pap er, w e improv e this result sho wing that the ab o ve infi- m um is alw a ys attained at d , whenev er d is a metric satisfying some sp ecial conditions. Theses metrics are called admissible metrics and alw a ys exist when X is a lo cally compact separable space. Mor eov er the suprem um and the minim um coincide with a natural extension of AKM-top ological en tr o p y , in tro duced in the second section. In the third section, this v ariational prin- ciple is pro v en. In the last section, w e apply the v ariationa l principle to determine the top ological en tropy of automorphisms of some Lie groups. First w e consider linear isomorphisms of a finite dimensional ve ctor space and characterize their rec urrent sets in terms of their m ultiplicative Jordan decomp ositions. W e show that the top ological entrop y of these linear isomorphisms alw ays v anishes. This sho ws that the classical form ula for its d - en tropy , where d is the euclidian metric, is just an upp er b ound for its null top ological en tropy . Remem b er tha t this classical form ula is giv en b y h d ( T ) = X | λ | > 1 log | λ | , where λ runs through the eigen v a lues of the linear isomorphism T (cf. The- orem 8.14 in [7]). Th us it might b e an inte resting problem to calculate the top ological en tropy o f a giv en auto mo r phism φ o f a general noncompact Lie group G , since the classical formula for its d -en tropy , w here d is some in v ari- an t metric, is as w ell just an upp er b ound for its top ological en tropy . W e conclude t his paper pro viding an answ er for this problem when G is a simply connected nilp oten t Lie g r oup . W e show tha t the top ological en tropy o f its automorphisms also v anishes. 2 2 T op ologi cal entrop y and admissible metrics W e start this sec tion pr esen ting an extension of the AKM-top ological en tropy in tro duced in [1]. Let X b e a top ological space and T : X → X b e a prop er map, i.e., T is a con tin uous map suc h that the pre-imag e by T of an y compact set is compact. An admi s s ible co ve ring of X is an op en and finite co v ering α of X suc h that, for each A ∈ α , the closure or the complemen t o f A is compact. Given an admissible co v ering α of X , for ev ery n ∈ N , we ha v e that the set g iv en by α n = { A 0 ∩ T − 1 ( A 1 ) ∩ . . . ∩ T − n ( A n ) : A i ∈ α } is also an admissible co vering of X , since T is a prop er map. Giv en an admissible cov ering α of X , w e denote b y N ( α n ) the smallest cardinality of all sub-co v erings of α n . Ex actly as in the compact case, it can b e show n that t he sequence log N ( α n ) is subadditiv e, whic h implies the existence of the follo wing limit h ( T , α ) = lim n →∞ 1 n log N ( α n ) . The top olo g i c al entr opy of the map T is thus defined as h ( T ) = sup α h ( T , α ) , where t he suprem um is tak en o ve r all admissible cov erings α o f X . W e note that, when X is already compact, the a b ov e definition coincides with the AKM definition, since t hus ev ery con tinuous map is prop er and ev ery open and finite co v ering of X is admissible. The follow ing res ult generalizes, for non-compact spaces, the relation of the top ological en tro pies of tw o semi-conjugated maps. Prop osition 2.1 L et T : X → X and S : Y → Y b e two pr op er maps , wher e X an d Y ar e top olo gic a l sp ac es. If f : Y → X is a pr op er surje ctive map such that f ◦ S = T ◦ f , then we have that h ( T ) ≤ h ( S ) . Pro of: Let α b e an admissible co v ering of X . Since f is