Infinite Ergodicity that Preserves the Lebesgue Measure

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). Considered systems connect the parameter region in which dynamical systems are exact and the parameter region in which systems are dissipative, and correspond to the critical points of the parameter in which weak chaos occurs (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 obey the Mittag-Leffler distribution of order $1/2$ by numerical simulation.


I. INTRODUCTION
Chaos theory has developed statistical physics through ergodic theory. In a chaotic dynamics, it is difficult to predict future orbital state from past information because the system is unstable, which is characterized by the sensitivity to initial conditions. However, from its mixing property, one can characterize the system statistically using the invariant density function. The relation between microscopic dynamics and density function is important when macroscopic properties are led from microscopic dynamics, and ergodicity plays a significant role in this derivation.
In the case of a dynamical system (X, T, µ) with a normalized ergodic invariant measure µ where X and T represent the phase space and a map, respectively, for an observable f ∈ L 1 (µ), a time average lim n→∞ T i (x) converges to the phase average X f dµ in almost all region 3 . In systems with a normalized ergodic measure, we can characterize their stability by Lyapunov exponent λ, which is defined as λ def = lim n→∞ 1 n n−1 i=0 log |T ′ (x i )| when log |T ′ (x i )| is a L 1 class function for the measure µ. Usually, since it is difficult to obtain the information at infinite time, we use numerical simulations or apply the ergodicity to calculate the Lyapunov exponent as λ = X log |T ′ (x)| dµ. For example, for logistic map x n+1 = ax n (1 − x n ), the Lyapunov exponent λ at a = 4 is obtained as λ = π{x 2 (1−α)+β} dx = log 1 + 2 α(1 − α) for 0 < α < 1. We know other systems whose invariant ergodic measure can be expressed explicitly 7,8 .
On the other hand, how about the case of infinite ergodic case? Consider the Boole transformation x n+1 = T (x n ) = x n − 1/x n 2,9 , which corresponds to α = β = 1 for the generalized Boole transformations where the dynamical system preserves the Lebesgue measure as an infinite ergodic measure. That means it holds that x dx where f is a L 1 function with respect to dx. For an observable log |T ′ |, although the usual time average lim n→∞ 1 n n−1 i=0 log |T ′ (x i )| converges to zero 1 , the phase average is as ∞ −∞ log 1 + 1/x 2 dx = 2π 1 , so that the time average does not consistent with the phase average.
In infinite ergodic systems, the Darling, Kac and Aaronson theorem says that if the observable f is positive L 1 function in terms of invariant measure µ, the time average using the return sequence a n converges in distribution 5 . In the case of the Boole transformation, by defining the return sequence a n def = √ 2n π , the distribution of 1 an n−1 i=0 log |T ′ (x i )| converges to the Mittag-Leffler distribution of order 1/2 5 . That is, interesting phenomena are observed which are different from the usual ergodic theory and the standard statistical mechanics. In infinite measure system, it is known that fol-lowing L 1 class observables converge to the Mittag-Leffler distribution such as Lempel-Ziv complexity 10 , the transformed observation function for the correlation function 11 , normalized Lyapunov exponent 1 , normalized diffusion coefficient 12 and that non-L 1 class observables such as time average of position 13 converges to generalized arc-sin distribution [14][15][16] or other distribution 17 . Infinite densities were observed in the context with the long time limit of solution of Fokker-Planck equation for Brownian motion 18,19 and semiclassical Monte Carlo simulations of cold atoms 20 .
In order to characterize the instability of systems with infinite measure, several quantities were invented such as Lyapunov pair 1 and generalized Lyapunov exponent [21][22][23] .
In relation to Lyapunov exponent, the change of stability of systems characterizes their dynamical properties and is important phenomenon. In particular, critical phenomena at which systems become unstable from stable called as routes to chaos has attracted a lot of interests in the fields of Hamiltonian dynamics 24 , intermittent systems [25][26][27][28] , logistic map 29 , a differential equation 30 , coupled chaotic oscillators 31 , a noise-induced system 32 , experiments(Belousov-Zhabotinskii reaction, Rayleigh-Bénard convection, and Couette-Taylor flow) 33 and optomechanics [34][35][36] .
For generalized Boole transformations, at the onset of chaos the Lyapunov exponent defined by the time average, converges to zero as α → 1. The point α c = 1 is referred to as the critical point at which Type 1 intermittency occurs 6 . Since the parameter dependence at the critical point diverges as lim α→1−0 ∂ ∂α 1 + 2 α(1 − α) = ∞, we know that it is difficult to obtain the correct Lyapunov exponent by numerical experiments. The bahavior at α = β = 1 is explained by the Boole transformation in which the Lyapunov exponent derived from the time average does not consist with that derived from the phase average although the system is ergodic.
The authors previously extended the generalized Boole transformations by defining countably infinite number of one-parameterized maps, which are called super generalized Boole (SGB) transformations 37 and showed that the Lyapunov exponent converges to zero from positive value as α → 1 and Type 1 intermittency occurs at α = 1 for countably infinite number of maps (SGB). The authors proved that the SGB transformations are exact when (K, α) are in Range A. However, the ergodic property at α = ±1 of SGB (K ≥ 3) was left unresolved. In this paper, we show that all the super generalized Boole (SGB) transformations at α = ±1 also preserve the Lebesgue measure and is proven to be ergodic as same as the Boole transformation. If we look at the foundation of statistical mechanics, the Liouville measure on R 2N is vitally important and this can be regarded as the Lebesgue masure which is invariant under Hamiltonian dynamical system with N degrees of freedom 38,39 . Thus, it is of great interest to investigate "ergodic" Lebesgue measure on R which is invariant under nonlinear transformations from the physical point of view.

