# Statistics of Poincaré recurrences for a class of smooth circle maps

###### Abstract

Statistics of Poincaré recurrence for a class of circle maps, including sub-critical, critical, and super-critical cases, are studied. It is shown how the topological differences in the various types of the dynamics are manifested in the statistics of the return times.

PACS:

## 1 Introduction

Statistics of the Poincaré recurrences, i.e. return times statistics, has recently gained renewed importance in the theory of dynamical systems, primarily due to the fact that it could be used as an indicator of the statistical properties of the system’s global dynamics on large parts of its phase space (see for example [1],[2]). For instance, the first return time can be used to calculate the metric entropy of a system with an ergodic invariant measure [3], and seems to be related to other generalized dimensions used to describe the fractal properties of the dynamics, at least for certain types of dynamical systems [4]. In this paper we study the Poincaré recurrences for maps of the circle, that can display quasi-periodic, bi-stable or chaotic dynamics, depending on the values of their two parameters.

Given a discrete dynamical system on a phase space with a transformation and a reference measure on , the first return time, in a measurable set , of a point , is defined by

(1) |

The first return time for the set , and the average return time for the set , are the following

(2) |

where is the conditional measure for any .

With and we define the probability that the normalized first return time in the set is larger than for the points in the set

(3) |

A limit probability measure may be associated to any point by considering a sequence of neighborhoods of with as . For ergodic systems the limit is the the same for almost every point and in this case the average return time in a domain is given by

(4) |

according to the well known Kac’s lemma. For some classes of hyperbolic dynamical systems [5] it has been proved that the return times spectrum follows the exponential-one decay law at almost every . If belongs to the dense set of the unstable periodic points then where and depend on the period. It is known that the properties of the distribution of the return times can give criteria for the existence of an equilibrium and the rates of mixing [7]. A polynomial decay law for was found for integrable area-preserving maps [8][9]. On the basis of numerical computations, the polynomial decay of return times spectra was also claimed to be a generic property of Hamiltonian systems with mixed phase space, where complicated self similar fractal structures are present [10],[1]. In fact, for such systems, a convex combination of the power law and the exponential law decays seems to provide a very good fit of and is theoretically justified [11].

Our aim is to study the statistics of the first return times for smooth perturbations (non necessarily invertible) of the uniform rotations on the circle ,

(5) |

where and and are the parameters of the map. The function is a trigonometric polynomial such that the maps are monotonic for and non-invertible for . The details of the dynamics and the structure of the bifurcation diagram have been thoroughly studied for the sine-circle map given by (see for example [14]- [26]). Other families of the form (4) are used to study the universality of the properties found for the sine-circle map.

The main results of our analysis of the recurrence times for the circle maps of the form (5) can be summarized as follows. For , where the map is invertible and diffeomorphic to a rotation, if the rotation number is diophantine, three return times are observed. This is in agreement with Slater’s theorem [12], which we extend from linear rotations to the diffeomorphisms of the circle, see section 2. At the critical value three return times are still observed and the average return times allow an effective reconstruction of the invariant measure, see section 3. In the super-critical case (section 4), we show the appearance of a continuous spectrum for maps that have chaotic orbits at (section 4.1), and for maps for beyond the corresponding accumulation point of period-doubling bifurcations (section 4.3). Also, the properties of the return times could be used to detect the existence of attracting periodic orbits (section 4.2).

Our conclusion is that spectrum reflects, the bifurcations in the topological properties of the dynamics. and is a useful tool to investigate them.

## 2 Dynamical properties of the circle map

Let us briefly recapitulate some of the properties of the circle maps (5). For our purposes, we distinguish three regions depending on the parameter : the sub-critical region , the weakly super-critical region and the strongly super-critical region .

### 2.1 Sub-critical and critical region

For the circle map is an orientation preserving homeomorphism of the circle, and for the map is a diffeomorphism. In any case, its topological properties are fixed by the rotation number defined by

(6) |

where is the lift of on the real line, and for the definition to be unique.

The map , for is conjugate to the linear rotation by the angle . The properties of the conjugation with the linear rotation depend on the arithmetic properties of . For a generic rotation number is a homeomorphism according to Denjoy’s theorem [14].