a proper map, it follo ws that f − 1 ( α ) = { f − 1 ( A ) : A ∈ α } 3 is an admiss ible cov ering of Y . W e claim that if β is a subs et of α n , then f − 1 ( β ) is a subset of f − 1 ( α ) n . In fact, B ∈ β if and only if B = A 0 ∩ T − 1 ( A 1 ) ∩ . . . ∩ T − n ( A n ) where A i ∈ α , for eac h i ∈ { 0 , . . . , n } . Th us we ha v e t ha t f − 1 ( B ) = f − 1 ( A 0 ) ∩ f − 1 ( T − 1 ( A 1 )) ∩ . . . ∩ f − 1 ( T − n ( A n )) = f − 1 ( A 0 ) ∩ S − 1 ( f − 1 ( A 1 )) ∩ . . . ∩ S − n ( f − 1 ( A n )) , where we used that f − 1 ◦ T − i = S − i ◦ T − 1 , since f ◦ S = T ◦ f . Thus it follo ws that f − 1 ( β ) ⊂ f − 1 ( α ) n . R ecipro cally , we claim that if γ is a subset of f − 1 ( α ) n , then γ = f − 1 ( β ), where β is some subset of α n . Pro ceeding analogously , if C ∈ γ , then C = f − 1 ( A 0 ) ∩ S − 1 ( f − 1 ( A 1 )) ∩ . . . ∩ S − n ( f − 1 ( A n )) (1) = f − 1 ( A 0 ∩ T − 1 ( A 1 ) ∩ . . . ∩ T − n ( A n )) , where A i ∈ α , f or eac h i ∈ { 0 , . . . , n } . Th us it follows tha t C = f − 1 ( B ), for some B ∈ α n , whic h implies that γ = f − 1 ( β ), where β is some subset of α n . If β is a sub-co ve ring of α n , then f − 1 ( β ) is a sub-cov ering of f − 1 ( α ) n . Recipro cally , if γ is a sub-co vering of f − 1 ( α ) n , then γ = f − 1 ( β ), where β is some sub-co ve ring of α n . In fact, w e ha v e already kno wn tha t γ = f − 1 ( β ), for some subs et β of α n . W e ha v e to sho w just that β is a co v ering o f X . But this is immediate, since γ = f − 1 ( β ) is a co ve ring of Y , f is surjectiv e and B = f ( f − 1 ( B )). Hence w e ha ve that N ( α n ) = N ( f − 1 ( α ) n ) a nd ta king logarithms, divid ing b y n and taking limits, it follows that h ( T , α ) ≤ h ( S, f − 1 ( α )) ≤ h ( S ) . Since α is an arbitra ry admissible co vering of X , w e get that h ( T ) ≤ h ( S ). No w w e in tro duce the concept of en tropy asso ciated to some giv en metric in tw o differen t w a ys. First w e remem b er the classical definition, introduced in [2 ]. Giv en a metric space ( X , d ) and a con tin uous map T : X → X , we consider the metric giv en by d n ( x, y ) = max { d ( T i ( x ) , T i ( y )) : 0 ≤ i ≤ n } , 4 whic h is equiv alen t to the original metric d . Giv en a subset Y ⊂ X , a subs et G is a n ( n, ε )-generator of Y if and only if, for ev ery y ∈ Y , there exists x ∈ G suc h t ha t d n ( x, y ) < ε . In other w o rds, the collection of all ε -balls of d n cen tered at p oin ts o f G is in fact a co ve ring o f Y . W e denote b y G n ( ε, Y ) the smallest cardinalit y of a ll ( n, ε )-generator s of Y and define g ( ε, Y ) = lim n →∞ sup 1 n log G n ( ε, Y ) . It can b e pro v ed that g ( ε, Y ) is monotone with resp ect to ε , so we can define h d ( T , Y ) = lim ε ↓ 0 g ( ε, Y ) . The d -entr opy of the map T is thus defined as h d ( T ) = sup K h d ( T , K ) , where the suprem um is tak en ov er all compact subsets K of X . The definition of d -en tropy whic h w e presen t in this pap er is giv en by h d ( T ) = sup Y h d ( T , Y ) , where no w the supre mum is tak en ov er all subsets Y of X instead of just the compact ones. Note that here the metric d lies at the superscript. Since w e ha v e that h d ( T , Y ) is monoto ne with resp ect to Y , it fo llo ws that h d ( T ) = h d ( T , X ) . W e immediately no t ice that alwa ys h d ( T ) ≤ h d ( T ) a nd, when X is itself compact, we get the equalit y . Denoting G n ( ε, X ) and g ( ε, X ), resp ective ly , b y G n ( ε ) and g ( ε ), we get that h d ( T ) = lim ε ↓ 0 g ( ε ) and that g ( ε ) = lim n →∞ sup 1 n log G n ( ε ) . Our first result is a generalization of the w ell know result whic h states that, if ( X , d ) is compact, then the top olo g ical and the metric entropies coincide. In the non-compact case, t his remains true but w e need to ask for some regularit y in the metric. Let ( X , d ) b e a metric space. The metric d is admissible if the follow ing conditions a r e v erified: 5 (1) If α δ = { B ( x 1 , δ ) , . . . , B ( x k , δ ) } is a co vering of X , for ev ery δ ∈ ( a, b ), where 0 < a < b , t hen there exis ts δ ε ∈ ( a, b ) suc h that α δ ε is admissible. (2) Eve ry admissible co ve ring of X has a Leb esgue n umber. W e observ e that, if ( X , d ) is compact, then d is automatically admissible. Prop osition 2.2 L et T : X → X b e a pr o p er map, wher e ( X , d ) is a metric sp ac e. If d is an admissible m e tric, then h ( T ) = h d ( T ) . Pro of: First w e claim that, for an y admiss ible co v ering α of X , it follo ws tha t g ( | α | ) ≤ h ( T , α ), where | α | is the maxim um of the diameters of A ∈ α . In fact, the elemen ts of α n are g iv en b y A 0 ∩ T − 1 ( A 1 ) ∩ . . . ∩ T − n ( A n ), where A i ∈ α . Let β be a sub-cov ering of α n . F or eac h B ∈ β , tak e an x ∈ B and consider G the set of all suc h p o in ts. W e claim that G is an ( n, | α | ) - generator set of X . In fact, let y ∈ X and take some A 0 ∩ T − 1 ( A 1 ) ∩ . . . ∩ T − n ( A n ) ∈ β containing y . T aking x ∈ G suc h that x ∈ A 0 ∩ T − 1 ( A 1 ) ∩ . . . ∩ T − n ( A n ), w e hav e that d ( T i ( x ) , T i ( y )) < | α | , for ev ery i ∈ { 0 , 1 , . . . , n } , since T i ( x ) , T i ( y ) ∈ A i . Hence, for eac h sub-co ve ring β o f α n , there exists an ( n, | α | )- generator set of X suc h that G n ( | α | ) ≤ # G ≤ # β . Th us w e get that G n ( | α | ) ≤ N ( α n ). T aking loga r it hms, dividing b y n and t a king limits, it follows as claimed that g ( | α | ) ≤ h ( T , α ). No w w e claim that, for all ε > 0, there exists δ ε ∈ ( ε/ 2 , ε ) suc h t ha t h ( T , α δ ε ) ≤ g ( ε/ 2), where α δ ε is an admissible co ve ring of balls with radius equals to δ ε . In fact, if G = { x 1 , . . . , x k } is an ( n, ε/ 2)-generator of X , for ev ery δ ∈ ( ε / 2 , ε ), we hav e that β δ = { B ( x i , δ ) ∩ . . . ∩ T − n ( B ( T n ( x i ) , δ )) : x i ∈ G } is a co v ering of X . T o see t his, giv en x ∈ X , there exists x i ∈ G suc h that d n ( x, x i ) < ε/ 2 < δ and th us w e ha v e tha t x ∈ B ( x i , δ ) ∩ . . . ∩ T − n ( B ( T n ( x i ) , δ )) . This implies that α δ = { B ( T l ( x i ) , δ ) : x i ∈ G, 0 ≤ l ≤ n } is a finite cov ering of X , for ev ery δ ∈ ( ε / 2 , ε ). Since d is an admissible metric, there exists δ ε ∈ ( ε / 2 , ε ) s uch that α δ ε is an admissible co ve ring of 6 balls with radius equals to δ ε . F urthermore w e hav e that β δ ε is a sub-co vering of α n δ ε . Hence, for eac h ( n, ε/ 2)- generator G of X , there exist δ ε ∈ ( ε/ 2 , ε ) and a sub-cov ering β δ ε of α n δ ε suc h that N ( α δ ε ) ≤ # β δ ε ≤ # G . Th us it follo ws that N ( α δ ε ) ≤ G n ( ε/ 2). T aking logarithms, dividing b y n and taking limits, w e get as claimed that there exists δ ε ∈ ( ε / 2 , ε ) suc h that h ( T , α δ ε ) ≤ g ( ε/ 2). Since | α δ ε | ≤ 2 δ ε ≤ 2 ε , w e hav e that g (2 ε ) ≤ g ( | α δ ε | ) ≤ h ( T , α δ ε ) ≤ g ( ε/ 2) . T aking limits with ε ↓ 0, it follo ws that h d ( T ) = lim ε ↓ 0 h ( T , α δ ε ) = sup ε> 0 h ( T , α δ ε ) . In order to complete the pro of, it r emains to show that the ab ov e supre- m