Limiting stochastic processes of shift-periodic dynamical systems

A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences xn+1 = F(xn) generated by such maps display rich dynamical behaviour. The integer parts ⌊xn⌋ give a discrete-time random walk for a suitable initial distribution of x0 and converge in certain limits to Brownian motion or more general Lévy processes. Furthermore, for certain shift-periodic maps with small holes on [0,1], convergence of trajectories to a continuous-time random walk is shown in a limit.


Introduction
Dynamical systems and their stochastic properties have been studied for more than 100 years, starting with the pioneering works of Poincaré [1], who connected probabilistic concepts with dynamics, conjecturing the Poincaré recurrence theorem. Major advances in the field were made in the 1930s by Birkhoff [2] and von Neumann [3], via the proof of the so-called ergodic theorems, concerning time averages of functions along trajectories. Birkhoff also first used topological methods for the study of dynamical systems. In these early years, differential equations were often the main focus of the study of dynamical systems. However, since the 1970s attention also turned to simple dynamical systems, generated iteratively from a map F : Ω → Ω via an equation (1:1) where Ω has been taken to be a low-dimensional set [4], such as interval [0,1]. It has been observed that even very simple maps and systems can give rise to complicated, seemingly random behaviour of trajectories, a phenomenon Yorke and Li named 'chaos' in their seminal paper [5]. A well-studied example of this phenomenon is given by the logistic map F(x; r) = rx(1 − x),

Shift-periodic maps
This paper studies the behaviour of iterative sequences given by (1.1) for functions F defined on the real line, which are periodic up to integer shifts. The key property of maps F is a shift-periodic formula given in the next definition as condition (i), together with a minor technical restriction (ii) on discontinuities of F. Note that, unlike in other works on this topic in the literature, we allow F to have singularities.
Definition 2.1. A shift-periodic map is a map F : R ! R 1 with the following properties: (ii) There exist 0 = t 0 < t 1 < · · · < t k = 1 so that for i = 1, 2, …, k map F is continuous and monotonic on (t i−1 , t i ).
Example 2.2. In §1, we discussed the climbing sine map F(x; a): = x + a sin(2πx) as an example of a map with translational symmetry possessing interesting dynamical properties. It satisfies the conditions of definition 2.1. More generally, a wide variety of shift-periodic maps can be constructed by choosing two polynomials P(x) and Q(x), the latter non-zero, and setting F(x) = x + P(sin(2πx))/ Q(cos(2πx)). For example, take P(x) = Q(x) = x to get F(x) = x + tan(2πx). In figure 2b, a sample trajectory of this map is plotted in red.
The nonlinearity of the climbing sine map and other climbing trigonometric functions however complicates the discussion of invariant densities and the behaviour of iterates in general. So below, in example 2.3, we discuss a piecewise linear map F(x; ɛ, δ) of a similar structure.  The map F(x; ɛ, δ) has one local maximum and one local minimum in interval [0,1], maps interval [0,1] to a larger interval [−δ/4, 1 + ɛ/4] for parameters δ > 0 and ɛ > 0, and is plotted in figure 1a (as a red solid line). Behaviour of its iterates (1.1) closely resembles that of a random walk. Choosing relatively small values δ = ɛ = 10 −2 , the first 10 4 iterations of map from example 2.3 are shown in figure 1b. Identifying intervals [i, i + 1) with integer valued lattice points {i} for i [ Z, we observe that sequence x n can be viewed as a random walk between these lattice points. More precisely, we can map sequence x n to an integer-valued sequence bx n c, which gives lattice positions of a random walker that is jumping from site {i} to neighbouring sites {i − 1} and {i + 1} with certain probabilities. Such behaviour is common among trajectories of shift-periodic royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 191423 maps and under certain conditions on F the jumps between sides are independent, as is traditionally required of random walks. This will be discussed in §3. The map in example 2.3 does not satisfy this independence condition, but attains the structure of a continuous-time random walk with independent waiting times in a suitable limit. Section 4 is dedicated to this result. Finally, a more general example of a shift-periodic map, in the spirit of the climbing tangent map, is illustrated in figure 1a as a black solid line and is formally defined in example 2.4.
In figure 2, we plot illustrative trajectories for two different values of κ. For large κ (figure 2a), the behaviour of iterations x n+1 = F(x n ; κ) resembles Brownian motion, while for small κ (figure 2b, blue dots) it resembles a Lévy flight. We write 'resembles' since we compare discrete dynamics with continuous time stochastic processes. In §3.6, we make these statements rigorous. To do this, we identify the index n in x n with time and introduce suitable scaling of time to get convergence to a continuous-time process. Before that, we study the random walk behaviour of a certain class of shift-periodic maps when time is left unscaled.

