Distributional behavior of time averages of non-L^{1} observables in one-dimensional intermittent maps with infinite invariant measures

# Distributional behavior of time averages of non-L1 observables in one-dimensional intermittent maps with infinite invariant measures

Takuma Akimoto Department of Mechanical Engineering, Keio University, Yokohama, 223-8522, Japan    Soya Shinkai Research Center for the Mathematics on Chromatin Live Dynamics (RcMcD), Hiroshima University, 739-8530, Japan    Yoji Aizawa Department of Applied Physics, Advanced School of Science and Engineering, Waseda University, Tokyo 169-8555, Japan
July 29, 2019
###### Abstract

In infinite ergodic theory, two distributional limit theorems are well-known. One is characterized by the Mittag-Leffler distribution for time averages of functions, i.e., integrable functions with respect to an infinite invariant measure. The other is characterized by the generalized arc-sine distribution for time averages of non- functions. Here, we provide another distributional behavior of time averages of non- functions in one-dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure. Observation functions considered here are non- functions which vanish at the indifferent fixed point. We call this class of observation functions weak non- function. Our main result represents a first step toward a third distributional limit theorem, i.e., a distributional limit theorem for this class of observables, in infinite ergodic theory. To prove our proposition, we propose a stochastic process induced by a renewal process to mimic a Birkoff sum of a weak non- function in the one-dimensional intermittent maps.

## I Introduction

In statistical physics, many observables result from time averages of the microscopic observation functions. Ergodic theory plays an important role in providing the asymptotic behavior of time-averaged observables in dynamical systems. Trajectories in chaotic dynamical systems cannot be predicted due to the sensitivity dependence of initial conditions. However, with the aid of the unpredictability, trajectories can be regarded as a stochastic process. Then, one can introduce a measure in dynamical systems. In fact, an invariant measure characterizes chaotic orbits. Birkhoff’s ergodic theorem tells us that time averages of an observation function converge to a constant for almost all initial conditions if the observation function is integrable with respect to an absolutely continuous invariant measure Birkhoff1931 (). On the other hand, when an invariant measure cannot be normalized (infinite measure), the asymptotic behavior of time-averaged observables is completely different from that stated by the Birkhoff’s ergodic theorem. In infinite measure systems (infinite ergodic theory), one of the most striking points is that a time-averaged observable does not converge to a constant but converges in distribution Aaronson1981 (); Aaronson1997 (); Thaler1998 (); Thaler2002 (); Akimoto2008 ().

In infinite ergodic theory, two different distributional limit theorems for time averages have been known. Distribution of time averages of an function, which is an integrable function with respect to an invariant measure , follows the Mittag-Leffler distribution Aaronson1981 (); Aaronson1997 (). This distributional limit theorem is based on Darling-Kac theorem in stochastic processes Darling1957 (). The other distributional limit theorem states that time averages of a non- function converges in distribution to the generalized arc-sin distribution Aizawa1989 (); Thaler1998 (); Thaler2002 (); TZ2006 (); Akimoto2008 (), which is based on Dynkin-Lamperti’s generalized arc-sine law Lamperti1958 (); Dynkin1961 (). In infinite ergodic theory, it is important to determine the distribution of time averages for arbitrary observation functions as well as arbitrary ensembles of initial points. Recently, one of the authors has shown that the distribution of time averages of functions depends also on the ratio of a measurement time and the time at which system started, i.e., aging distributional behavior Akimoto2013 (). Here, we provide another distributional behavior that is in-between the above two distributional limit theorems.

Infinite ergodic theory has attracted the interest from not only mathematics but also physics community Gaspard1988 (); Barkai2003 (); Akimoto2007 (); Akimoto2008 (); Korabel2009 (); Akimoto2010a (); Akimoto2010 (); Akimoto2012 (). This is because distributional behaviors of time-averaged observables are ubiquitous in phenomena ranging from fluorescence in nano material Brokmann2003 () to biological transports Golding2006 (); Weigel2011 (); Tabei2013 (). Theoretical studies on distributional behaviors of time-averaged observables have been extensively conducted using stochastic models with divergent mean trapping-time distributions such as continuous-time random walks He2008 (); Miyaguchi2013 (), random walk with static disorder Miyaguchi2011 (), and dichotomous processes Margolin2006 (). The distribution function of time-averaged observables depends on the type of observation function. In particular, the distribution of time-averaged mean square displacement follows the Mittag-Leffler distribution He2008 (); Miyaguchi2013 (), while that of the ratio of occupation time of on state in dichotomous processes follows the generalized arc-sine distribution Margolin2006 (). Although distributional limit theorems in stochastic processes have been elucidated, it will be possible to construct another distributional limit theorem of time-averaged observables by introducing another type of the observation function in stochastic models with divergent mean trapping-time distributions. In fact, one of the authors has shown a novel distributional behavior for time-averaged mean square displacements in stored-energy-driven Lévy flight Akimoto2013a (); Akimoto2014 ().

