Infinite Ergodicity that Preserves the Lebesgue Measure
aa r X i v : . [ n li n . C D ] S e p Infinite Ergodicity that Preserves the Lebesgue Measure
Ken-ichi Okubo a) and Ken Umeno b) Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University,Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501 Japan (Dated: 8 September 2020)
We proved that for the countably infinite number of one-parameterized one dimensional dynamical systems,they preserve the Lebesgue measure and they are ergodic for the measure (infinite ergodicity). Consideredsystems connect the parameter region in which dynamical systems are exact and the parameter region inwhich systems are dissipative, and correspond to the critical points of the parameter in which weak chaosoccurs (the Lyapunov exponent converges to zero). These results are the generalization of the work by R.Adler and B. Weiss. We show that the distributions of normalized Lyapunov exponent for these systems obeythe Mittag-Leffler distribution of order 1 / As a typical characteristics of ergodicity, theequality of the time average and the space averageis pointed out. However, there exist phenomenain which the time average is not equivalent to thespace average in infinite ergodic systems . TheBoole transformation is known as a one dimen-sional map , which is proven that the transfor-mation preserves the Lebesgue measure (infinitemeasure) and is ergodic. In this paper it is proventhat countably infinite number of one parameter-ized one dimensional maps which are generalizedfrom the Boole transformation exactly preservethe Lebesgue measure and are ergodic at certainparameters. Additionally we show that in thesemaps the normalized Lyapunov exponent obeysthe Mittag-Leffler distribution of order / as wellas the Boole transformation. I. INTRODUCTION
Chaos theory has developed statistical physics throughergodic theory. In a chaotic dynamics, it is difficult topredict future orbital state from past information becausethe system is unstable, which is characterized by the sen-sitivity to initial conditions. However, from its mixingproperty, one can characterize the system statistically us-ing the invariant density function. The relation betweenmicroscopic dynamics and density function is importantwhen macroscopic properties are led from microscopicdynamics, and ergodicity plays a significant role in thisderivation.In the case of a dynamical system (
X, T, µ ) with a nor-malized ergodic invariant measure µ where X and T rep-resent the phase space and a map, respectively, for an ob-servable f ∈ L ( µ ), a time average lim n →∞ n P n − i =0 f ◦ a) Electronic mail: [email protected]; Presentadress:Department of Information and Physical Sciences, GraduateSchool of Information Science and Technology, Osaka University1-5 Yamadaoka, Suita, Osaka 565-0871 Japan b) Electronic mail: [email protected] T i ( x ) converges to the phase average R X f dµ in almost allregion . In systems with a normalized ergodic measure,we can characterize their stability by Lyapunov exponent λ , which is defined as λ def = lim n →∞ n P n − i =0 log | T ′ ( x i ) | when log | T ′ ( x i ) | is a L class function for the measure µ . Usually, since it is difficult to obtain the informa-tion at infinite time, we use numerical simulations or ap-ply the ergodicity to calculate the Lyapunov exponentas λ = R X log | T ′ ( x ) | dµ . For example, for logistic map x n +1 = ax n (1 − x n ), the Lyapunov exponent λ at a = 4is obtained as λ = R − x ) dxπ √ x (1 − x ) = log 2 , andfor the generalized Boole transformations x n +1 = αx n − βx n , α > , β > , λ is obtained analytically as λ = R ∞−∞ (cid:16) α + βx (cid:17) √ β (1 − α ) π { x (1 − α )+ β } dx = log (cid:16) p α (1 − α ) (cid:17) for 0 < α <