um is e qual to h ( T ). F or any admiss ible co ve ring α of X , tak e ε a Lebesgue n um b er of this co v ering. W e claim that N ( α n ) ≤ N ( α n δ ε ), where α δ ε is given ab ov e. In fact, since ev ery elemen t of α n δ ε if giv en b y B 0 ∩ . . . ∩ T − n ( B n ), where { B 0 , . . . , B n } are balls of radius δ ε < ε , there exist { A 0 , . . . , A n } ⊂ α suc h that B i ⊂ A i , for a ll i ∈ { 0 , . . . , n } . Th us w e ha v e t ha t B 0 ∩ . . . ∩ T − n ( B n ) ⊂ A 0 ∩ . . . ∩ T − n ( A n ) , sho wing tha t , for eac h sub-co ve ring β of α n δ ε , there exists a sub-co v ering γ of α n suc h that N ( α n ) ≤ # γ ≤ # β . Hence w e get as claimed that N ( α n ) ≤ N ( α n δ ε ). T aking log a rithms, dividing by n and taking limits, it follo ws that h ( T , α ) ≤ h ( T , α δ ε ) ≤ sup ε> 0 h ( T , α δ ε ) , whic h sho ws that h ( T ) = sup ε> 0 h ( T , α δ ε ) , completing the pro of. F rom no w o ne we assume that X is a lo cally compact space. Th us w e ha v e asso ciat ed to X its one p oin t compactification, whic h we w ill denote b y e X . W e hav e that e X is defined as the disjoin t union of X with {∞} , where ∞ is some p o in t not in X called the p oint at the infinity . The top ology in e X consists b y the former op en sets in X and b y the sets A ∪ {∞} , where 7 the the complemen t of A in X is compact. If T : X → X is a prop er map, defining e T : e X → e X by e T ( e x ) =  T ( e x ) , x 6 = ∞ ∞ , x = ∞ , w e hav e that e T is also prop er map, called the extension of T to e X . T o see this, w e only need to ve rif y that e T is con tin uous at ∞ . If A ∪ {∞} is a neigh b orho o d of ∞ , then the complemen t of A is compact and we ha v e that e T − 1 ( A ∪ {∞} ) = T − 1 ( A ) ∪ {∞} is also a neigh b orho o d of ∞ , since T is prop er and th us the complemen t of T − 1 ( A ) is a lso compact. The following result show s that the restriction to X of an y metric on e X is alw a ys a dmissible and that their resp ectiv e en tropies coincide. Prop osition 2.3 L et T : X → X b e a pr op er map, wher e X is a lo c al ly c omp act sep ar able sp a c e. L et d b e the metric given by the r estriction to X of some metric e d on e X , the one p oin t c omp a ctific ation of X . Then it fol lows that d is an admissible metric and that h d ( T ) = h e d  e T  , wher e e T is the extension of T to e X . I n p articular, we get that h ( T ) = h  e T  . Pro of: First w e sho w that G n ( ε ) is finite for eve ry ε > 0. Let e G = { e x 1 , . . . , e x k } ⊂ e X b e an ( n, ε/ 2)-generator set of e X . By the density of X in e X , it follo ws that there ex ist { x 1 , . . . , x k } ⊂ X , suc h that e d n ( x i , e x i ) < ε/ 2. If x ∈ X ⊂ e X , we hav e that e d n ( x, e x i ) < ε/ 2, fo r some e x i ∈ e G . Hence it follo ws that d n ( x, x i ) ≤ e d n ( x, e x i ) + e d n ( x i , e x i ) < ε/ 2 + ε / 2 = ε, sho wing that G = { x 1 , . . . , x k } is an ( n, ε )-generator set of X . If w e c ho ose e G suc h that # e G = e G n ( ε/ 2), then w e get as claimed that G n ( ε ) ≤ e G n ( ε/ 2) < ∞ . 