Discrete-time random walks
While the iterative formula (1.1) uniquely determines the next iterate x n+1 from the knowledge of x n , figures 1b and 2 suggest that the next value bx nþ1 c is determined from bx n c only with a certain probability. The goal of this section is to formalize this observation for certain shift-periodic maps by studying the connections between the dynamics of (1.1) and random walks, defined below.
where we assume Y 0 = 0. We say (Y n ) n[N is a discrete-time random walk if Z n , n [ N, are independent and identically distributed. Now, as briefly noted in §2, the increments bx n c À bx nÀ1 c of sequence bx n c, n [ N, are not generally independent. For instance, if the parameters δ and ɛ of map F(x; ɛ, δ) in example 2.3 are chosen sufficiently small, consecutive jumps between unit intervals [i, i þ 1), i [ Z, are not possible. A key issue here is that the local extrema of F(x; ɛ, δ) do not take integer values. This motivates the next definition, introducing a restriction on shift-periodic maps for which an appropriate distribution of the initial values guarantees that the behaviour of a sequence generated by such a shift-periodic map will be that of a random walk.
Definition 3.2. Let F be a shift-periodic map with 0 = t 0 < t 1 < · · · < t k = 1 such that F is continuous and monotonic on (t i−1 , t i ). We then say that F has integer spikes if F additionally satisfies conditions (iii) and (iv) below: royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 191423 (iii) jF(x) À F(y)j . jx À yj holds for all distinct x, y [ (t iÀ1 , t i ) where i ¼ 1, 2, . . . , k: and (iv) lim and lim Condition (iii) is a technical restriction, which is sometimes called expanding in the literature [28], although this terminology is usually reserved for stronger conditions [29]. It will be important in lemma 3.7 later. Condition (iv) ensures that {F} admits a Markov partition on [0,1], a partition of the unit interval into a collection of subintervals such that the fractional parts of F map any such interval onto a union of other intervals in the partition. Precisely, there exist disjoint intervals ( The intervals can be chosen so that F is continuous and monotonic on each of them. This ensures that each trajectory of the associated dynamical system can be described by a sequence of symbols corresponding to the different intervals, and often significantly simplifies the analysis of dynamical behaviour, particularly in the case of finite Markov partitions. For instance, the invariant density of piecewise linear maps admitting a finite Markov partition is easy to compute since its Frobenius-Perron operator can be described in matrix form [29]. In this section, we allow infinite partitions which will play an important role in the proof of lemma 3.7, where we will make use of the aforementioned representation of trajectories in terms of sequences of intervals. Theorem 3.3. Let F : R ! R 1 be a shift-periodic map with integer spikes and let U be uniformly distributed on [0,1]. Then there exists a homeomorphism h : [0,1] → [0,1] so that for Y n ¼ bF n (h(U))c and Y 0 = 0, the stochastic process (Y n ) n[N is an integer-valued discrete-time random walk and we have We take two different approaches to proving this statement. First, we consider shift-periodic maps Caution must be taken with the interpretation of the latter result, as here the measure of h(U) might not be absolutely continuous with respect to the Lebesgue measure. In that case, typical trajectories might not display the discussed random walk structure.
To investigate random walk behaviour, it will be beneficial to split the map into its fractional and integer parts, considering it as a skew product of maps on [0,1] and Z. We will introduce this concept in the next subsection.

Skew-products
Let F be a shift-periodic map. We first define the 'restricted map', F r : [0,1] → [0,1], given by whenever bF(F k r (x))c Ó {À1, 1} for any k ∈ {0, 1, …, n − 1}. Else we take ϕ F (x, n) = 0. We call ϕ F the cocycle associated with F. Note that the conditions placed on shift-periodic map F, in particular (i), ensure that whenever {F(x), F 2 (x), …, F n (x)} does not intersect with {∞, − ∞} then F n So for sequence x n defined by iterative formula (1.1) with starting value description of the sequence is valid away from a set of Lebesgue measure 0. We can rephrase theorem 3.3 in the following way: Theorem 3.3.* Let F:R ! R 1 be a shift-periodic map with integer spikes and let U be uniformly distributed on [0,1]. Then there exists a homeomorphism h : [0,1] → [0,1] so that for Y n = ϕ F (h(U), n) and Y 0 = 0, the stochastic process (Y n ) n[N is an integer-valued discrete-time random walk and we have P(Y n À Y nÀ1 ¼ m) ¼ p m , where p m is a constant satisfying Remark. Cocycle ϕ F (x, n) allows us to associate with F a skew-product F s induced by cocycle ϕ F . Let . Skew-products in general frequently appear as models in physics, for example, see [30] for a recent discussion of diffusion and Lévy-type behaviour of different systems from the point of view of skew-products, including a version of the Pomeau-Manneville map [13] mentioned in §1. Skew-products are also used to investigate recurrence of random walks in [31,32]. Later, in the proof of lemma 3.7, we will relate the trajectories of piecewise monotone maps to those of piecewise linear maps via conjugacy. Similarly, in [33], skewproducts with piecewise monotone fibres are related to those with piecewise linear fibres via a semiconjugacy, though in [33] the map is only allowed to have a finite number of monotonic branches. For more general discussions of cocycles and skew-products see [34].  Proof of lemma 3.4. Since F has integer spikes, F r linear on (a i ,b i ) with F r ((a i ,b i )) = (0,1) implies that