1. We know other systems whose invariantergodic measure can be expressed explicitly .On the other hand, how about the case of infi-nite ergodic case? Consider the Boole transforma-tion x n +1 = T ( x n ) = x n − /x n , which corre-sponds to α = β = 1 for the generalized Boole trans-formations where the dynamical system preserves theLebesgue measure as an infinite ergodic measure. Thatmeans it holds that R ∞−∞ f ( x ) dx = R ∞−∞ f (cid:0) x − x (cid:1) dx where f is a L function with respect to dx . Foran observable log | T ′ | , although the usual time aver-age lim n →∞ n P n − i =0 log | T ′ ( x i ) | converges to zero , thephase average is as R ∞−∞ log (cid:12)(cid:12) /x (cid:12)(cid:12) dx = 2 π , so thatthe time average does not consistent with the phase av-erage.In infinite ergodic systems, the Darling, Kac andAaronson theorem says that if the observable f is pos-itive L function in terms of invariant measure µ , thetime average using the return sequence a n converges indistribution . In the case of the Boole transformation, bydefining the return sequence a n def = √ nπ , the distributionof a n P n − i =0 log | T ′ ( x i ) | converges to the Mittag-Lefflerdistribution of order 1 / . That is, interesting phenom-ena are observed which are different from the usual er-godic theory and the standard statistical mechanics.In infinite measure system, it is known that fol-lowing L class observables converge to the Mittag-Leffler distribution such as Lempel-Ziv complexity ,the transformed observation function for the correlationfunction , normalized Lyapunov exponent , normalizeddiffusion coefficient and that non- L class observablessuch as time average of position converges to general-ized arc-sin distribution or other distribution .Infinite densities were observed in the context with thelong time limit of solution of Fokker-Planck equation forBrownian motion and semiclassical Monte Carlo sim-ulations of cold atoms .In order to characterize the instability of systemswith infinite measure, several quantities were inventedsuch as Lyapunov pair and generalized Lyapunovexponent .In relation to Lyapunov exponent, the change of sta-bility of systems characterizes their dynamical proper-ties and is important phenomenon. In particular, criti-cal phenomena at which systems become unstable fromstable called as routes to chaos has attracted a lot ofinterests in the fields of Hamiltonian dynamics , in-termittent systems , logistic map , a differentialequation , coupled chaotic oscillators , a noise-inducedsystem , experiments(Belousov-Zhabotinskii reaction,Rayleigh-B´enard convection, and Couette-Taylor flow) and optomechanics .For generalized Boole transformations, at the on-set of chaos the Lyapunov exponent defined by thetime average, converges to zero as α →
1. Thepoint α c = 1 is referred to as the critical point atwhich Type . Since the pa-rameter dependence at the critical point diverges aslim α → − (cid:12)(cid:12)(cid:12) ∂∂α (cid:16) p α (1 − α ) (cid:17)(cid:12)(cid:12)(cid:12) = ∞ , we know thatit is difficult to obtain the correct Lyapunov exponentby numerical experiments. The bahavior at α = β = 1is explained by the Boole transformation in which theLyapunov exponent derived from the time average doesnot consist with that derived from the phase average al-though the system is ergodic.The authors previously extended the generalized Booletransformations by defining countably infinite number ofone-parameterized maps, which are called super general-ized Boole (SGB) transformations and showed that theLyapunov exponent converges to zero from positive valueas α → Type α = 1 forcountably infinite number of maps (SGB). The authorsproved that the SGB transformations are exact when( K, α ) are in Range A. However, the ergodic property at α = ± K ≥
3) was left unresolved. In this pa-per, we show that all the super generalized Boole (SGB)transformations at α = ± R N is vitally im-portant and this can be regarded as the Lebesgue masurewhich is invariant under Hamiltonian dynamical systemwith N degrees of freedom . Thus, it is of great in-terest to investigate “ ergodic ” Lebesgue measure on R which is invariant under nonlinear transformations fromthe physical point of view. II. SUPER GENERALIZED BOOLETRANSFORMATIONS