8 No w assume that α δ = { B ( x 1 , δ ) , . . . , B ( x k , δ ) } is a co v ering of X , for ev ery δ ∈ ( a, b ), whe re 0 < a < b . Since, f or eac h fixed δ , the n um b er of ba lls are finite, it follows that there exists δ ε ∈ ( a, b ) suc h that ∞ / ∈ e S ( x i , δ ε ), for ev ery i ∈ { 1 , . . . , k } , where e S ( x, r ) is the sphere in e X of radius r cen tered in x . D enoting b y e B ( x, r ) the op en ball in e X of radius r cen tered in x , it remains tw o alternativ es: 1) the p oin t ∞ is inside e B ( x i , δ ε ) or 2) the po in t ∞ is not in the closure of e B ( x i , δ ε ). In the first case, the complemen t of B ( x i , δ ε ) in X is equal to the complemen t of e B ( x i , δ ε ) in X , whic h is compact. In the second alternative , there exist a o p en neighborho o d U of ∞ whic h has empt y in tersection with e B ( x i , δ ε ). Th us B ( x i , δ ε ) = e B ( x i , δ ε ) is in the complemen t of U in X , whic h is compact. This s hows that the closure o r the complemen t of B ( x i , δ ε ) in X is compact, sho wing that α δ ε is already admissible. W e show now that ev ery admissible co vering o f X has a Leb esgue n um b er. If α is admissible co v ering of X , there exists an op en co v ering e α o f e X suc h that A ∈ α if and only if t here exists e A ∈ e α , with A = e A ∩ X . In fact, if α is an a dmissible co ve ring of X , there exists at least one A ∈ α with compact complemen t in X and suc h that its closu re in X is not compact. If w e define e A = A ∪ {∞} , w e ha v e that e A is an op en neigh b orho o d of {∞} in e X . Thus defining e α = n e A o ∪ { B ∈ α : B 6 = A } , it follows as claimed that e α is an o p en co v ering of e X suc h that A ∈ α if and only if there exists e A ∈ e α , with A = e A ∩ X . If ε is a Leb esgue n umber for e α , then w e claim that ε is a also a Leb esgue n um b er for α . T o see this, if B ( x, ε ) is a ball in X , there exists e A ∈ e α suc h that B ( x, ε ) ⊂ e A . Th us it follo ws that B ( x, ε ) ⊂ A , where A = e A ∩ X ∈ α , completing the pro o f that d is an admissible metric. Finally w e show that h d ( T ) = h e d  e T  . Let G = { x 1 , . . . , x k } b e an ( n, ε/ 2)-generator set of X . By the densit y of X in e X , if e x ∈ e X , there exists x ∈ X suc h that e d n ( e x, x ) < ε/ 2, since d n is t op ologically equiv a len t to d . Th us w e ha v e that d n ( x, x i ) < ε/ 2, for some x i ∈ G . Hence it follow s that e d n ( e x, x i ) ≤ e d n ( e x, x ) + d n ( x, x i ) < ε/ 2 + ε / 2 = ε, sho wing that G = { x 1 , . . . , x k } is an ( n, ε )- g enerator set of e X . Cho osing G suc h that # G = G n ( ε/ 2), we get that e G n ( ε ) ≤ G n ( ε/ 2). Since w e ha v e also sho wn ab ov e that G n ( ε ) ≤ e G n ( ε/ 2), taking log arithms, dividing by n and 9 taking limits, w e get that g ( ε ) ≤ e g ( ε/ 2 ) and e g ( ε ) ≤ g ( ε/ 2) . Therefore it follo ws that h d ( T ) = lim ε ↓ 0 g (4 ε ) ≤ lim ε ↓ 0 e g (2 ε ) = h e d  e T  ≤ lim ε ↓ 0 g ( ε ) = h d ( T ) , sho wing that h d ( T ) = h e d  e T  . The last statemen t no w f o llo ws applying Prop osition 2.2. 3 The v ariational princ iple In this section, w e presen t a f ull extension for non-compact sets o f the w ell kno wn v a riational principle in volvin g en tropies. W e start with a prop er map T : X → X and remem b er the concept of entrop y asso ciated to some give n T -in v a rian t pro babilit y on X . A T -in variant pr ob ability on X is a Borel measure µ suc h that µ ( X ) = 1 and µ ( T − 1 ( A )) = µ ( A ), for all Borel subsets A ⊂ X . W e denote b y P T ( X ) the set of all T -in v ar ian t probability on X . A Borel partition A of X is a partition of X where all of its elemen ts are Borel sets. Giv en a finite Borel partition A of X , fo r ev ery n ∈ N , w e ha v e that the set giv en b y A n = { A 0 ∩ T − 1 ( A 1 ) ∩ . . . ∩ T − n ( A n ) : A i ∈ A } is also a finite Borel partition of X , since T is con tin uous. F or a giv en finite Borel partition A of X , w e define its asso ciated n -entrop y as H ( A n ) = X B ∈A n φ ( µ ( B )) , where φ : [0 , 1] → R is the contin uous f unction g iv en by φ ( x ) =  − x log( x ) , x ∈ (0 , 1] 0 , x = 0 . It can b e sho wn that the sequence H ( A n ) is subadditiv e, whic h implies the existence of the follo wing limit h ( T , A ) = lim n →∞ 1 n H ( A n ) . 