Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors

Skew product systems with monotone one-dimensional fibre maps driven by piecewise expanding Markov interval maps may show the phenomenon of intermingled basins. To quantify the degree of intermingledness the uncertainty exponent and the stability index were suggested by various authors and characterized (partially). Here we present an approach to evaluate/estimate these two quantities rigorously using thermodynamic formalism for the driving Markov map.


Introduction
Motivated by numerical observations by Alexander, Yorke, You and Kan [1] and Sommerer and Ott [30], the term intermingled basins was introduced to the mathematics literature by Kan in [16]: " [. . . ] any open set which intersects one basin in a set of positive measure also intersects each of the other basins in a set of positive measure", and he showed the existence of an open set of diffeomorphisms with two intermingled basins on a three-dimensional manifold. The degree of intermingledness was soon quantified by the uncertainty exponent [23] originally introduced in [13]. In [22] this exponent is calculated numerically for a certain model of point particle motion subject to friction and periodic forcing in a two-dimensional potential (which does not display intermingled but riddled basins), and the authors support their findings theoretically by calculations carried out for a simple piecewise linear model, that has the form of a skew product over two-leg base map with two full linear branches and for which the dynamics are essentially equivalent to that of a random walk with two regimes of transition probabilities.
Later, a related quantity denoted stability index was introduced in [29] and further studied in [20]. While the uncertainty exponent measures the degree of intermingledness averaged over points that are equidistant to one of the attractors, the stability index measures intermingledness close to individual points, so it resembles a local dimension.
In the present paper we provide rigorous derivations of the two exponents mentioned before for nonlinear generalizations of the piecewise linear system. We use thermodynamic formalism and large deviations theory for the base map in order to study a nonlinear version of the tworegime random walk. The class of systems we investigate includes examples from [5] and also some of the maps from [14] that serve as models of indeterminate competition in two species systems. For a particular family of such systems the numerical analyses in the spirit of [22] was carried out in [26].
In the remainder of this section we introduce the class of skew product danamical systems studied in this paper and some basic concepts related to them. In Section 2 we recall the characterization of intermingledness through normal Lyapunov exponents, define the stability index and the uncertainty exponent and state our three main theorems, whose proofs are deferred to Section 3.

Invariant graphs and their Lyapunov exponents
Denote by P(S) the space of all S-invariant probability measures on (Ω, B) and by P e (S) the family of all ergodic ν ∈ P(S). As (Ω, B) is assumed to be standard Borel, non-ergodic invariant probability measures can be decomposed into ergodic ones.
If ν ∈ P(S) and the identity holds for ν-almost every ω we call ϕ a ν-a.e. invariant graph.
observe that ϕ − c and ϕ + c are invariant graphs.
Part a) of the following proposition can be found in [5]. In a slightly different setting, both parts were proved in the earlier paper [15]. The reader can easily adapt the proof of part b) to the present setting. Proposition 1.6. Let F ∈ F s , ν ∈ P e (S). Then exactly one of the following three possibilities occurs: e. so that (ν ×m)((Ω×J)\B + ) = 0, and (ν × m)(B − ) = 0.
In particular, there are at most three ν-equivalence classes of invariant graphs, namely ϕ − , ϕ + and ϕ c .
The fact that there are at most three ν-equivalence classes of invariant graphs is, of course, due to the assumption of negative Schwarzian derivative.

Stability index and uncertainty exponent for intermingled basins
In order to keep the notation as simple as possible, we specialize already from here on to one-dimensional mixing Markov maps at the base: Hypothesis 1. Ω = I is a finite interval, and S : I → I is a piecewise expanding and piecewise C 1+Hölder mixing Markov map with finitely many branches. It is a well known fact, often called the folklore theorem [6], that S has a unique invariant probability measure µ ac equivalent to Lebesgue measure m on I. 1

Intermingled basins
An early reference for intermingled basins (with quadratic fibre maps) is the paper [16].  5]). If λ µac (ϕ − ) < 0 and λ µac (ϕ + ) < 0 and if there are ν − , ν + ∈ P e (S) This is precisely the situation we are looking at, so we state this as a further hypothesis.
where C(ω) and D(ω) are determined such that f ω (±1) = ±1. Then Sf ω < 0.  [2] suggests that ϕ c is continuous. (We say "suggests", because the baker map at the base is not a homeomorphism as required in that Lemma.) In those cases where ϕ c is a continuous curve bounded away from ϕ − and from ϕ + , the box dimension of the graph of ϕ c is known: Bedford [4] studied systems that include our examples and showed that the box dimension is the unique zero of the pressure function t → p ω → −(t − 1) log |S (ω)| + log f ω (ϕ c (ω))) , provided this zero is bigger than 1.