In this paper, we provide a novel distribution for time averages of a class of non- functions in one-dimensional maps with indifferent fixed points having infinite invariant measures. The value of the observation function at the indifferent fixed point is zero. Because the observation function is non-, the generalized arc-sine distribution can be applied to those observation functions. However, it only gives a trivial result that time averages converge to zero. Our distributional limit theorem gives a non-trivial broad distribution of normalized time averages. In other words, we refine the distribution of normalized time averages of such observation functions by introducing a normalizing sequence. The proof is based on a stochastic process induced by a renewal process proposed here, which mimics a Birkhoff sum of a non- function.

## Ii From dynamical system to stochastic process: partial sums of non-L1(m) functions

A dynamical system considered here is a transformation which satisfies the following conditions for some : (i) the restrictions and are and onto, and have -extensions to the respective closed intervals; (ii) on ; ; (iii) is regularly varying at zero with index , (). For example, a transformation,

 Tα(z)=z⎛⎝1+(z1+z)1−αα−z1−αα⎞⎠−α1−α (mod 1), (1)

satisfies the conditions (). It is known that an invariant measure of the map is given by Thaler1983 (). Thus, the invariant measure cannot be normalized for . While this dynamical system has zero-Lyapunov exponent, the dynamical instability can be characterized as a sub-exponential instability Gaspard1988 (); Korabel2009 (); Akimoto2010a ().

For , the following ordinary differential equation can be used to describe the dynamics Manneville1980 (); Geisel1984 ():

 d~ztdt=a~z1+1/αt, (2)

where we use as the solution of the ordinary differential equation (2) with an initial condition . The solution is given by

 ~zt=~z0(1−tτ)−α, (3)

where

 τ=α/(~z1/α0a) (4)

is a characteristic time scale that a trajectory with initial point escapes from . In fact, a time when becomes unity denoted by , i.e., , is given by . In what follows, we use a sequence defined by with () and (see Fig. 1). Trajectory is reinjected to from . Because this dynamical system has a sub-exponential dynamical instability, the reinjection points, , can be regarded as a random variable, and it is known that the reinjection points are almost uniformly distributed on . Because the distribution of is determined by that of , by assuming the probability density function (PDF) of is uniform on , we have the PDF of residence times on :

 w(τ)∼Ac|Γ(−α)|τ−1−αas τ→∞ (z0→0), (5)

where depends on not only and but also details of the map . We note that the mean residence time diverges when an invariant measure cannot be normalized (). Here, we give a rigorous result that a normalized Birkoff’s sum can be represented by the trajectory generated by Eq. (2).

###### Lemma 1.

For , there exists such that where .

###### Proof.

By Eqs. (3) and (4), we have

 ~z1−z1 = cN⎛⎝1−c1/αNaα⎞⎠−α−cN−1>cN+ac1+1/αN−cN−1≃0, (6)

and

 z2−~z1 ≃ cN−1+ac1+1/αN−1−cN⎛⎝1−c1/αNaα⎞⎠−α (7) = cN−1−cN+a(c1+1/αN−1−c1+1/αN)+o(c1+1/αN). (8)

Because the sequence is given by for Thaler1983 (), we have

 cN−1−cN∼(αaN)α[(1−1N)−α−1]∼a(αaN)α+1=O(N−1−α), (9)

and

 c1+1/αN∼(αaN)α+1. (10)

It follows that , and . We assume and for . Then, we have

 ~zt+1−zt+1 ≃ ~zt⎛⎝1−~z1/αtaα⎞⎠−α−zt−az1+1/αt (11) > ~zt+a~z1+1/αt−zt−az1+1/αt, (12)

and

 zt+2−~zt+1 ≃ zt+1+az1+1/αt+1−~zt⎛⎝1−~z1/αtaα⎞⎠−α (13) = zt+1−~zt+a(z1+1/αt+1−~z1+1/αt)+o(N−1−α). (14)

Because we assume and , we have , and . It follows by mathematical induction that there exits such that for . ∎

Here, we consider the following bounded continuous observation function, (), which is not an function for ; we call this type of functions as weak non- functions. In particular, we study statistical properties of partial sums of this type of observables,

 St=t−1∑k=0f(zk) (15)

to elucidate the ergodic properties (Note that is the time average).