Piecewise linear maps
We now wish to show that for any m 1 , . . . , m n [ Z we have P(Z 1 ¼ m 1 , . . . , Z n ¼ m n ) ¼ p 1 . . . p n . For n = 1, the statement holds by definition of p m . For n > 1 note that F r is linear on (a i ,b i ) with F r ((a i ,b i )) = (0,1), so F r (U) is uniformly distributed on the unit interval conditional on U ∈ (a i ,b i ). Then P( The statement then follows by induction. This gives us independence of random variables Z n , n [ N, and further P(Z n ¼ m) ¼ p m . The lemma holds. ▪

Absolutely continuous invariant measures
As mentioned in our earlier discussion, it is sometimes possible to deduce the behaviour of trajectories of a shift-periodic map F from a given absolutely continuous invariant measure μ of F r . Since μ is absolutely continuous, x 7 ! m([0,x]) is continuous, and if this map is additionally strictly increasing, it has an inverse denoted by h. Crucial now is the integer spike condition: we may split the unit interval into subintervals (a i ,b i ), i ∈ I on which F r is monotonic with F r ((a i ,b i )) = (0,1). If h −1°F r°h is additionally linear on h −1 ((a i ,b i )), our earlier lemma 3.4 becomes applicable.
Proposition 3.5. Let F : R ! R 1 be a shift-periodic map with integer spikes and let U be uniformly distributed on is countable. Suppose that μ is an absolutely continuous invariant measure for F r and that the map gives a random walk with transition probabilities given by equation  Proof. Define a shift-periodic map G : In figure 3a, we present an example of a map F which satisfies the conditions of proposition 3.5. It is constructed by conjugating the piecewise linear map G(x) = F(x; 4, 4), given in example 2.3, with homeomorphism h(x) = x(1 + x)/2. More precisely, we take Its invariant distribution, corresponding to an absolutely continuous invariant measure, is h(U ). Although, starting from a piecewise linear map and a given h, we can use equation (3.4) to construct many other examples satisfying proposition 3.5, the statement of theorem 3.3 is more general, as we instead need to construct h.

Topological conjugacy
We will now relate the trajectories generated by an arbitrary shift-periodic map F with integer spikes to those of a shift-periodic map G which is linear between integer function values. Below is a formal definition of topological conjugacy, a construction which has already appeared in proposition 3.5 and in equation (3.4). Baldwin [28] describes all classes of topologically conjugate maps on [0,1] which are continuous and piecewise monotonic. In [28,33], it is assumed that piecewise monotonic maps have only finitely many monotonic branches. In [35], conjugates between maps with infinitely many monotonic branches are considered, though piecewise differentiability is assumed. A slight adaptation of Baldwin's proof establishes the Lemma below, which is valid for maps with infinitely monotonic branches and does not require differentiability.
Define a corresponding linearized map g : [0,1] → [0,1] in the following way: g is linear on for any i ∈ I and further g(x) = 0 for any x ∈ Λ. Then f and g are topologically conjugate.  royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 191423 length greater or equal n, we say cj n ¼ b if the first n entries of b and c coincide. For a finite sequence b of length n with entries in I < L then write We now make the following observations: suppose x ≠ y. Because of condition (ii) it is clear that for some n [ N the interval ( f n (x), f n ( y)) or ( f n ( y), f n (x)) intersects with Λ. We then easily deduce a f (x) ≠ a f ( y).
Let again b denote a finite sequence. From above we then note that J f b is empty or a singleton if at least one of the entries of b is in Λ. Otherwise J f b is an open interval. This can easily be shown by induction, and is a consequence of f((a i , b i )) = (0, 1) and of monotonicity of f on (a i , b i ). Furthermore, for another finite where b runs over all finite sequences of length n in I < L. Note that the least upper bound on the lengths of intervals J f b , where b sequence with n entries, goes to 0 as n goes to infinity. h n thus converges uniformly to a continuous, monotonically increasing map h. It is in fact strictly increasing: Note that for x < y there must exist z 1 , z 2 ∈ [0,1] with x < z 1 < z 2 < y so that a g (z 1 ) and a g (z 2 ) have some entries contained in Λ. This implies that (h n (z 1 )) and (h n (z 2 )) have constant tails and so, as h n (x) < h n (z 1 ) < h n (z 2 ) < h(z 2 ), taking the limit as n → ∞ we obtain h( Assume the latter holds. There is a point y with a g (y) ¼ a f (h(x)) and h n is eventually constant and equal h at y. Then a g (y) and a f (h(y)) agree up to arbitrary length and a g ( As h is continuous and strictly increasing, it is also a homeomorphism on [0,1] with a g (x) ¼ a f (h(x)). Then also a g (g(x)) ¼ a f (f(h(x))) and so a g (g(x)) ¼ a((h À1 f g)(x)). Therefore, (h −1°f°h )(x) = g(x) and thus f and g are topologically conjugate. ▪ While the topological conjugacy described above sends f to the linearization g which keeps the endpoints of monotonic branches fixed, the construction itself is valid for any map g which has the same number and orientation of monotonic branches as f, or, for an f with infinitely many branches, when g has the same number of limit points of [0, 1]n< i[I (a i , b i ), corresponding to singularities.