10 The µ -entr opy of the map T is thus defined as h µ ( T ) = sup A h ( T , A ) , where the suprem um is take n o v er all finite Borel par t it ion A o f X . The next result w as first pro v ed in the Lemma 1.5 of [4], where they us ed the fact that the suprem um o f the measure en t r opies taken o v er all in v arian t proba bilities is equals to tha t one just tak en o v er all ergodic in v ariant probabilities. W e presen t here an elemen ta ry pro of of the quoted result whic h dos not use this fact. In the follo wing P T ( X ) denotes the set of a ll T -inv ar ian t probabilities on X . Lemma 3.1 L e t T : X → X b e a pr op er map and e T : e X → e X b e its extension i n the one p oi n t c omp actific ation e X of the lo c al ly c omp act sep ar ab l e sp ac e X . T hen it fo l lows that sup µ h µ ( T ) = sup e µ h e µ  e T  , wher e the supr ema ar e taken, r esp e ctively, over P T ( X ) and over P e T ( e X ) . Pro of: If µ ∈ P T ( X ), definig e µ  e A  = µ  e A ∩ X  , it is immediate that e µ ∈ P e T ( e X ), since X and {∞} are e T -in v ariant sets . It is also immediate that h µ ( T ) = h e µ  e T  , sho wing tha t sup µ h µ ( T ) ≤ sup e µ h e µ  e T  . No w let e µ ∈ P e T ( e X ). If e µ ( ∞ ) = 1, it is immediate that h e µ  e T  = 0, sinc e X and {∞} are e T -in v a rian t sets. Thus we can assume that e µ ( ∞ ) = c < 1. It is also immediate that µ =  1 1 − c  e µ | B ( X ) ∈ P T ( X ) . W e claim that h e µ  e T  ≤ h µ ( T ). If e A = { e A 1 , . . . , e A l } is a measurable par- tition of e X , defining A i = e A i ∩ X , it follow s that A = { A 1 , . . . , A l } is a 11 measurable partition of X . W e ha v e that , B ∈ A n if and only if there exists e B ∈ e A n , with B = e B ∩ X . This follows , b ecause e T − j  e A i  ∩ X = T − j ( A i ) for any j ∈ { 0 , . . . , n } and an y i ∈ { 1 , . . . , l } , since X a nd {∞} are e T - in v aria n t sets. It follo ws that H ( e A n ) = X e B ∈ e A n φ  e µ  e B  = φ  e µ  e B ∞  + X e B 6 = e B ∞ φ  e µ  e B  , where ∞ ∈ e B ∞ . Since e µ  e B  = (1 − c ) µ ( B ), for eac h e B 6 = e B ∞ , it follo ws that H ( e A n ) = b + X B ∈A n φ ( aµ ( B )) , where a = 1 − c and b = φ  e µ  e B ∞  − φ ( aµ ( B )) . Hence H ( e A n ) = b − X B ∈A n aµ ( B ) log ( aµ ( B )) = b − a X B ∈A n µ ( B )(log ( a ) + log ( µ ( B )) ! = b − a log ( a ) X B ∈A n µ ( B ) ! + a X B ∈A n φ ( µ ( B )) ! = b + φ ( a ) + aH ( A n ) . It follo ws that H ( e A n ) ≤ d + H ( A n ), since a = 1 − c ≤ 1 and b + φ ( a ) ≤ d = 2 max { φ ( x ) : x ∈ [0 , 1] } . Dividing b y n and taking limits, we get that h e µ  e T , e A  ≤ h µ ( T , A ) ≤ h µ ( T ) . Since e A is arbitrary , w e ha v e that h e µ  e T  ≤ h µ ( T ) ≤ sup µ h µ ( T ) . 