Furthermore if is a Diophantine irrational the conjugation is a diffeomorphism [15]. In such a case the topological and metric properties of the linear rotation extend to the map by using

(7) |

Slater’s theorem [12], stating that for any irrational linear rotation, and any connected interval there are at most three different return times, one of them being the sum of the others, extends from linear rotations to the map . Two of the three return times are always the consecutive denominators in the continued fraction expansion of the irrational rotation number , and one of the return times is a sum of the other two. Two points in a connected interval are mapped into and of the connected interval

A critical map is only a homemorphism of the circle but is still characterized by a unique rotation number. Our calculations indicate that, at least, for sufficiently irrational rotation numbers there are still only three return times, like in the sub-critical case.

### 2.2 Weakly super-critical region

When the map ceases to be invertible and it does not have a unique rotation number. For a given quasi-periodic, chaotic, at most two stable periodic orbits, and orbits asymptotic to the latter can coexist. The points in the parameters plane that correspond to sub-critical maps with rational rotation numbers form domains, the tongues, that can be extended above the critical line where they start to overlap. The boundaries of the tongues, correspond to tangent bifurcation, and can be found by solving for and the equations

(8) |

where is the -th application of the map, its derivative (see for example [19]).

The union of all tongues at has full measure, and for any map belongs to intersection of two tongues and that correspond to Farey neighbours with sufficiently high and . Actually, is in the intersection of the tongues that correspond to all rationals deeper in the Farey tree and between and . To predict the dynamics of the map one has to know a very complicated fine structure of bifurcations inside the overlapping tongues. The lines in the plane that correspond to maps with irrational rotation numbers become, for , domains such that a map in such a domain have at least one orbit with the corresponding irrational rotation number, but also has orbits with other rotation numbers consistent with the tongues that overlap, and chaotic orbits [20]. Weakly super-critical maps of the form (5) with chaotic orbits have been used to study the quasi-periodic rout to chaotic dynamics [18], [21],[22].

### 2.3 Strongly super-critical region

Besides the maps with the small that have chaotic orbits, there are such that maps inside a tongue will have strongly chaotic behaviour only if , where is the value of at which the period doubling bifurcations inside tongue accumulate, which could be quite large. For example, for the critical value of is estimated to be [16]. We shall see that the properties of return times change abruptly at , and can be used to estimate .

## 3 Return times for sub-critical and critical dynamics

The numerically observed return times for and diophantine rotation numbers are in agreement with a straightforward extension of Slater’s theorem. The same result is found at the critical value . For generic interval and each quadratic irrational that we have studied, there are again only three return times, one is the sum of the other two and two always coincide with the denominators of the corresponding two successive approximants of , see for example figure 1a. We are fairly confident that the conclusions are valid for any quadratic and the maps in the class (5), but we can not claim anything for homeomorphisms with a generic rotation number (see the reference [28]).

The three return times, and their relative weights depend on the location and the size of the interval. However, there is a sequence of intervals, obtained by partitioning the circle with the iterates of an initial point that is best suited for the analysis of the return times at a point of the map with a given irrational rotation number . One considers the trajectory formed by the successive iterates of a point , where are rational continued fraction approximates of the .

The -th and the -th iterate form the boundary of an interval which contains only the initial point and no other points of the considered part of the trajectory. The points generated by iterates will subdivide the intervals generated by iterates. One obtains a sequence of intervals that converge to the point on the orbit of . Calculating the return time for such sequence of intervals is best adopted to the calculation of the return time at the point . Suppose that the -th iterate is to the left of which is to the left of the -th iterate. Then, each of the points that are on the left side of have a unique first return time equal to , and points to the right of also have a unique first return time equal to . Thus, there are only two return times equal to and . Furthermore, the union of the intervals formed by iterates gives a partition of the circle, and the return times into various intervals of a partition are the same and , although the relative weights are obviously different for a nonlinear map. However, as , and for an irrational number with a constant tail of the continued fraction expansion, the relative weights of the two return times become independent of , which implies the existence of a point spectrum independent of (see Appendix)

(9) |

where

(10) |

are the renormalized return times in the limit and are denominators of the continued fraction approximants of . Since the map is smoothly conjugated for and diophantine to a linear rotation, the above properties follow from a proposition we prove in this case in the Appendix.