Proof of theorem 3.3
The proof of theorem 3.3 now follows from the previous results: Let U be uniformly distributed on [0,1]. Let F : R ! R 1 be a shift-periodic map with integer spikes. The conditions placed on F, in particular integer spikes, ensure that F r satisfies the requirements of lemma 3.7. Let g be the corresponding linearized map, so that F r is topologically conjugate to g. Define a shift-periodic map G : R ! R 1 by setting form a random walk with transition probabilities Hence the same must be true for Y n ¼ f F (h(U), n), n [ N and theorem 3.3 holds.
royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 191423 Proof. Let G be as in the proof of theorem 3.3. Since G r is linear on each (a i , b i ) and h((a i , b i )) = (0, 1), we see that U is an invariant distribution with respect to G r . So we deduce As noted below the proof of lemma 3.7, the choice of h in the proof of theorem 3.3 is rather arbitrary in the sense that there are infinitely many piecewise linear maps which F r conjugates to, and for any of these maps, theorem 3.3 is satisfied. The choice of h in lemma 3.7 has the nice property that p m ¼ l({x [ [0, 1] : bF(x)c ¼ m}). However, h(U) is not necessarily absolutely continuous, and neither is this necessarily the case for any of the alternative conjugacies.
As an example, consider again F(x) defined in equation (3.4) with h(x) = x(1 + x)/2. We know that h is a smooth map which conjugates F r to the piecewise linear map G r . The map h(x) − x = x(x − 1)/2 is plotted in figure 3c as a blue dashed line. However, the construction in lemma 3.7 gives a much more complicated choice of h, shown in figure 3c in red. For this choice of h, the measure corresponding to distribution h(U) is no longer absolutely continuous with respect to the Lebesgue measure, and the results of theorem 3.3 do not translate into observations of typical trajectories.
The red line in figure 3c illustrates an example in which lemma 3.7 fails to give us a more natural choice of h, which exists for this example (blue dashed line). For other shift-periodic maps, however, there is no choice of h for which the measure corresponding to h(U) is absolutely continuous. An example is the Pomeau-Manneville map introduced in §1. Choosing its parameters so that F(x) = x + 6x 2 on [0,1/2), it satisfies the conditions of theorem 3.3, but has derivative equal to 1 at x = 0 and x = 1. Whenever a trajectory comes close to one of these points, a long sequence of constant integer parts follows until jumps reoccur. The long-term dynamics of a typical trajectory do not display the discussed random walk structure. The map h constructed in lemma 3.7 is shown in figure 3 as a green line.

Alpha-stable processes
The results in the previous subsection showed how iterates Y n ¼ bF n (X)c of shift-periodic maps with integer spikes can be viewed as sums of independent and identically distributed random variables Y n − Y n−1 , for suitable choices of initial distribution X. A lot is known about the behaviour of continuous stochastic processes arising as the limit of such partial sum processes under suitable scaling, for a summary see for example [36]. We will briefly give an overview of such results for independent random variables and the implications for our shift-periodic maps.
First, we make our notion of a scaling, passing from a discrete to a continuous process, precise. Take Y n ¼ bF n (h(U))c, choose translation-scaling and space-scaling constants a n and b n , and set V (n) (t) ¼ 1 b n (Y bntc À a n t), where n [ N: Since for integer spike maps as in theorem 3.3 the generated random variables Y bntc behave like a random walk, we can apply functional central limit theorems (FCLTs) to investigate the behaviour of the limit of V (n) (t). The classical example is Donsker's theorem, which treats the convergence of processes of the form (1=b n )( P bntc k¼1 Z k À a n t) to the Wiener process when the independent random variables Z k follow a normal distribution. This convergence is with respect to the Skorohod metric on the space of right-continuous functions with existing left limits [36,37]. We refer to such a space of functions on [0,∞) as D([0, 1), R).
A standard example of such a Lévy motion is the Wiener process for α = 2. Lévy motions allow the following generalization of Donsker's theorem.
Then the stochastic processes defined by converge, with respect to the Skorohod metric on D([0, 1), R), to the Lévy motion V(t; α, β).
Proof. This is a simple consequence of combining Uchaikin's version of generalized CLT [39] with the discussion of FCLTs arising from CLTs in Whitt [36]. ▪ We can now rephrase this theorem in terms of our sequences of iterates of shift-periodic maps. Choose α, β, a n and b n as in theorem 3.11. Let h be as described in theorem 3.3, so that h(U) is an invariant distribution of F r . Define V (n) (t) by (3.6). Then these stochastic processes converge, with respect to the Skorohod metric on D([0, 1), R), to the Lévy motion V(t; α, β).
Note that while we required c + > 0 or c − > 0 in corollary 3.12, effectively excluding continuous shift-periodic maps, the random variables Y n ¼ bF n (h(U))c À bF nÀ1 (h(U))c generated by a continuous shift-periodic map F with integer spikes have finite second moments, so that Donsker's theorem covers this case and we get Brownian motion upon an appropriate scaling.
Let us briefly return to map F(x; κ) in example 2.4. Simple calculations show that the constant κ in corollary 3.12 is the same as parameter κ of map F(x; κ). The corollary then tells us that for appropriately chosen a n and b n the stochastic process V (n) (t), as defined in equation (3.6), behaves in a limit like a Lévy motion Y(t; α, β). Here α = min{κ, 2}. In particular, we get a Wiener process for κ ≥ 2. This can also be observed in figure 2.
More general results concerning generalized CLTs and convergence to Lévy motions can be found in the literature, relaxing the independence condition to various mixing conditions. For recent results see, κ a n b n royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 191423 for example, [40][41][42][43][44]. It is possible to extend statements like in corollary 3.12 to sequences of iterates generated from more general shift-periodic maps, but the conditions from papers [40][41][42][43][44] translate to quite technical restrictions.
In the next section, we look at continuous stochastic processes from a different point of view. Instead of scaling a sequence of iterates by multiplication with a space-scaling constant, we scale the parameters ɛ and δ of map F(x; ɛ, δ) from example 2.3. The resulting process is still confined to Z, but continuous in time, giving dynamical behaviour different from what we have seen in this subsection.