12 Since e µ ∈ P e T ( e X ) is also arbitrary , it follows tha t sup e µ h e µ  e T  ≤ sup µ h µ ( T ) , completing the pro of. No w w e pro v e the v ariational principle for entropies in a lo cally compact separable space. In Lemma 1.5 of [4], it was prov ed tha t sup µ h µ ( T ) = inf d h d ( T ) . Here w e impro v e this result sho wing that the a b ov e infimum is alwa ys at- tained at d , whenev er d is an admissible metric. Moreov er the suprem um and the minim um coincide with the top olo gical entrop y in tro duced in the Section 2. Theorem 3.2 L e t T : X → X b e a c ontinuous map, w h er e X is a lo c al ly c omp act s ep ar able sp ac e. Then it fol low s that sup µ h µ ( T ) = h ( T ) = min d h d ( T ) , wher e the minimum is attaine d whenever d is an admissible metric. Pro of: The the v ariational principle for compact spaces states that h  e T  = sup e µ h e µ  e T  . By Prop osition 1.4 of [4], w e hav e tha t sup µ h µ ( T ) ≤ inf d h d ( T ) . Applying Lemma 3.1, it follo ws that h  e T  = sup µ h µ ( T ) ≤ inf d h d ( T ) . Applying Prop ositions 2.2 and 2.3, w e get tha t h d ( T ) ≤ h d ( T ) = h ( T ) = sup µ h µ ( T ) ≤ inf d h d ( T ) , where, in the first t w o t erms, d is an y admissible metric. 13 4 T op ologi cal en tropy of automorphi sms In this section, we compute the top ological en tro py for automorphisms of simply connected nilpotent Lie groups. W e start with linear isomorphisms of a finite dimensional v ector space. F or this, we need to determine the recurren t set o f a linear isomorphism in t erms of it s m ultiplicativ e Jordan decomp osition (see Lema 7.1, pag e 430 , of [5]). If T : V → V is a linear isomorphism, where V is a finite dimensional v ector space, t hen we can write T = T H T E T U , where T H : V → V is diagonalizable in V with p o sitive eigen- v alues, T E : V → V is an isometry relative to some appropriate norm and T U : V → V is a linear isomorphism whic h can b e decomp o sed in to the sum of the iden tit y map plus some nilp o t ent linear map. The linear maps T H , T E and T U comm ute, are unique a nd called, resp ectiv ely , the hyp e rb olic , the el liptic and the unip otent comp onents of the m ultiplicativ e Jordan decomp o- sition of T . In the next result, we pro ve that the recurren t set R ( T ) o f T is giv en as the in tersection of the fixed p oin ts of the h yp erb o lic a nd unip oten t comp onen ts. Using this c haracterization, w e also sho w that the top o logical en tropy of T alw a ys v anishes. Prop osition 4.1 L et T : V → V b e a line ar isomorphism, wher e V is a finite dim e n sional ve ctor sp ac e. Then the r e curr ent set of T is given by R ( T ) = fix ( T H ) ∩ fix ( T U ) . F urthermor e, it fol lows that h ( T ) = 0 . Pro of: Let P T : P V → P V b e the map induc ed by T in the pro jectiv e space of V and, for a subspace W ⊂ V , denote b y P W its pro jection in P V . By Prop osition 2 of [3], w e hav e tha t R ( P T ) = fix ( P T H ) ∩ fix ( P T U ) . By linearit y , it is immediate that P h R ( T ) i ⊂ R ( P T ), where h A i denotes the linear subspace generated b y A ⊂ V . Thus it follo ws that R ( T ) ⊂ eig( T H ) ∩ eig( T U ) , where eig( S ) denotes the union o f the eigenspaces of a linear map S : V → V . Since T U is unip otent, all of its eigen v alues are equals to one, implying that eig( T U ) = fix( T U ). Hence R ( T ) ⊂ eig( T H ) ∩ fix( T U ). Now let v ∈ R ( T ) 14 b e such that T H v = λv , for some λ > 0 . Since the multiplic ative Jordan decomp osition is comm utat ive, it follo ws that | T n v | = | T n E T n H T n U v | = | T n H v | = λ n | v | , where w e used the fact that T E is an isometry relativ e to some appropriate norm | · | . This shows that λ is equals to one and th us that R ( T ) is con ta ined in fix ( T H ) ∩ fix( T U ). Since fix( T H ) ∩ fix ( T U ) is in v ariant b y T E , w e can cons ider the restriction of T t o this set, whic h is just the restriction of T E . Thu s we get that