Approximation of the measure: The distribution of the return times in various intervals can be illustrated using the mean return time. In fact, for any homeomorphism of the circle there is a unique invariant ergodic measure which is absolutely continuous with respect to the Lebesgue measure on the circle, and the density of this measure can be obtained using return times, as is indicated by the Kac lemma. In figures 1b,c,d we plot density of a coarse-grained mean return time, i.e. the ratio between uniform average of the return times into an interval of a partition of the circle divided by the Lebesque measure of the interval, for a sub-critical and the critical sine-circle map. For a sufficiently fine partition this quantity illustrates the density of the unique invariant ergodic measure for the considered map. The main computational cost of this method is due to the computational time, complementary to the perturbative method where the main requirement is for a sufficient storage space. In order to achieve an resolution of by the return times calculations one needs roughly iterations of the map. Here, is the number of intervals of the partition, with is roughly the average return time and is the number of points in each of the intervals. The same space resolution is archived by using Fourier components of the conjugation function in the perturbation method. For percent accuracy, i.e. , both methods can be easily implemented, but accuracy represents a more challenging task for both the methods.

For the density of the invariant measure is a smooth function (see figure 1b,c), and for the critical maps the density becomes singular (figure 1d) [25],[26]. Fractal properties of the ergodic measures for the critical circle maps (5) have been studied. There is strong evidence [26] that the class of critical maps with the same fractal spectrum of the invariant measure are characterized only by the rotation number (actually probably only by the tail in it’s continued fraction expansion [27]) and by the type of the singularity that induces the critical behaviour.

## 4 Return times in the super-critical cases

A super-critical map could have chaotic orbits, at most two stable attracting orbits and orbits asymptotic to these. Our numerical computations support a conclusion that for any there is sufficiently large , such that the distribution of the first return times is given by exponentially fast decay. Furthermore, there are maps which have chaotic orbits for with arbitrary small but non-zero , and maps that show chaotic behaviour only for sufficiently large . This is manifested by two different roots to the exponential decay of the distribution of the first return times.

The following three typical situations can be clearly distinguished by studying the properties of the return times.

### 4.1 Quasi-periodic route

Consider, first, a weakly-super-critical map with , where is arbitrary small but non-zero, and with such that there is an orbit with an irrational rotation number. For example, suppose that the rotation number of the orbit through point is numerically equal to the golden mean . We can use the return times into a sequence of nested intervals that shrinks to , or analogous sequences that shrink on other points to study the dynamics.

Results of such analysis are shown in figures 2a,b,c,d which illustrate the dynamics of the same map but as revealed by the statistics of the return times into a sequence of nested intervals of decreasing size. Here and , leading to an orbit with whose continued fraction expansion is , and which has denominators of the convergents .

The fact that the map is not topologically equivalent to a uniform rotation is manifested already at the resolution given by the interval of finite size. For example, for the interval (fig. 2a) there are four return times , where the fourth is not a denominator of any of the convergents to . However, it is the denominator of the Farey neighbor of the approximant . This signals that, for the -tongue and the -tongue have common part of their interiors, and that the point belongs to both tongues. The interval that will detect the existence of two intersecting tongues at must be smaller than the distance between and its -th iterate. The first return times into a larger interval are at most 144 for all its points, i.e. with the resolution weaker than the points of the first 144 iterates of the map looks as a smooth rotation.

Further analysis of the return times for the same map but into smaller intervals reveals intersections of tongues at the deeper levels of the Farey tree between and . This is illustrated in figures 2b,c which show the return times into the interval , and a very large number of return times into the interval . The statistics of the return times into is illustrated in figure 2d, by plotting the logarithm of the probability density of the return time larger that versus the normalized time . The distribution is given by exponential decay with an exponent that is numerically close to 1. Thus, on the sufficiently small scale, the map has the distribution of the return times characteristic of strongly chaotic systems.

Increasing , and changing so that there is always an orbit with rotation number , moves the point into the domain were the intersections of tongues, and the chaotic behaviour, can be detected using larger intervals. Figures 3a,b,c, show the effects of more intersected tongues on the distribution of return times into the same interval . For , the return times and their relative weights are such that the distribution is given by exponential decay with the exponent that is again numerically equal to 1. In fact, for any one can find pairs which imply non-periodic orbit through the point , and the the density for such map is . This is illustrated in figure 4a for and and the corresponding . For all these maps and for all tested the distribution is always .

Numerical evidence, presented in figures 2 and 3, suggests the following conjecture: Suppose that a point lies on a quasi-periodic or on a chaotic orbit of a map , for arbitrary and consider a sequence of nested intervals containing whose length approaches zero by increasing . The number of different return times also increases with , and asymptotically, as shrink to , the distribution of return times becomes continuous. Furthermore, the density of probability with respect to the Lebesgue measure on the circle of the normalized first return time larger than is given by exponential decay.