Continuous-time random walks
In §2, our investigation of shift-periodic maps was motivated by the parameter-dependent map F(x; ɛ, δ) from example 2.3, see figure 1. For most choices of parameters ɛ and δ, sequence bx n c cannot have two consecutive jumps, so does not strictly behave like a random walk according to our definition. However, it does display apparent similarities. We will describe in this section in what sense random walk behaviour appears in trajectories generated by this map.
Before we give a precise statement of theorem 4.4, we discuss the invariant density of F r (x; ɛ, δ). This will be necessary both for motivation and proof of the theorem.

Invariant density
To calculate the invariant density of F r (x; ɛ, δ), we apply a general result on invariant densities of piecewise linear maps due to Góra [45]. where we leave function β(x, n) undefined when the derivatives do not exist. This is the case only for finitely many points in (0, 1) for each n. We further define two cumulative derivatives at c 1 = 1/4 and c 2 = 3/4 by The invariant density of F(x; ɛ, δ) is given by where K is a normalization constant, chosen so that f i integrates to 1 over [0,1] and D L 1 , D L 2 , D R 1 , D R 2 are constants dependent on ɛ, δ with D R i ! 1 and D L i ! 1 as ɛ, δ → 0, for i ∈ {1, 2}. Proof. This is just an application of Góra's results on invariant densities of eventually expanding maps in [45].
is the solution of (−S T +I )D T = (1, 1, 1, 1) T where S = (S i,j ) is a matrix with entries dependent on F n r (c) and β(c, n), c [ {c L 1 , c R 1 , c L 2 , c R 2 }, converging to 0 as the parameter ɛ and δ of F n r (x; 1, d) converge to 0. For more details on S see p. 7 of [45]. ▪ For ɛ, δ = 0 map F(x; ɛ, δ) is linear between grid lines, so of the type discussed in §3, and has invariant density equal 1. So one might expect that f i (x; ɛ, δ) → 1 as ɛ, δ → 0. This convergence is not uniform on [0, 1], but we can make the important observation below: Proof. Let d > 0. For any fixed n [ N, we have that F n r (c 1 ) ! 0 and F n r (c 2 ) ! 1 as ɛ, δ → 0, so that the length of the interval on which 1(x; F n r (c j ), À b L (c j , n)) = 0 or 1(x; F n r (c j ), b R (c j , n)) = 0 also goes to zero, royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 191423 for j = 1, 2. Noting that |β L (c j , n)| ≥ 2 n and |β R (c j , n)| ≥ 2 n , then the integral of the weighted sum of four infinite sums in equation (4.1) over [0,1] goes to 0 as ɛ, δ → 0. Adding 1 to this integral, we get K. So we have K → 1. By the same argument, for sufficiently small ɛ > 0 and δ > 0 we have F n r (c 1 ) , d and F n r (c 2 ) . 1 À d for all n ∈ {1, 2, …, N}, so that we achieve bound