In this section, let us define the super generalized Boole(SGB) transformations . At first, define a function F K : R \ A → R \ A such as F K (cot θ ) def = cot( Kθ ) (1)where K ∈ N \{ } and A represents a set of point x ∈ R such that for finite iteration n ∈ Z , F nK ( x ) reaches thesingular point. F K corresponds to the K -angle formula of cot func-tion. For example, F ( x ) = ( x − x ) corresponds to thecot(2 θ ) = (cid:0) cot θ − θ (cid:1) .Then, super generalized Boole transformations S K,α : R \ B → R \ B are defined as follows. x n +1 = S K,α ( x n ) def = αKF K ( x n ) , (2)where | α | > K ∈ N \{ } and B represents a set ofpoint x ∈ R such that for finite iteration n ∈ Z , S nK,α ( x )reaches the singular point.Let us define Range A as “0 < α < K =2 N ” or “ K < α < K = 2 N + 1” where N ∈ N .When ( K, α ) are in Range A, the super generalized Boole(SGB) transformations are exact and when α >
1, theany orbits diverge to the infinity and the SGB transfor-mations do not preserve measure over real line .In the following one can extend the Range A to thenewly defined Range B such that “0 < | α | < K =2 N ” or “ K < | α | < K = 2 N + 1” where N ∈ N .Let us define the Range A’ as (cid:26) − < α < K = 2 N, − < α < − K in the case of K = 2 N + 1 , where N ∈ N . In the following extension from α to | α | can be proven in the similar way as the reference 37. Ifthe density function at the time n ( f n ( x )) is denoted as f n ( x ) = 1 π γx + γ ,f n +1 ( x ) is given by f n +1 ( x ) = 1 π | α | KG K ( γ ) x + | α | K G K ( γ )according to the Perron-Frobenius equation where G K ( γ )corresponds to the the K-angle formula of the cothfunction . Then the scale parameter γ is changed ina single iteration as γ
7→ | α | KG K ( γ ) . Then, by changing the parameter from α to | α | , we canprove straightforwardly that the SGB transformations { S K,α } preserve the Cauchy distribution and the scaleparameter can be chosen uniquely when the parameters( K, α ) are in Range B. In terms of exactness, it holds that¯ S ′ K,α ( θ ) < K, α ) are in the Range A’. Then,¯ S K,α ( θ ) is also the monotonic function. Thus, we canprove that the SGB transformations { S K,α } are exactwhen the parameters ( K, α ) are in Range B by consid-ering the intervals { I j,n } . In the case of α < −
1, onestraightforwardly sees that orbits diverge to the infinityand the SGB transformations do not preserve measureover real line.From above discussion, we know that the SGB trans-formations are exact when the parameters (
K, α ) are inrange B and the systems are disspative when | α | > α = ± α = ±
1? Since the statisticalproperties drastically change before and after the valueof α = ±
1, the erogidic property of the critical
SGBtransformations at α = ± K = 2 , α = 1 preserves the Lebesgue measure and areergodic . In the following section, we show that all theSGB transformations at α = ± K ∈ N \{ } . Table I shows the explicitform of S K, ± for K = 2 , , , -30-20-100102030 -4 -2 0 2 4 X n + X n aaaa FIG. 1:
Return maps of S , , S , and S , . The function f ( x ) = x represents the set of fixed points. III. INFINITE ERGODICITY FOR α = 1 , − Theorem III.1.
The SGB transformations at α = ± preserve the Lebesgue measure.Proof. The goal is to prove that (cid:12)(cid:12)(cid:12) S − K, ± I (cid:12)(cid:12)(cid:12) = | I | (3) for any interval I where |·| denotes the length of a in-terval. It is sufficient to verify this for intervals of I = (0 , η ) , η > I = ( η, , η < .(I) In the case of α = 1.(In the following, we will prove Eq. (3) holds for η > η < S K, increasesmonotonically. We have that x n +1 = S K, ( x n ) (4)and for x n +1 = 0, x n = cot θ n , θ n ∈ arccot( R \ B ) ⊂ [0 , π ]satisfies the following relation: Kθ n = π mπ, m ∈ Z θ n = π K + mK π, ≤ π K + mK π ≤ π. (5)The range of possible values for m is m = 0 , , , · · · , K −
1. Then for x n such that x n +1 = 0, it follows that x n = cot (cid:16) π K + mK π (cid:17) , m = 0 , , , · · · , K − . (6)For x n such that x n +1 = K cot( Kθ n ) = η , it follows that Kθ n = cot − (cid:0) ηK (cid:1) + mπθ n = K cot − (cid:0) ηK (cid:1) + mK π, ≤ K cot − (cid:0) ηK (cid:1) + mK π ≤ π. (7)Here, since 0 < cot − (cid:16) ηK (cid:17) < π , (8)the range of possible values for m is given by − < − π cot − (cid:16) ηK (cid:17) ≤ m ≤ K − π cot − (cid:16) ηK (cid:17) < K, (9)that is m = 0 , , , · · · , K −