###### Lemma 2.

For and , there exists such that

 ∣∣ ∣∣l∑k=0f(zk)−∫l0f(~zt)dt∣∣ ∣∣<ΔI, (16)

where .

###### Proof.

First, we define and as

 Imax=N∑k=N−lf(ck) and Imin=N∑k=N−lf(ck+1), (17)

and

 ΔI ≡ Imax−Imin=f(cN−l)−f(cN+1). (18)

By Lemma 1,

 Imin

It follows

 ∣∣ ∣∣l∑k=0f(zk)−∫l0f(~zt)dt∣∣ ∣∣<ΔI. (20)

Here, we decompose the function into an part and a non- part, where an part, , is defined by on and on , and a non- part, , is defined by on and on . By the Aaronson’s distributional limit theorem, converges in distribution for all because is an function for all . It follows that for a sequence such that as , the normalized time averages, , converge to zero: as .

For the dynamical systems defined above, a trajectory is trapped in the interval for a long time and then escapes to the other interval for small . Let us consider -th such trapping state. We note that the -th trapping time denoted by is approximately given by , where is the -th reinjection point. We will show that a partial sum during the -th trap in , , can be replaced as with , where

 ∫τe0f(~zt)dt∼Bτγk[1−(1−τeτk)γ], (21)

for , , is the elapsed time since the beginning of the trapping, and is a constant given by . Because we assume that is uniformly distributed on or equivalently assume Eq. (5), the PDF of is given by

 l(x)∼AδBαγγ|Γ(−α)|x−1−αγ(x→∞). (22)

We note that the constant depend on .

###### Lemma 3.

For , the asymptotic behavior of the normalized time average, , is given by

 Stat∼1atNt∑k=1I(τk)+∫τe0f(~zt)dtat, (23)

where , is the number of reinjections to until time and is the -th trapping time on and .

###### Proof.

A partial sum is given by

 St=t−1∑k=0fLδ(zk)+t−1∑k=0fRδ(zk), (24)

where the second term contributes to a Mittag-Leffler distribution but it can be ignored when we consider a normalized time averages of weak non- functions, because the order of the normalizing sequence is greater than that of the return sequence. In fact, the normalizing sequences for and are given by and , respectively (). Therefore, it is sufficient to consider the first term only. By Lemma 2, for and (), there exists a constant such that

 ∣∣ ∣∣τk∑i=0fLδ(zi)−I(τk)∣∣ ∣∣<ε and ∣∣ ∣∣τe∑i=0fLδ(zi)−∫τe0f(~zt)dt∣∣ ∣∣<ε, (25)

where the constant does not depend on but depend on . For , we have

 1at∣∣ ∣∣t−1∑k=0fLδ(zk)−Nt∑k=1I(τk)−∫τe0f(~zt)dt∣∣ ∣∣<ε(Nt+1)at. (26)

Because Akimoto2010 (), the left-hand-side goes to zero as . ∎

In the following sections, we will show that there exists a sequence such that the normalized time average, , converges in distribution:

 1att−1∑k=0f(zk)⇒Yα,γasn→∞, (27)

where the Laplace transform of the random variable is given by Eq. (45). We note that the sequence is given by , which is not the so-called return sequence in infinite ergodic theory Aaronson1997 (). In particular, the order of the return sequence is given by , which is smaller than that of , i.e., as .

## Iii Continuous accumulation process

To analyze the partial sum [Eq. (15)], we generalize a renewal process. Renewal process is a point process where the time intervals between point events are independent and identically distributed (i.i.d) random variables Cox (). Because residence times near the indifferent fixed point in intermittent maps are considered to be almost i.i.d. random variables, one can apply renewal processes to study dynamical systems Akimoto2010b ().

Here, we consider a cumulative process by introducing an intensity of each renewal event Cox (), where intensity is correlated with the time interval between successive renewals. This process can be characterized by the total intensity until time , whereas renewal processes are characterized by the number of renewals in the time interval , denoted by . Let be the time intervals between successive renewals, which are i.i.d. random variables with PDF . We assume that the -th intensity is determined by the -th interevent time (IET) as , where and . Thus, the longer the IET between renewals becomes, the larger the intensity is. Furthermore, we propose a continuous accumulation process induced by a renewal process to consider a Birkhoff sum. In the continuous accumulation process, the intensity is gradually accumulated according to a function in between the two successive renewals, where is the elapsed time after the -th renewal (see Fig. 2). Here, we use the following intensity function:

 I(τe,τk)=I(τk)[1−(1−τeτk)γ]. (28)

This intensity function mimics an increase of the Birkhoff sum in dynamical systems [see Eq. (21)]. As shown in the previous section, the following stochastic variable (the integrated intensity up to time ),

 Xt=Nt∑k=1I(τk)+I(t−tNt,τNt+1), (29)

where , is related to the Birkhoff sum of the non-integrable function. The case in which and is exactly equivalent to the usual renewal process, because .