II. SUPER GENERALIZED BOOLE TRANSFORMATIONS
In this section, let us define the super generalized Boole (SGB) transformations 37 . At first, define a function F K : R\A → R\A such as where K ∈ N\{1} and A represents a set of point x ∈ R such that for finite iteration n ∈ Z, F n K (x) reaches the singular point.
F K corresponds to the K-angle formula of cot function. For example, Then, super generalized Boole transformations S K,α : R\B → R\B are defined as follows.
where |α| > 0, K ∈ N\{1} and B represents a set of point x ∈ R such that for finite iteration n ∈ Z, S n K,α (x) reaches the singular point.
Let us define Range A as "0 < α < 1 and K = 2N " or " 1 K 2 < α < 1 and K = 2N + 1" where N ∈ N. When (K, α) are in Range A, the super generalized Boole (SGB) transformations are exact and when α > 1, the any orbits diverge to the infinity and the SGB transformations do not preserve measure over real line 37 .
In the following one can extend the Range A to the newly defined Range B such that "0 < |α| < 1 and K = 2N " or " 1 K 2 < |α| < 1 and K = 2N + 1" where N ∈ N. Let us define the Range A' as where N ∈ N. In the following extension from α to |α| can be proven in the similar way as the reference 37. If the density function at the time n (f n (x)) is denoted as according to the Perron-Frobenius equation where G K (γ) corresponds to the the K-angle formula of the coth function 37 . Then the scale parameter γ is changed in a single iteration as Then, by changing the parameter from α to |α|, we can prove straightforwardly that the SGB transformations {S K,α } preserve the Cauchy distribution and the scale parameter can be chosen uniquely when the parameters (K, α) are in Range B. In terms of exactness, it holds that S ′ K,α (θ) < 0 when (K, α) are in the Range A'. Then, S K,α (θ) is also the monotonic function. Thus, we can prove that the SGB transformations {S K,α } are exact when the parameters (K, α) are in Range B by considering the intervals {I j,n }. In the case of α < −1, one straightforwardly sees that orbits diverge to the infinity and the SGB transformations do not preserve measure over real line.
From above discussion, we know that the SGB transformations are exact when the parameters (K, α) are in range B and the systems are disspative when |α| > 1. However, the ergodic property at the critical point α = ±1 was unsettled.
Then, what happens at α = ±1? Since the statistical properties drastically change before and after the value of α = ±1, the erogidic property of the critical SGB transformations at α = ±1 is important. As we know, the Boole transformation which corresponds to the case of K = 2, α = 1 preserves the Lebesgue measure and are ergodic 2 . In the following section, we show that all the SGB transformations at α = ±1 preserve the Lebesgue measure for any K ∈ N\{1}. Table I shows the explicit form of S K,±1 for K = 2, 3, 4, 5 and 6. Proof. The goal is to prove that for any interval I where |·| denotes the length of a interval. It is sufficient to verify this for intervals of I = (0, η), η > 0 and I = (η, 0), η < 0 2 . (I) In the case of α = 1.
Thus, for K = 2N , the following relation holds: (ii)Case K = 2N + 1. We have Much as in (i), because the term corresponding to m = l negates the term corresponding to m = K − 1 − l, l = 0, · · · , K−3 2 , it follows that Thus, we have that In the following discussion, we calculate Eq. (18). Let the K roots x n of the equation η = S K,1 x n be denoted ξ i , i = 0, · · · , K − 1. Because the map S K,1 (x) corresponds to the K-angle formula of the cot function, η is given by Because by definition ξ i is a root of the above Kthdegree equation, it follows that (x n −ξ 0 )(x n −ξ 1 ) · · · (x n − ξ K−1 ) = 0. According to the relation between the roots and coefficients of a Kth-degree equation, we have that Therefore, because Eq. (3) holds.
Consider the case η > 0 as in (I). Because the map S K,−1 decreases monotonically, For the map S K,−1 , the following relation holds:

th and the smaller order terms)
Kx K−1 n +(K − 3 th and the smaller order terms) x K n + ηx K−1 n + (K − 2 th and the smaller order terms) = 0. (24) According to the relation between the roots and coefficients of a Kth-degree equation, we have the relation: (25) Therefore, it follows that At α = ±1, the SGB transformations preserve the Lebesgue measure for any K ≥ 2. Thus, for the SGB transformations the measure for the whole set cannot be normalized to unity. Then we define the ergodicity for the system with the infinite measure as follows (according to Definition III.1): Definition III.1 (ergodicity 40 ). Let (X, A , µ) be a measure space. S : X → X is a measurable transformation on the measure space (X, A , µ). The transformation S is called ergodic if every invariant set A ∈ A is such that either µ(A) = 0 or µ(X\A) = 0.
Proof. For the map S K,±1 substituting cot(πθ n ) into x n ∈ R\B, one has the induced mapS K,±1 : X def = 1 π arccot(R\B) → X such that The Figure 2 shows the relation between R\B and X in the range of −10 < x n < 10.
Since we eliminate countably infinite number of points whose measure is 0 from (0, 1) to obtain the set X, X ⊂ (0, 1). The mapS K,±1 has topological conjugacy with the map S K,±1 , so that the ergodic properties ofS K,±1 are the same as those of S K,±1 . In terms of absolute value of the derivative ofS K,±1 , it holds that I j,n ⊂ (η j,n , η j+1,n ) , η j,n < η j+1,n , n ∈ N, 0 ≤ j ≤ K n − 1, η 0,n = 0 and η K n ,n = 1, S n K,±1 (I j,n ) = X. Figure 3 illustrates the case of {I j,1 } for K = 3, 4, and 5 at α = 1. Since the absolute value of the derivativeS ′ K,±1 on any I j,n is larger than ( ∵ {θ| cot 2 (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.
From the definition of I p,q , it follows that Next, for any set A such that µ(A c ) = 0, it follows that Then, for any A such that µ(A) = 0 and µ(A c ) = 0, it follows that 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 5 , for infinite measure m, for a conservative, ergodic, measure presrving map T and for a function f such as f ∈ L 1 (m), f ≥ 0, X f dm > 0 where X is a set on which the map T is defined, normalized time average of f converges to the normalized Mittag-Leffler distribution such as 1,12,41 1 a n n−1 where a n is the return sequence and Y γ is a random variable which obeys the normalized Mittag-leffler distribution of order γ. In the case of the Boole transformation, the return sequence is obtained as a n = √ 2n π 5 . In the case of this SGB transformations at α = ±1, consider f as log dSK,±1 dx and we clarify whether the notmalized Lyapunov exponent converges to the normalized Mittag-Leffler distribution by numerical simulation.

V. CONCLUSION
In this paper, we showed the statistical ergodic property of one dimensional chaotic maps, the super generalized Boole (SGB) transformations S K,α at α = ±1. That is, for infinite number of K, we proved that the S K,±1 preserve the Lebesgue measure and that the dynamical systems are ergodic for K ≥ 2. In the case of K = 2 (the Boole transformation), Adler and Wiss proved its ergodicity in unbounded region 2 but in our method, we proved the ergodicity by transforming the unbounded domain to the bounded domain using topological conjugacy. In the previous work 37 , the authors proved that the SGB transformations are exact for 0 < α < 1(K = 2N ) or 1 K 2 < α < 1 (K = 2N + 1), N ∈ N and they are dissipative for α > 1. The result of this paper connects these two regions in the same way of the generalized Boole transformations 6 . Then, we demonstrated that the normalized Lyapunov exponents actually obey the Mittag-Leffler distribution of order 1 2 for (K, α) = (3, 1), (4, 1), (5, 1), (3, −1), (4, −1) and (5, −1). In this numerical experiments, the form of Mittag-Leffler distribution does not depend on the value of K although there is a relation between c and K. Owing to these results, we obtain a class of countably infinite number of critical maps in the sense of Type 1 or Type 3 intermittency which preserve the Lebesgue measure and are proven to be ergodic with respect to the Lebesgue measure.
In previous works various indicators were proposed to characterize the instablity when the corresponding Lyapunov exponent is zero such as generalized Lyapunov exponent 21,22 and Lyapunov pair 1 . It is fully expected that the these infinite critical SGB transformations will be used as represented indicator maps in order to detect chaotic criticality since the ergodic properties are exactly obtained.