### 4.2 (Bi)-stability

Possible existence of attracting periodic orbits can be detected by studying the return times into various intervals of a single partition of the circle. Although there could be no bounded invariant density in this case the return times are still well defined. The return times into different intervals depend on whether the stable orbit have points in the considered interval or not. If there is an attracting periodic orbit with no points in the interval than, obviously, there are points in the interval that will never come back, i.e. with the first return time , indicating the existence of the attracting orbit. In this case there will be only those return times that correspond to orbits that re-visit the interval at least once before being attracted to the attracting periodic orbit. On the other hand, for examples of chaotic maps, illustrated in the figure 4a, the return time statistics for increasingly fine partitions confirms that these maps have no attracting periodic orbits.

### 4.3 Period-doubling route

In order to study how the period-doubling route to an ultimately chaotic map is manifested in the properties of the return times, consider the maps for various . Results are illustrated by various curves in figure 4. The return times into intervals at for any show the existence of stable periodic orbits, as described in the previous subsection. Suddenly, at the accumulation points of the period-doubling cascade , the distribution of the return times becomes continuous. For such critical , the distribution at the point is given by the exponential decay , with and . Furthermore, still for the return times into intervals at of finite size are given by double exponential curves, which converge to the single exponential as the intervals shrink to . For example, for the interval the distribution is well approximated by

(11) |

represented by the dash-ed line in figure 4b.

Other curves in figure 4 represent the distributions of the return times for examples of strongly super-critical maps, and for intervals of decreasing size around different points. The curves with the unique slope in figure 4b are the distributions for maps in the tongues with for beyond the accumulation of period doublings, i.e. for and with for . In all the cases, the distribution is given by exponential decay that converges to , where and is on an unstable periodic orbit. Non-linear curves in figure 4b represent the distributions for the map at the accumulation of period doublings and on the indicated intervals of decreasing size. The curves in 4a represent for examples of strongly super-critical maps at not on a periodic point, when with . Shown are examples with and the corresponding as explained earlier, and also examples of maps with and large . All this is consistent with the statistics of the return times for other examples of strongly chaotic maps.

## 5 Summary and conclusions

We have analyzed the circle map numerically for a wide range of values in the parameter space and different sets of initial conditions. The results can be summarized as follows.

Sub-critical and critical region: For three return times are observed and this is theoretically and numerically justified. For diophantine rotation numbers this result can be presented as a corollary of Slater’s theorem, since the map is diffeomorphic to a linear rotation, to which such a theorem applies. For the case of a special sequence of nested intervals including a given point we provide a very simple proof of Slater’s theorem, showing that there are only two return times and proving the existence of a limit point spectrum . The critical dynamics is also clear since three return times for a generic interval and for each quadratic irrational rotation number are observed. For and diophantine a piece-wise constant approximation to the invariant measure is obtained from the average return times from a uniform partition of the circle with a very simple procedure and an accuracy comparable to other methods.

Super-critical region: The dynamics of a map in the weak super-critical case is dictated by the tongues with a nonempty intersection that contains the point . This is manifested, and could be detected, in the distribution of the first return times by appearance of more than three return times, which correspond to the rationals on the Farey tree in-between the the two major overlapping tongues that contain . In the case that the interval contains a point on a non-periodic orbit than there is a sub-interval such that the distribution, with respect to the uniform distribution of initial points, of the return times into this sub-interval is typical for strongly chaotic systems, i.e. the exponential decay with exponent equal to unity.

In the intermediate region two return times or the exponential-one spectrum are observed depending on the existence of attracting periodic orbits. The way when the size of containing approaches depends on and it is convenient to distinguish various routes.

Quasi -periodic route: For any non-periodic point of the map with where is arbitrary small, the spectrum is exponential-one. Choosing a finite interval the spectrum appears as continuous for a sufficiently small .

Bi-stability: For there are values of such that the map has attracting periodic orbits. For any interval not intersecting one of these orbits one of the return times is , since many points do not return being attracted by the periodic orbit. For the attractive periodic orbits are present for most values of .

Period-doubling route: For any rational there is a critical corresponding to the accumulation of period-doublings in every tongue. The transition from (bi)-stability to chaoticity is manifested abruptly in the spectrum . For the spectrum is continuous and there are intervals such that it can be fitted with a double exponential: in the limit the spectrum becomes exponential.