Here, we consider the case that the mean interevent times of renewals diverges . In particular, we use Eq. (5) as the PDF of IETs. Thus, the survival probability is given by

 W(τ)≡1−∫τ0w(τ′)dτ′∼AΓ(1−α)τ−α(τ→∞), (30)

and the PDF of , denoted by , is given by Eq. (22). Because the mean intensity diverges for , the renewal theory cannot be straightforwardly applied.

## Iv Theory of a continuous accumulation process

### iv.1 Generalized renewal equation

Distribution of can be derived by a generalized renewal equation, which is similar to a generalized master equation for the continuous-time random walk (CTRW) Shlesinger1982 (). First, we define a joint PDF of the IET and the intensity increment as , and we also use , where for and 1 otherwise. Let be the PDF of at time when a renewal occurs, then we have

 Q(x,t) = ∫∞0dx′ψ(x′,t)δ(x−x′) (31) +∫x0dx′∫t0dt′ψ(x′,t′)Q(x−x′,t−t′).

The conditional PDF of at time on the condition of , denoted by is given by

 P(x,t;τ) = ∫x0dx′∫t0dt′Ψ(x′,t′;τ)Q(x−x′,t−t′)+Ψ(x,t;τ). (32)

It follows that the PDF of at time reads

 P(x,t)=∫∞0w(τ)P(x,t;τ)dτ. (33)

Here, we assume that a renewal occurs at time , i.e., ordinary renewal process Cox (). Using the double Laplace transform with respect to time () and (, defined by

 ^P(k,s)≡∫∞0dt∫∞0dxe−st−kxP(x,t), (34)

we have

 ^P(k,s)=∫∞0w(τ)^Ψ(k,s;τ)1−^ψ(k,s)dτ, (35)

where

 ^ψ(k,s)≡∫∞0dτ∫∞0dxe−sτ−kxψ(x,τ)=∫∞0e−sτe−kI(τ)w(τ)dτ, (36)

and

 ^Ψ(k,s;τ)≡∫∞0dt∫∞0dxe−st−kxΨ(x,t;τ)=∫τ0e−st−kI(t,τ)dt. (37)

In what follows, we use the asymptotic behaviors of the Laplace transforms of and , i.e., and for .

### iv.2 Moments of Xt

#### iv.2.1 First moment

The Laplace transform of , denoted by , is given by . As shown in the Appendix A, the leading order of is given by

 ⟨Xs⟩∼⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩BM1(α,γ)|Γ(−α)|1s1+γ,(γ>α)B|Γ(−α)|1s1+αlog(1s),(γ=α)⟨I⟩A1s1+α,(γ<α) (38)

where . The inverse Laplace transform reads

 ⟨Xt⟩∼⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩BM1(α,γ)|Γ(−α)|Γ(1+γ)tγ,(γ>α)B|Γ(−α)|Γ(1+α)tαlogt,(γ=α)⟨I⟩AΓ(1+α)tα.(γ<α) (39)

We note that the asymptotic behavior of is determined by and . In other words, it does not depend on .

#### iv.2.2 Second moment

The Laplace transform for the second moment of , denoted by is given by . As shown in the Appendix B, the leading order of is given by

 ⟨X2s⟩ ∼ ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩B2M2(α,γ)|Γ(−α)|1s1+2γ,(γ>α)2B2|Γ(−α)|n1s1+2αlog2(1s),(γ=α)2⟨I⟩2A21s1+2α,(γ<α) (40)

where . The inverse Laplace transform reads

 ⟨X2t⟩ ∼ ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩B2M2(α,γ)|Γ(−α)|Γ(1+2γ)t2γ,(γ>α)2B2|Γ(−α)|nt2α(logt)2,(γ=α)2⟨I⟩2A2t2α.(γ<α) (41)

#### iv.2.3 nth moment

As shown in the Appendix C, the leading order of the Laplace transform of with for is given by