From maps to continuous-time random walks
Now consider X distributed according to the invariant distribution of F r (x; ɛ, δ) on [0,1] and X n ¼ bF n (X; 1, d)c. For now we say a jump occurs when X n ≠ X n+1 . By corollary 4.2 the invariant density of F r (x; ɛ, δ) is close to 1 at the spikes 1/4 and 3/4 of the map for small parameters ɛ and δ. Further, direct calculation shows that the length of the subset of [0,1] mapped outside the unit interval by F(x; ɛ, δ) is : This calculation suggests that the probability of a jump is about 3(ɛ + δ)/16. However, successive jump probabilities are not independent, since for small parameters successive jumps are impossible. We fix this issue by introducing a scaling in time, together with a scaling of the parameters. This scaling gives us behaviour resembling a continuous-time random walk, defined below.
# : Let T m,1 , T m,2 , … denote the jump times of Y m (t), that is, where T m,0 = 0. Then for any k ≥ 0 we have Let Y(t) be a continuous-time random walk with waiting times, T j − T j−1 , exponentially distributed with mean 1/γ. Then theorem 4.4 says that for Y m (t) the probability of the (k + 1)-th jump occurring at time bmt k c=m þ t, given that the first k jumps occurred at times bmt 1 c=m, bmt 2 c=m, . . . , bmt k c=m, converges to the probability of the (k + 1)-th jump of Y(t) occurring at time t k + τ, given that the first k jumps occurred at t 1 , t 2 , …, t k .

Conditionally invariant distribution
While we discussed invariant distributions of F r (x; ɛ, δ) above, to prove theorem 4.4, it will often be more convenient to work with conditional distributions on [0,1] which are invariant with respect to F(x; ɛ, δ) conditioned on the event that the iterative sequence stays in [0,1]. The use of conditionally invariant densities is common in investigations of dynamical systems with holes, that is, trajectories generated by maps F : Ω → F(Ω) with V ⫋ F(V) until the point of escape from Ω. Early results are due to Pianigiani & Yorke [46], who motivated the discussion with the example of a billiard table with chaotic trajectories, and introduced the concept of conditionally invariant measures. This idea has been investigated further by other authors, for example, Demers and Young studied escape rates through the small holes in [47].
In our case, it will be useful to work from the point of view of a dynamical system with holes whenever no jump is occurring. For a probability density f of a distribution on [0,1], the density after application of F(x; ɛ, δ), conditional on not mapping outside [0,1], is given by the Frobenius-Perron operator [46] denote the inverses of F(x; ɛ, δ) restricted to (0, 1/4), (1/4, 1/2), (1/2, 3/4), (3/4, 1), respectively, and normalization constant C is chosen so that P 1,d integrates to 1 over the unit interval.
Proof. Suppose f is a density which is constant equal ν on (0, 1/2) and 1 − ν on (1/2, 1). It satisfies P 1,d (f) ¼ f if and only if ν satisfies the following equation: This equation is obtained from the first line of equation (4.4), the left corresponds to normalization constant C multiplied by ν. Solving this quadratic equation, we obtain for all ɛ, δ ≥ 0 a unique solution ν with both ν ≥ 0 and 2 − ν ≥ 0, and also the unique conditionally invariant density f (x; ɛ, δ) described in lemma 4.5. This solution linearizes to In particular, f c (x; ɛ, δ) → 1 as ɛ, δ → 0. ▪