1. Then θ n and x n are givenby θ n = K cot − (cid:0) ηK (cid:1) + mK π, m = 0 , , , · · · , K − ,x n = cot (cid:8) K cot − (cid:0) ηK (cid:1) + mK π (cid:9) (10)where η = K cot( Kθ n ). Because the S K, increasesmonotonically and the cot function decreases monotoni-cally for θ ∈ [0 , π ], the interval that is mapped from (0 , η )by S − K, is K − [ m =0 (cid:18) cot (cid:16) π K + mK π (cid:17) , cot (cid:26) K cot − (cid:16) ηK (cid:17) + mK π (cid:27)(cid:19) . (11)Then we have that (cid:12)(cid:12)(cid:12) S − K, (0 , η ) (cid:12)(cid:12)(cid:12) = K − X m =0 (cid:20) cot (cid:26) K cot − (cid:16) ηK (cid:17) + mK π (cid:27) − cot (cid:16) π K + mK π (cid:17)(cid:21) . (12) TABLE I: S K, ± ( x ) for K = 2 , , , K = 2 3 4 5 6 S K, ± ( x ) ± (cid:18) x − x (cid:19) ± x − x x − ± x − x + 14 x − x ± x − x + 5 x x − x + 1 ± x − x + 15 x − x − x + 6 x In the following discussion, we consider K − X m =0 cot (cid:16) π K + mK π (cid:17) .(i) Case K = 2 N . For K − X m =0 cot (cid:16) π K + mK π (cid:17) , addingthe terms corresponding to m = 0 and m = K −
1, weobtain cot (cid:0) π K (cid:1) + cot (cid:0) π K + K − K π (cid:1) = cot (cid:0) π K (cid:1) + cot (cid:0) π − π K (cid:1) = 0 . (13)Adding the terms corresponding to m = l and m = K − − l , l = 0 , · · · , K −
1, we obtaincot (cid:18) (2 l + 1) π K (cid:19) + cot (cid:18) π − (2 l + 1) π K (cid:19) = 0 . (14)Thus, for K = 2 N , the following relation holds: K − X m =0 cot (cid:16) π K + mK π (cid:17) = 0 . (15)(ii)Case K = 2 N + 1. We have K − X m =0 cot (cid:16) π K + mK π (cid:17) = K − X m =0 cot (cid:16) π K + mK π (cid:17) + cot (cid:18) K − K π (cid:19) + K − X m = K +12 cot (cid:16) π K + mK π (cid:17) = K − X m =0 cot (cid:16) π K + mK π (cid:17) + K − X m = K +12 cot (cid:16) π K + mK π (cid:17) . (16)Much as in (i), because the term corresponding to m = l negates the term corresponding to m = K − − l , l =0 , · · · , K − , it follows that K − X m =0 cot (cid:16) π K + mK π (cid:17) = 0 . (17)Thus, we have that (cid:12)(cid:12)(cid:12) S − K, (0 , η ) (cid:12)(cid:12)(cid:12) = K − X m =0 cot (cid:26) K cot − (cid:16) ηK (cid:17) + mK π (cid:27) . (18)In the following discussion, we calculate Eq. (18). Letthe K roots x n of the equation η = S K, x n be denoted ξ i , i = 0 , · · · , K −
1. Because the map S K, ( x ) corre-sponds to the K -angle formula of the cot function, η isgiven by η = x n +1 = S K, ( x n )= K x Kn + ( K − Kx K − n + ( K − x n +1 = η . Then it follows that x Kn − ηx K − n +( K − . (20) Because by definition ξ i is a root of the above K th-degree equation, it follows that ( x n − ξ )( x n − ξ ) · · · ( x n − ξ K − ) = 0. According to the relation between the rootsand coefficients of a K th-degree equation, we have that η = K − X m =0 ξ m = K − X m =0 cot (cid:26) K cot − (cid:16) ηK (cid:17) + mK π (cid:27) . (21)Therefore, because (cid:12)(cid:12)(cid:12) S − K, (0 , η ) (cid:12)(cid:12)(cid:12) = η, (22)Eq. (3) holds.(II) In the case of α = − η > S K, − decreases monotonically, (cid:12)(cid:12) S − K, − (0 , η ) (cid:12)(cid:12) = K − X m =0 (cid:20) cot (cid:16) π K + mK π (cid:17) − cot (cid:26) K cot − (cid:16) − ηK (cid:17) + mK π (cid:27)(cid:21) = − K − X m =0 cot (cid:26) K cot − (cid:16) − ηK (cid:17) + mK π (cid:27) . (23) For the map S K, − , the following relation holds: η = − K x Kn + ( K − Kx K − n + ( K − x Kn + ηx K − n + ( K − . (24)According to the relation between the roots and coeffi-cients of a K th-degree equation, we have the relation: − η = K − X m =0 ξ m = K − X m =0 cot (cid:26) K cot − (cid:18) − ηK (cid:19) + mK π (cid:27) ∴ − K − X m =0 cot (cid:26) K cot − (cid:18) − ηK (cid:19) + mK π (cid:27) = η. (25)Therefore, it follows that (cid:12)(cid:12)(cid:12) S − K, − (0 , η ) (cid:12)(cid:12)(cid:12) = − K − X m =0 cot (cid:26) K cot − (cid:18) − ηK (cid:19) + mK π (cid:27) = η (26)and Eq. (3) holds.At α = ±
1, the SGB transformations preserve theLebesgue measure for any K ≥
2. Thus, for the SGBtransformations the measure for the whole set cannot benormalized to unity. Then we define the ergodicity forthe system with the infinite measure as follows (accordingto Definition III.1):
Definition III.1 (ergodicity ) . Let ( X, A , µ ) be a mea-sure space. S : X → X is a measurable transformationon the measure space ( X, A , µ ) . The transformation S is called ergodic if every invariant set A ∈ A is such thateither µ ( A ) = 0 or µ ( X \ A ) = 0 . Theorem III.2.