 ⟨Xns⟩ = ^P(n)(0,s) (42) ∼ ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩BnMn(α,γ)|Γ(−α)|1s1+nγ(γ>α)(−1)nn![1−^ψ(s)]s[1−^ψ(0,s)]n+1{^ψ′(0,s)}n(γ≤α) ∼ ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩BnMn(α,γ)|Γ(−α)|1s1+nγ,(γ>α)n!Bn|Γ(−α)|n1s1+nα{log(1s)}n,(γ=α)n!⟨I⟩nAn1s1+nα,(γ<α). (43)

where is given by (S14). The inverse Laplace transform reads

 ⟨Xnt⟩∼⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩BnMn(α,γ)|Γ(−α)|Γ(1+nγ)tnγ(γ>α)n!Bn|Γ(−α)|nΓ(1+nα)tnα(logt)n(γ=α)n!⟨I⟩nAnΓ(1+nα)tnα(γ<α). (44)

It follows that converges in distribution to , where

 ⟨ezYα,γ⟩=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∞∑k=0Mk(α,γ)zkk!M1(α,γ)kΓ(1+kγ)(γ>α)∞∑k=0Γ(1+α)kzkΓ(1+kα)(γ≤α), (45)

and . We note that the distribution of the normalized random variable does not depend on for (in this case the observation function is in the dynamical system) and the distribution is called the Mittag-Leffler distribution of order , whereas the distribution of a scaled sum for converges to a time-independent non-trivial distribution, which is not the Mittag-Leffler distribution. Figure 3 shows the PDFs of for different and .

## V Distribution of time averages of weak non-L1(m) functions

In the previous section, we have shown that the normalized random variable converges to in distribution. Because can be represented by for (Lemma 3), we have the following proposition for the distribution of time average of a weak non- function.

###### Proposition 1.

For a transformation satisfying the conditions (i), (ii), and (iii), and a bounded continuous observation function with and , there exists sequence such that the normalized time average, , converges in distribution:

 1ann−1∑k=0f(zk)⇒Yα,γasn→∞, (46)

where the Laplace transform of the random variable is given by Eq. (45).

###### Remark 1.

The sequence is given by for , which is not the return sequence, where means an average with respect to the initial point .

###### Proof.

By Lemma 3, the distribution of time averages of in the dynamical system considered here can be regarded as that in a continuous accumulation process. The result in section 4 implies the proposition. ∎

To demonstrate our proposition, we use the map with Thaler2000 () defined by Eq. (1) The asymptotic behavior of for is given by . Thus, . The invariant density of this map is exactly known as Thaler2000 ()

 ρα(x)=Cαx1α+Cα(1+x)1α, (47)

where is a multiplicative constant. By numerical simulations, we have confirmed that the asymptotic behaviors of moments of are well described by the theory (44) as shown in Fig. 4. Figure 5 shows that PDF of is in good agreement with the PDF of in the corresponding continuous accumulation process.

Because the ensemble average of with respect to an infinite measure diverges, one cannot obtain a relation between the time average and the ensemble average with respect to the infinite measure. However, we have shown that converges to a constant, and the constant is determined by , and . Because these constants are determined by the asymptotic behaviors of and , the normalizing sequence, , in the proposition can be determined by the asymptotic behaviors of and . In other words, the sequence does not depend on the details of a map and except for a small behavior. We have numerically confirmed them (not shown).

## Vi Conclusion

For one-dimensional intermittent maps with infinite invariant measures, we have shown a novel distributional behavior for time averages of weak non- functions. The distribution refines the generalized arc-sine distribution of time average for weak non- functions because the normalizing sequence is not but is proportional to (). Therefore, the distribution is not the generalized arc-sine distribution nor the Mittag-Leffler distribution. In other words, we have made an important first step for a foundation of the third distributional limit theorem in infinite ergodic theory. Recently, distributional behaviors in intermittent maps with more than two indifferent fixed points has been studied Shinkai2007 (); Korabel2012 (); Korabel2013 (). This kind of extension will be interesting for a future work. The proof of our proposition is based on the theory of the continuous accumulation process proposed here. Our result is summarized in Fig. 6. This novel distributional limit theorem is related to a distributional behavior of time-averaged diffusion coefficients in a model of anomalous diffusion like stored-energy-driven Lévy flight Akimoto2013a ().

## acknowledgement

ã This work was inspired by the conference of “Weak Chaos, Infinite Ergodic Theory, and Anomalous Dynamics.” We are indebted to T. Miyaguchi and H. Takahashi for his helpful comments. This work was partially supported by Grant-in-Aid for Young Scientists (B) (Grant No. 26800204).

## Appendix A First moment

The Laplace transform of is given by

 ⟨Xs⟩ =−∂^P(k,s)∂k∣∣ ∣∣k=0 =−^ψ′(0,s)∫∞0W(τ)e−sτdτ[1−^ψ(0,s)]2+∫∞0w(τ)[∫τ0I(t,τ)e−stdt]dτ1−^ψ(0,s)