The stability index
As in [19], which was strongly motivated by [29], we define a local stability index σ(ω, x) of a point (ω, x) ∈ I × J w.r.t. B − and B + in the following way: Let where where Of course, the limits in (3) need not exist for every (ω, x), but observe that lim inf →0 log Σ ± (ω,x) log 0 always. Hence σ + (ω, x) and σ − (ω, x) are non-negative, and at most one of them can be strictly positive. Observe also that σ ± (F (ω, x)) = σ ± (ω, x) for all (ω, x) ∈ I × J. (This is essentially Theorem 2.2 of [29].) Remark 2.4. Denote by m ± the 2-dimensional Lebesgue measure on Ω × J restricted to B ± Then σ ± (ω, x) + 2 is just the local dimension of the measure m ± at (ω, x).
In order to obtain precise quantitative information about σ ± , we require some regularity of f ω (x): A first step towards evaluating σ ± is the following result which is similar to Theorem 2.1 in [19]. Its proof is provided in Section 3.2.
Theorem 1. Let F ∈ F s and assume Hypotheses 1 -3. There are t − * , t + * > 0 (both defined by thermodynamic formalism, see below) such that The numbers t ± * are uniquely determined as positive zeros of the pressure functions (To be more precise, we mean the pressure function of the topological Markov chain that encodes S.) Indeed, p ± (0) = 0 and (p ± ) (0) = λ µac (ϕ ± ) < 0 by Hypothesis 2 (see e.g. [25,17]), so that the convex functions p − and p + have unique positive zeros provided sup s>0 p + (s) > 0 and sup s>0 p − (s) > 0. This latter property is guaranteed by the existence of the measures ν − and ν + from Hypothesis 2.
Remark 2.5. This formula for t ± * is in good agreement with the formula for the corresponding quantity η in [22,23]. As the authors work with a diffusion approximation and restrict to negative λ µac (ϕ ± ) close to zero, t ± * can be approximated by the positive zero of the log-Laplace transform of a normal distribution with mean λ µac (ϕ ± ) = (p ± ) (0) and variance where we used the convention σ 2 = 2D as in [22,23]. Numerical studies in [22,23] and later in [26] confirm this approximate formula for the exponent.
Theorem 2. Let F ∈ F s and assume Hypotheses 1 -3. For any Gibbs measure ν ∈ P e (S) The same formula holds for general Gibbs measures ν ∈ P e (S), except possibly in the case Remark 2.6. a) The requirement that ν ∈ P e (S) is a Gibbs measure can be considerably relaxed, see [19,Remark 5].

The uncertainty exponent
Let F ∈ F s be as before. As in [13,22,23] we define the uncertainty exponent of F in the following way: Choose x ∈ int(J) and denote by Π ,x the probability that two points chosen at random with distance at most on the line I × {x} belong to different basins. More formally, Theorem 3. Let F ∈ F s and assume Hypotheses 1 -3. For every x ∈ int(J) we have where φ is determined by the pressure function Details on φ are provided in Equations (34) and (37) in Section 3.4.
Remark 2.7. One cannot expect this estimate to be sharp, because the asymptotics of Π ,x depend crucially on the large deviations behaviour of the time a trajectory needs to enter the final regime, when it is attracted to either ϕ − or ϕ + . But this is governed (at best) by λ µac (ϕ c ), an exponent of which we do not know more than the bounds 0 < λ µac (ϕ c ) < min{−λ µac (ϕ − ), −λ µac (ϕ + )} (because of negative Schwarzian derivative). As this is the exponent of a measure supported by the graph of the highly discontinuous function ϕ c , it is unlikely to be characterizable by thermodynamic quantities related to the base transformation. This is only one reason why it would be difficult to compare our φ to the approximate expression for φ derived in [22,23] or to the numerical results provided in for φ provided in [22,23,26]. Further issues are that the authors of [22,23] treat riddled basins, which can be modeled by a single random walk, while we need for the intermingled case two random walks (one for each attractor) that are moreover negatively correlated.
We return to this in Remark 3.10 after the proof of Theorem 3, where we derive an approximate formula for φ that yields a value twice as large as that used in [22,23,26]. This might explain why not only in [22, Figure 13], but also in [26, Figure 6b (small κ)], the numerically observed values for φ are roughly twice as large as the values suggested by the formula used there.