The SGB transformations at α = ± are ergodic.Proof. For the map S K, ± substituting cot( πθ n ) into x n ∈ R \ B , one has the induced map ¯ S K, ± : X def =1 π arccot( R \ B ) → X such that θ n +1 = ¯ S K, ± ( θ n ) = 1 π cot − {± K cot( πKθ n ) } . (27)The Figure 2 shows the relation between R \ B and X inthe range of − < x n < ,
1) to obtain the set X , X ⊂ (0 , S K, ± has topological conjugacy withthe map S K, ± , so that the ergodic properties of ¯ S K, ± are the same as those of S K, ± . In terms of absolutevalue of the derivative of ¯ S K, ± , it holds that (cid:12)(cid:12) ¯ S ′ K, ± ( θ ) (cid:12)(cid:12) = K (cid:8) ( πKθ ) (cid:9) K cot ( πKθ ) + 1 > , ∀ θ ∈ X. (28)Take the contraposition for Definition III.1 and we willshow thatfor any A s.t. µ ( A ) = 0 , and µ ( A c ) = 0 ⇒ A is not invariant . (29) θ n x n a FIG. 2:
The Relation between x n and θ n . In a way similar to the proof of the mixing propertyin generalized Boole transformations and the exactnessin super generalized Boole transformations , we definethe open intervals { I j,n } for which the following relationshold: I j,n ⊂ ( η j,n , η j +1 ,n ) , η j,n < η j +1 ,n ,n ∈ N , ≤ j ≤ K n − ,η ,n = 0 and η K n ,n = 1 , ¯ S nK, ± ( I j,n ) = X. (30)Figure 3 illustrates the case of { I j, } for K = 3 , , and 5at α = 1. θ n + θ n (a) K = 3 θ n + θ n (b) K = 4 θ n + θ n (c) K = 5 FIG. 3:
Solid lines correspond to the transformation ¯ S K, which has exact topological conjugacy with the su-per generalized Boole transformation S K, , where K = 3 , θ n +1 = θ n . Since the absolute value of the derivative ¯ S ′ K, ± on any I j,n is larger than ( ∵ { θ | cot ( πKθ ) = ∞} / ∈ X ) unity,the length of the interval I j,n becomes infinitesimal as n → ∞ . Then, for any set A such that µ ( A ) = 0, it follows that ∃ p, q s.t. I p,q ⊂ A. (31)From the definition of I p,q , it follows that¯ S qK, ± I p,q = X, ∴ ¯ S qK, ± A = X. (32)Next, for any set A such that µ ( A c ) = 0, it follows that A = X. (33)Then, for any A such that µ ( A ) = 0 and µ ( A c ) = 0, itfollows that ∃ q ∈ N s.t. ¯ S qK, ± A = X and A = X. (34)This means that the set A is not invariant. Therefore,Theorem III.2 holds. IV. NORMALIZED LYAPUNOV EXPONENT
According to the Darling-Kac-Aaronson theorem , forinfinite measure m , for a conservative, ergodic, mea-sure presrving map T and for a function f such as f ∈ L ( m ) , f ≥ , R X f dm > X is a set onwhich the map T is defined, normalized time average of f converges to the normalized Mittag-Leffler distribution such as a n n − X k =0 f ◦ T k → (cid:18)Z X f dm (cid:19) Y γ , (35)where a n is the return sequence and Y γ is a random vari-able which obeys the normalized Mittag-leffler distribu-tion of order γ . In the case of the Boole transformation,the return sequence is obtained as a n = √ nπ .In the case of this SGB transformations at α = ± f as log (cid:12)(cid:12)(cid:12) dS K, ± dx (cid:12)(cid:12)(cid:12) and we clarify whether the not-malized Lyapunov exponent converges to the normalizedMittag-Leffler distribution by numerical simulation.We have that log (cid:12)(cid:12)(cid:12) dS K, ± dx (cid:12)(cid:12)(cid:12) ≥ . In the folloing, weassume such condition as a n ∝ n , log (cid:12)(cid:12) S ′ K, ± (cid:12)(cid:12) ∈ L ( µ ) . (36)as the case ( K, α ) = (2 , .We calculate the normalized Lyapunov exponents suchas λ = c ( K ) √ n n − X i =0 log (cid:12)(cid:12)(cid:12)(cid:12) dS K, ± dx ( x i ) (cid:12)(cid:12)(cid:12)(cid:12) (37)where c are the normalization constants to make themean values equal to unity. Figures 4a, 4b, 4c 5a, 5band 5c show the density function of the normalized Lya-punov exponents for ( K, α ) = (3 , , (4 , , , − , −