R ( T ) = fix( T H ) ∩ fix ( T U ), since the restriction of T E to fix( T H ) ∩ fix ( T U ) is an isometry whose orbits ha v e compact closure. No w to sho w that h ( T ) = 0, w e first note that, b y the P oincar´ e recurrence theorem, fo r ev ery µ ∈ P T ( X ), w e hav e that h µ ( T ) = h µ  T | R ( T )  , since R ( T ) is a closed set. By the v ariatio na l principle, it f o llo ws that h ( T ) = h  T | R ( T )  ≤ h d  T | R ( T )  , for ev ery metric d . Since T | R ( T ) = T E | R ( T ) is an isometry relativ e to some appropriate metric d , w e get that h d  T | R ( T )  = 0, completing the pro of. W e note that the classical form ula for the d -en tr o p y of a linear isomor- phism, where d is the euc lidian metric, is just an upp er b ound for its null top ological e ntrop y . Th us it might b e a n inte resting problem to calculate the top ological en tro py of a giv en automorphism φ of a noncompact Lie group G , since the classical fo rm ula for its d -en trop y , where d is s ome in v ariant metric, is as w ell just an upp er bound for its topolo gical en tro p y . Remem b er that the classical form ula is given by h d ( φ ) = X | λ | > 1 log | λ | , where λ runs through the eigen v alues of d 1 φ : g → g , the differen tia l of the automorphism φ at the iden tit y elemen t of G . The follow ing theorem gives a n answ er for the ab o ve problem when G is a simply connected nilp oten t Lie group. Theorem 4.2 L e t φ : G → G b e an automorphism , wher e G is a simply c onne cte d nilp otent Lie gr oup. Then it fol lows that h ( φ ) = 0 . 15 Pro of: If G is a connected and simply connected nilp ot en t Lie group, w e ha v e that the exp onential map is a diffeomorphism b et we en g and G (see Theorem 1.127, page 107, of [6 ]) . Since φ (exp( X )) = exp(d 1 φX ), we get that φ and d 1 φ are conjugated maps. By Prop osition 2.1, it follow s that h ( φ ) = h (d 1 φ ) = 0. W e note that, if the fundamen tal group of G is not t r ivial, it is p ossible the existence o f an automorphism with p ositiv e top ological entrop y , ev en when G is ab elian. In fact, the map φ ( z ) = z 2 is an automorphism o f the ab elian Lie gro up S 1 and it is w ell kno wn tha t it has top ological entrop y h ( φ ) = log(2) > 0. W e hav e that the canonical homomorphism π : R → S 1 , giv en b y π ( x ) = e ix , is a semi-conjugation b et we en φ and the automorphism e φ of the univ ersal co v ering R of S 1 , giv en by e φ ( x ) = 2 x . Although h  e φ  = 0, w e can not apply Prop osition 2.1 to conclude that h ( φ ) = 0, since the canonical homomorphism π is a prop er map if a nd only if the fundamental group of G is finite. References [1] R. Adler, A. Ko nheim and H. MacAnd rew : T op olo gi c al entr opy . T r ans. Americ. Math So c. 114 (1965), 309- 319. [2] R. Bow en : T op olo gic al en tr opy for nonc om p act sets . T rans. Americ. Math So c. 184 (1973), 125- 136. [3] T. F erraiol, M. P atr˜ ao, L. San tos and L. Seco : Multiplic ative Jor dan de c omp osition and dynamics on flag manifo l d s . preprin t (2008). [4] M. Handel and B. Kitc hens: Metrics and entr op y for non-c omp a ct sp ac es . Isr. J. Math., 91 (1995), 253-271 . [5] Helgason, S. D iffer ential Ge ometry, Lie Gr oups and Symmetric Sp ac es . Academic Press, New Y ork, (1978). [6] Knapp, A. W. : Lie G r o ups Bey o nd an Introduction, Progress in Math- ematics, v. 140, Birkh¨ auser, Boston (2004). [7] P . W alters : A n intr o duction to er go d i c the ory . G TM, Springer-V erlag, Berlin, 1981. 16