Distortion estimates
Remark 3.1. Denote by U n (ω) the family of all interval neighbourhoods U of ω ∈ I such that S n |U : U → S n U is a diffeomorphism. It is a well known fact (e.g. [7]) that under Hypotheses 1 and 2 there is a distortion constant D 1 such that for all n > 0, all ω ∈ I, all U ∈ U n (ω) and [24,Lemma 2.6] The following lemma is an immediate consequence of (8).
Another consequence is: and if h : | for all z ∈ S n (U ) (and similarly for ϕ + ).
Denote κ k := |ϑ| (f k ω ) (x) e 2kδ and observe that κ k 1 by the assumptions of the lemma. Then we have in view of (8) Hence, for k = n 0 , . . . , n, For k = n 0 , . . . , n it follows inductively that D|H k | α Dκ α k e −kδα De −n 0 δα < δ (recall that κ k 1). Hence |H n | κ n e −nδ = |ϑ| (f n ω ) (x) e nδ , which is the upper estimate in (10). For the lower estimate observe that the reasoning leading to (11) also yields and that D|H i | α < δ for i = 0, . . . , n − 1, as we just showed. Remark 3.5. As the proof of this lemma did not make use of negative Schwarzian derivative, it applies as well to the inverse branches (f n ω ) −1 .

Proof of Theorem 1
We follow closely [19, Sections 4.1 and 7]: For t ∈ R denote by L ± t the transfer operators and let ρ(L ± t ) be its spectral radius. Then p ± (t) = log ρ(L ± t ), and this is a strictly convex differentiable function of t, see e.g. [25].
We prove only the identity for the limit that evaluates to t − * , the other one is proved in the same way. So fix any t ∈ (0, t − * ) and choose δ > 0 such that ρ(L − t )e 4tδ < 1. There is a constant C = C t,δ > 0 such that Lemma 3.6. Let t ∈ (0, t − * ) and δ > 0 be as chosen above. Then Ce −ntδ ϑ t for all ϑ > 0 and n 1.
Proof. As t > 0, we have the usual Cramér type estimate for each n 1: where we used (13) for the last inequality.
For the upper bound we need another preparatory lemma. Proof. Fix z ∈ J, n 1 and an interval I ⊂ I, and denote by U the family of all maximal (Such an interval need not exist for each β > 0.) Denote for the moment the inverse of S n | U by ρ = ρ U : S n (U ) → U . Then, for each U ∈ U, where we used (9) for the second inequality and the monotonicity of f n ρ(ω) for the last one. In view of the distortion bound (7) this implies For the last inequality we used that S n (U ) contains at least one Markov interval when n n 0 and n 0 is sufficiently large. Choosing n 0 = n 0 ( ) even larger, if necessary, the at most two U ∈ U which are not fully contained in I are disjoint toĨ. Hence It remains to prove that γ is strictly positive: Let ϑ := e −D β and suppose for a contradiction that there is some Markov interval K of S such that ϕ c (ω) ϕ − +ϑ for all ω in a full measure subset K 0 of K. As S is a mixing Markov map, there is k ∈ N such that m(I\S k (K 0 )) = 0. Hence, for m-a.e.ω ∈ I there is ω ∈ K 0 such that ϕ c (ω) = f k ω (ϕ c (ω )) inf ω∈I f k ω (ϑ) > ϕ − , which is incompatible with Proposition 2.2.
In order to relate the expression f n ω (z) − ϕ − to (f n ω ) (ϕ − ), we invoke a rather immediate consequence of the fact that maps with negative Schwarzian derivative satisfy a variant of the one-sided Koebe principle [8, Section IV.1, Property 4 and Corollary 2]: for all ω ∈ I, z ∈ J and n 1.

Proof of Theorem 2
Let ν ∈ P e (S) be a Gibbs measure. Then ν-a.a. points ω ∈ I are regular in the following sense: The limits exist, and there are sequences of integers n 1 < n 2 < . . . and of reals 1 > 2 > · · · 0 such that the symmetric k -neighbourhoods V k (ω) of ω satisfy and such that it is enough to evaluate along the sequence k 0, see [19, Remark 5(a) and beginning of Section 5].
In a first step we reduce the evaluation of the area quotient in (16) to the evaluation of quotients of one-dimensional measures.
Proof. The lower bound follows from the inclusion U k (ω, , which holds for x < ϕ + and sufficiently small k > 0. In view of the distortion bound (7) for S we have There it is proved that Therefore, observing also that inf k m(S n k (V k (ω))) > 0, we have for where we used Theorem 1 (in its strengthened form (15)) for the equality. In a similar way one derives the lower bound and we note that corresponding statements hold for σ − (ω, x).