Strongly super-critical region For the map is chaotic and the periodic orbits are unstable. The spectrum is exponential-one except for a set of points of measure zero corresponding to unstable periodic orbits where the exponential decay speed is different from 1. The results for the super-critical dynamics indicate that the analysis of the return times spectra in the super-critical case could be a useful tool for a better understanding of the transition from the weakly to the strongly chaotic regime. It would be interesting to follow in details the pattern of intersections of tongues and period doubling bifurcations inside each tongue, leading to the strongly super-critical case, by using the return times spectra and the way they are approached when a sequence of nested intervals squeezing to a point is considered. To conclude the computation of the return times spectrum is a simple and effective way to explore a dynamical system and its bifurcations.

Acknowledgements

N.B. would like to acknowledge worm hospitality of the Department of Physics of the University of Bologna, and INFN for financial support.

## 6 Appendix

Letting be the linear map conjugated to and be the image of the initial point we consider a partition of the circle to which corresponds another partition . The order in these partitions is preserved since the maps are diffeomorphic for and diophantine Let the continued fraction expansion of be given by

(12) |

and be the corresponding rational approximations of order . The odd and even approximants are upper and lower bounds to , converging monotonically to it. The following recurrence relations hold

(13) |

and the following inequalities hold

(14) |

Denoting the linear map iterates of a point by

(15) |

the odd and even sequences and converge monotonically from below and from above to . The intervals if is even and if is odd, form a nested sequence of intervals squeezing to as and from the previous inequalities the following inclusions hold.

(16) |

The intervals enjoy the same properties for any diophantine since the map is orientation preserving and is a diffeomorphism. The sequence is a nested monotonic sequence of intervals approaching exponentially fast. According to Kac’s lemma the average return times for the intervals and are given by the inverse of their length, which increases to exponentially fast with .

According to Slater’s theorem for a generic interval and a linear map with irrational , there are three return times, the last one being the sum of the first two. The sequences of nested intervals and enjoy this property. For the intervals defined above the return times are only two and we give a sketch of the proof since it quite simple.

Proposition: The return times in the interval for the linear map with an irrational are two. If is even (odd) and for the return time is () if and () if .

Proof The proofs for even or odd are analogous so we consider only the even . In order to prove the results we suppose is sufficiently far from the identified ends 0 and 1 that the order preserving relation becomes the inequality between real numbers. For initial conditions near 0, the torus defined as the interval with identified ends should be considered to have the same correspondence. The Lagrange’s theorem states that in the interval the minimum of , where is the integer part, is reached for . As a consequence, the two points closest to for correspond to . We consider the sequence of nested intervals and for a fixed a point we denote by the points of its orbits and examine two possible cases.

Case 1 If then: for , .

To prove the first point we notice that

(17) |

and

(18) |

since both belong to . To prove the second property we notice that the points closest to for are met for and and we show that both are out of .

(19) |

To prove we show that . Indeed

(20) |

Consequently we obtain

(21) |

the equal sign holding for the golden mean.

Case 2 If then: for , .

The fist point is proved observing that

(22) |

and

(23) |

Concerning the last point we notice that we already proved (two equations above) that for any . We prove also that . Indeed from

Existence of the spectrum: The return times in the intervals are given by if and if . As a consequence in we consider the return times in the relative measures of points that come back to and to will be given by the Lebesgue measure of these intervals normalized to the Lebesgue measure of . As a consequence the weights for the return times and are given by

(24) |

We notice that these weights for a quadratic irrational are independent of . Indeed in this case and letting

(25) |

Indeed taking into account that , the above relation is verified if , which is the case due to the recurrence for . As a consequence the average time is

(26) |

The normalized times in the limit become

(27) |

As a consequence the limit point spectrum exists and is given by

(28) |

We believe that the spectrum exists for any irrational . The existence of a spectrum for a generic nested sequence of intervals, squeezing to , remains an open question.

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FIGURE CAPTIONS

Figure 1a,b,c,d:Illustrate the sub-critical and critical cases with :(a)Three return times for , and densities of the unique invariant ergodic measure for (b) (c) and (d) .

Figure 2a,b,c,d: Illustrate the distribution of the return times into a sequence of sub-intervals for the fixed weakly super-critical map .

Figures 3a,b,c,d:Illustrate the return times into the interval for weakly super-critical maps with (a) (b) (c) (d) shows .vs. for , with the slope .

Figures 4a,b: Strongly super-critical dynamics: (a) .vs. with slope for ,,, and for ,;

b).vs. with slope , for when ; and , with approximation (14) (dash-ed line). Other curves represent .vs. at unstable periodic points for other examples with . The maximal slope is .