Convergence to the conditionally invariant density
In this subsection, we make a first step towards proving theorem 4.4. We show for some initial densities k that P n 1,d (k) does not only converge to the corresponding conditionally invariant density f c , but that there is an upper bound on the convergence speed which works for all ɛ > 0 and δ > 0.
Pianigiani and Yorke extensively studied the existence of and convergence to conditionally invariant densities for expanding maps in [46]. While their approach does not give us the desired bound on convergence speed, one of their results, the lemma stated below, will be very useful in our proof.
, 1 and sup n kP n F (1)k 1 , 1: Then there exists some L such that for n ≥ 1 kP n F (f) À P n F (g)k 1 Lkf À gk 1 is satisfied. More precisely, we may take Proof. We can apply [46, proposition 1]. The original statement requires that f, g ∈ K where Ð 1 0 f(x)dx ¼ 1}, but the proof also works for the assumptions of lemma 4.6. Note that in this case, we define the infinum by inf [0,1] We now have all the necessary tools to prove theorem 4.4. The key idea is to approximate densities by piecewise constant densities.
Proof. By [46, theorem 3], densities P n 1,d (k) converge to invariant density f c as n → ∞. Lemma 4.7 will now show that this convergence is uniform over all choices of k. So let k be an arbitrary function satisfying the conditions of the Lemma. Density P n 1,d (k) is constant for each n [ N on both (0, 1/2) and (1/2, 1). First, we bound ratio , where x is the value of k appearing in equation (4.5). Note that on (0, 1/2) where c 1 (x) is the normalization constant C from formula (4.4). Moreover, we obtain P 1,d (k) ¼ 2 À a 1 (x)=c 1 (x) on (1/2, 1) from normalization. Furthermore, on (0, 1/2), where c 2 (x) is again the normalization constant C from formula (4.4) and a 2 (x) is a linear polynomial equal to c 1 (x)c 2 (x)P 2 1,d (k). Note that for fixed parameters ɛ and δ denominators c 1 (x) and c 1 (x)c 2 (x) can also be written as linear polynomials of x and are non-zero. With this notation r can be expressed as a quotient of two polynomials. As a quotient of a cubic and quadratic polynomial, an explicit calculation shows that denominator and enumerator have the same positive root and that this root has multiplicity 1 and is equal to the value of the conditionally invariant density ν from lemma 4.5. So r can be extended to a continuous function in x, ɛ and δ for x ≥ 0, ɛ ≥ 0, δ ≥ 0. For ɛ = δ = 0 a calculation gives r(x, 0, 0) = −2. By continuity we can choose some ω > 0 such that ɛ, δ < ω and x ∈ [0,2] implies |r(x, ɛ, δ)| > 3/2. By repeatedly applying this result, But as each P n 1,d (k) is constant on (0, 1/2) and equal to 2 À P n 1,d (k)(1=4) on (1/2 − 1), it follows that kP n 1,d (k) À f c ( Á ; 1, d)k 1 6(2=3) n for each n. ▪ Densities with three constant pieces are more convenient for approximating a density conditional on a jump having just occurred, something we will look at in the later parts of the proof of theorem 4.4. So we now focus our attention on such densities. where where k 1 , k 2 , k 3 are non-negative constants. We define further map Ψ : where b is as defined in equations (4.6)-(4.7).
Corollary 4.10. Let S ≥ 0 and s > 0. Let K S be as described in lemma 4.9. There exists L > 0 and ω > 0 such that for any 0 < ɛ, δ < ω, g ∈ K S , density f with inf [0,1] f > s and n ≥ 1 Lkf À gk 1 : Proof. By lemma 4.9 there exists ω > 0 such that 0 < ɛ, δ < ω and g ∈ K S imply P n 1,d (g) [ K S for each n ≥ 1. Using the last paragraph of the proof of lemma 4.9, then sup n kP n 1,d (g)k 1 2 þ S. Also, since P 1,d (1) is constant on both (0, 1/2) and (1/2, 1), we have kP n 1,d (1)k 1 2. By applying lemma 4.6, we find that for 0 < ɛ, δ < ω, density f with inf [0,1] f > s satisfies Proof. First, use lemma 4.9 to observe that for any S 1 ∈ (0, S) there exist ω 1 > 0 and N 1 [ N such that for any k ∈ K S and 0 < ɛ, δ < ω 1 we have P N1 1,d (k) [ K S1 . But S 1 can be chosen small enough that there is some s > 0 such that for any k ∈ K S and N 2 ≥ N 1 + 1 we have inf [0,1] P N2 1,d (k) . s. Apply corollary 4.10 with f ¼ P N2 1,d (k) to find some L > 0 and ω 2 > 0 such that when 0 < ɛ, δ < ω 2 , N 2 ≥ N 1 + 1 and g ∈ K S then jjP N2þn 1,d (k) À P n 1,d (g)jj 1 LjjP N2 1,d (k) À gjj 1 : royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 191423 Let τ > 0. Then pick m > max{M 3 , N/τ} and consider the probability that no jump occurs until time τ for Y 0 m (t), Write A m for the subset of [0,1] mapped outside the unit interval by F m . As in equation (4.2), denote by ℓ(ɛ/m) + ℓ(δ/m) the length of A m . Write f c,m = ν m on (0, 1/2) and f c,m = 2 − ν m on (1/2, 2). Using equation (4.14), we get the following upper estimate Since m ℓ(ɛ/m) + m ℓ(δ/m) → γ as m → ∞, where γ is given by equation ( as m → ∞. So P(T 0 m,1 . t) is bounded above by a product converging to exp[−γ(1 − μB(d))τ] and below by a product converging to exp[−γ(1 + μB(d))τ] as m → ∞. But μ > 0 was arbitrary, so P(T 0 m,1 t) ! 