1) and (5 , − distributed ac-cording to the normalized Mittag-Leffler distribution oforder . f ( λ ) λ f(x) (a) K = 3 (cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) f ( λ ) λ f(x) (b) K = 4 (cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16) f ( λ ) λ f(x) (c) K = 5 FIG. 4:
Relation between the density functions of nor-malized Lyapunov exponent and normalized Lya-punov exponent in SGB transformation for K =3 , α = 1). The number of initial points is M = 10 and the number of iteration is N = 10 .Initial points are distributed to obey the normal dis-tribution whose mean and variance are 0 and 1, re-spectively. The bar graph represents the numericalsimulation of the normalized Lyapunov exponentsand the solid line represents the normalized Mittag-Leffler distributions of order . (cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22) f ( λ ) λ f(x) (a) K = 3 (cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28) f ( λ ) λ f(x) (b) K = 4 (cid:29)(cid:30)(cid:31) !" f ( λ ) λ f(x) (c) K = 5 FIG. 5:
Relation between the density functions of nor-malized Lyapunov exponent and normalized Lya-punov exponent in SGB transformation for K =3 , α = − M = 10 and the number of iteration is N = 10 .Initial points are distributed to obey the normal dis-tribution whose mean and variance are 0 and 1, re-spectively. The bar graph represents the numericalsimulation of the normalized Lyapunov exponentsand the solid line represents the normalized Mittag-Leffler distributions of order . Figure 6 shows the relation between normalization con-stant c ( K ) and K at α = ±
1. We can see that c ( K ) tendsto decrease as K increases. At ( K, α ) = (2 , c ( K ) = √ ≃ .
354 from a n = √ nπ . Figure 6is consistent with this result and from the fact that thepoints at ( K, α ) = (2 , − , (3 ,
1) and (3 , −
1) are on g ( K )and that R ln | S ′ , − ( x ) | dx = R ln | S ′ , ± ( x ) | dx = 2 π , weconjecture that for S , − , the return sequence a n is givenby a n = √ nπ and that for S , ± , a n = √ nπ . no r m a li z ed c on s t an t c K abf(x)
FIG. 6:
The relation between normalization constant c ( K )and parameter K . The function g ( K ) is rewrittenas g ( K ) = √ K . V. CONCLUSION
In this paper, we showed the statistical ergodic prop-erty of one dimensional chaotic maps, the super general-ized Boole (SGB) transformations S K,α at α = ±
1. Thatis, for infinite number of K , we proved that the S K, ± preserve the Lebesgue measure and that the dynamicalsystems are ergodic for K ≥
2. In the case of K = 2(the Boole transformation), Adler and Wiss proved itsergodicity in unbounded region but in our method, weproved the ergodicity by transforming the unbounded do-main to the bounded domain using topological conjugacy.In the previous work , the authors proved that the SGBtransformations are exact for 0 < α < K = 2 N ) or K < α < K = 2 N + 1), N ∈ N and they are dissi-pative for α >
1. The result of this paper connects thesetwo regions in the same way of the generalized Booletransformations . Then, we demonstrated that the nor-malized Lyapunov exponents actually obey the Mittag-Leffler distribution of order for ( K, α ) = (3 , , (4 , , , − , −
1) and (5 , − K although there is a rela-tion between c and K . Owing to these results, we obtaina class of countably infinite number of critical maps inthe sense of Type
Type and Lyapunov pair . It is fully expected that the these infinite critical SGB transformations willbe used as represented indicator maps in order to detectchaotic criticality since the ergodic properties are exactlyobtained. ACKNOWLEDGMENTS
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