Proof of Theorem 3
As S is an irreducible and aperiodic Markov map, there are s, q ∈ N such that S s (K) = I (possibly except for the endpoints) for all Markov intervals K ∈ K, and such that each point in I has at most q preimages under S s .
Denote byñ (ω) the smallest integer such that |(S n ) (ω)| > e D −1 . We denote the maximal monotonicity and continuity interval of Sñ (ω) around ω by Z (ω). Observe that the distortion estimate (7) guarantees that the length of each Z (ω) is between e −2D and . In particular, Z (ω) ⊆ B (ω) where B (ω) denotes the -neighbourhood of ω in I. From each collection (Z (ω) : ω ∈ I) one can extract a Moran cover U of I, that is a disjoint (modulo endpoints) collection of sets Z (ω) covering all of I (modulo endpoints), see e.g. [27,Section 13]. By construction, each Z ∈ U is a maximal monotonicity interval of some S n Z .
With these conventions we have and similarly Observing that m(Z) = 2 for all Z ∈ U , we conclude where n (ω) = n Z + s when ω ∈ Z ∈ U .
In the next step we compare (f As all branches f ω have negative Schwarzian derivative, we have For z ∈ R and > 0 define ζ (z) = min 0, z − c | log | + log κ 2 | log | . Then the last inequality can be rewritten as Similarly, Therefore, observing (22) and the monotonicuty of Φ ± , We determine the asymptotics of this integral when → 0 using Varadhan's integral lemma. This needs some preparation: Consider the three-variate process We will determine the large deviations of this process and related ones. As log f ω (ϕ ± ) and log |S (ω)| are Hölder continuous on each set I i ×J, one can use transfer operators to prove the existence, convexity and smoothness of the asymptotic log-Laplace transform Indeed, ψ coincides with the pressure function (6) defined in Theorem 3. 3 Here are some particular values of ψ that help to understand its geometry: Negative Schwarzian derivative of the branches implies furthermore: This is a direct consequence of Propossition 1.4, because the partial derivatives are precisely the exponents for the equilibrium state associated with (λ − , λ + , λ S ). In particular, ψ is unbounded below and above. Let ψ * (x) := sup λ∈R 3 ( λ, x − ψ(λ)) be the Legendre-Fenchel transform of ψ. The Gärtner-Ellis theorem guarantees that the large deviations principle with rate n and with good convex rate function ψ * holds for the sequence (Y n ) n∈N [10, Theorem 2.3.6(c)]. Even more holds in the present situation: For t ∈ R denote Y n (t) := t Y [nt] . Fix m ∈ N and 0 = t 0 < t 1 < · · · < t m 1. Then the processỸ n := (Y n (t 1 ), Y n (t 2 ) − Y n (t 1 ), . . . , Y n (t m ) − Y n (t m−1 )) satisfies the LDP in (R 3 ) m with good rate function This is exactly condition (A-1) from [9]. Observe that ψ * m is the Legendre-Fenchel transform of ψ m (λ) : We show in Lemma 3.11 at the end of this section that ψ m is the asymptotic log-Laplace transform of (Ỹ n ) n∈N and that it is convex and smooth. Then the Gärtner-Ellis theorem guarantees the LDP as claimed in (31).
As 0 < inf ω∈I log |S (ω)| sup ω∈I log |S (ω)| < ∞, one can apply, as a next step, the time change techniques from [11,: Theorem 1 from [9] together with Theorem 1 from [12] allow to apply Theorem 4 (and the subsequent further discussion) from [11] and to conclude 4 5 : The process (n /| log | · Y n ) >0 satisfies the large deviation principle with rate | log | and good convex rate function Next we apply Varadhan's lemma [10,Theorem 4.3.1] to the process (n /| log | · Y n ) and the continuous function in order to evaluate namely: The following lemma, whose proof is deferred to the end of this section, relates this to the estimate of (25) we are looking for: Lemma 3.9. Let c, κ > 0. Then for each C > 0, As log f ω (x) and log |S (ω)| are uniformly bounded in ω and x, the process | log | · H(n /| log | · Y n (ω) (ω)) is uniformly bounded, so that (33) together with Lemma 3.9 implies: 4 Formally only the case of bivariate processes (Yn)n∈N is treated in [11]. A look at that paper, hovever, reveals that the results from [9] and [12] referred to in [11] apply to our situation as well and that the proof of Theorem 4 in [11] carries over without any changes. 5 A more attentive look at [11] reveals that the results from that paper apply immediately only to the process (ñ /| log | · Yn ) >0. However, the distortion property (7) and the uniform expansion of S immediately imply that the differenceñ (ω) − n (ω) is bounded uniformly in and in ω. In particular, both processes are exponentially equivalent so that one can be replaced by the other without changing the large deviations behaviour. A related question is also discussed after Theorem 2 of [11].