1 À exp ( À gt) as m → ∞. ▪ Now we will prove an alternate version of theorem 4.4 for Y 0 m (t). Lemma 4.13 established such a statement already for the first jump. The key component in the general proof will be the following lemma, which helps in describing how the densities develop after a jump, conditional on no further jump occurring, provided we start off close to the conditionally invariant density f c,m . Lemma 4.14. Let , 1 4 denote the subset of (0, 1/4) mapped outside of [0,1] by F m . Take 0 < μ < 1/4 and let g m be a density with kg m À f c,m k 1 , m: Let V m be distributed according to that density and g w denote the density corresponding to the distribution of Proof. Since V m [ A 1 m , we have F r (V m ; ɛ/m, δ/m) ∈ [0,ɛ/(4m)]. On (ɛ/(4m), 1) we then have g w m ¼ 0, while we have , for x [ 0, 1 4m , ( 4 :16) where c 0 is a constant so that g w m integrates to 1 over [0,1]. Let u m denote the smallest integer such that F umþ1 r (1=4; 1=m, d=m) ¼ (4 þ 1=m) um 1=(4m) ! 1=4. Then u m ¼ dlog (m=1)= log (4 þ 1=m)e. From equation (4.4), we deduce that for n ≤ u m , density P n m (g w m ) is obtained from g w m via a scaling of the form where c n is a constant dependent on m, such that P n m (g w m )(x) integrates to 1 over [0,1]. By lemma 4.5, there exists M 1 [ N such that m ≥ M 1 implies 1/2 < f c,m < 3/2. Since μ < 1/4, we obtain a bound g m ( y) ≥ , m and f c,m constant on (0, 1/2). Combining this with equation (4.17) gives us that the values of P um m (g w m ) on I um are contained in a subinterval of (0, ∞) of length 32μ. But applying equation (4.4) again, we see that for P umþ1 m (g w m ) the unit interval can be split into three subintervals (0, b 1 ), (b 1 , b 2 ) and (b 2 , 1), where b 1 = 1/2 or b 2 = 1/2, on each of which the values of P umþ1 m (g w m ) are contained in an subinterval of (0, ∞) of length 32μ/C. Here C is the normalization constant from equation (4.4). From f c,m < 3/2 and μ < 1/4 we get g m (y) < 2 and P um m (g w m )(x) 2=(c 0 c um ) 32, since we deduced earlier from equation (4.18) that c 0 c um ! 1=16. For m large enough, say m ≥ M 2 > M 1 , we will have ℓ(ɛ/m) + ℓ(δ/m) < 1/64, where ℓ is defined as in equation (4.2). Considering the integral of P um m (g w m ) over the subset of [0,1] mapped outside the unit interval by F m , we obtain a lower bound of 1 − 32/64 = 1/2 on C. So the values of P umþ1 m (g w m ) on each of (0, b 1 ), (b 1 , b 2 ), (b 2 , 1) are contained in intervals of length 64μ. Using (4.4), calculations show that we can find a bound, independent of choice of g m with kg m À f c,m k 1 , 1=4, on the range of values of P umþ1 m (g w m ) over all of (0, 1/2) and (1/2, 1) respectively. Call this bound S. We now may choose a piecewise constant map k m ∈ K S , as defined in Definition 4.8, so that kP um m (g w m ) À kk 1 , 32m: Recalling that f c,m > 1/2, we get g m > 1/4, and equation (4.17) gives us lower bound of 1/4 on P um m (g w m ) and by equation (4.4) a lower bound on P umþ1 Between time bt j mc=m and bt jþ1 mc=m, the density of F bmtc r (X 0 m ; 1=m, d=m) develops as given by applying operator P m . Provided that μ < B −k /4 and m is large enough so that N þ S(m) , bmt jþ1 c À bmt j c for j = 1, 2, …, k, we can iteratively apply lemma 4.15 with g m ¼ P bmt jþ1 cÀbmt j cÀ1 m (g (j) m ), where g (j) m is the density after the j-th jump has occurred, at time bmt j c=m. Then for large enough m and n ≥ N + S(m) we in particular find jjP n m (g (k) m ) À f c,m jj 1 mB(d)B k : (4:21) But P n m (g (k) m ) describes the densities of F bmtc r (X 0 m ; 1=m, d=m) conditional on T 0 m,k ¼ bmt k c=m, . . . , T 0 m,1 ¼ bmt 1 c=m and no further jump occurring. Choose τ > 0. Then where A m is defined as in lemma 4.15. Since equation (4.21) holds, we can use the same arguments as in the proof of lemma 4.13 to find lower and upper bounds on for n ≥ 1, implying that P n m (g (k) m ) stays close to a piecewise constant function in K S as soon as jumps are possible. This tells us that an upper bound b on the densities P n m (g (k) m ) can be found, valid for all n ≥ u m and all m large enough. But then Let Y m (t) and Y 0 m (t) be as defined in theorem 4.4 and definition 4.12 respectively, recalling that Y 0 m depends on a choice of 0 < d < 1/2. Write E and E 0 for events (T m,k ¼ bmt k c=m, . . . , T m,1 ¼ bmt 1 c=m) and (T 0 m,k ¼ bmt k c=m, . . . , T 0 m,1 ¼ bmt 1 c=m), respectively. By definition of initial distributions of Y m and Y 0 m , X m and X 0 m , with underlying densities f i,m and f 0 i,m , we have that We have already noted earlier, in the proof of lemma 4.13, that sup m kf i,m k 1 , 1 and sup m kf 0 i,m k 1 , 1. Conditioning on the events X m , X 0 m ∈ [d, 1 − d] and X m , X 0 m Ó [d, 1 À d], we get Since 0 < d < 1/2 was arbitrary, we apply lemma 4.16 and let d → 0 to conclude the proof. □

Discussion
We have seen in §3 that the behaviour of the trajectories of a shift-periodic map F with integer spikes (definition 3.2) can be described in terms of discrete-time random walks for suitable initial distributions, and we observed a variety of interesting stochastic processes in scaling limits. Omitting the integer spike condition, the class of maps considered appears to be too large to allow for nontrivial statements about trajectories which apply to all shift-periodic maps. Some of the potential royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 191423