Rigorous bounds on the Hausdorff dimension of Feigenbaum attractors
RRigorous bounds on the Hausdorff dimension of Feigenbaumattractors
Andrew Burbanks, Andrew Osbaldestin, Judi ThurlbyFebruary 2021
Abstract
We calculate rigorous bounds on the Hausdorff dimension of the attractor at the accumulation of the period-doubling cascade for families of maps with quadratic, cubic, and quartic critical point. To do this, we expressthe attractors as the limit sets of appropriate Iterated Function Systems constructed using rigorous boundson the corresponding renormalisation fixed point functions. We use interval arithmetic with rigorous directedrounding modes to show that the respective dimensions lie in subintervals of the intervals (0 . , . . , . . , . In [4], Feigenbaum showed that the period-doubling attractor, A , can be represented as the limit set of anIterated Function System (IFS) defined by the two contractive maps:Ψ : x (cid:55)→ α − x, (1)Ψ : x (cid:55)→ g − ( α − x ) , (2)on the interval I = [ α − , g is the fixed point of Feigenbaum’s renormalisation operator [3] R : g ( x ) (cid:55)→ αg ( g ( α − x )) , with α = g (1) − . (3)Note that here, and in all that follows, g − denotes the inverse of the restriction of g to the interval J =[ g ( g (1)) , αg ( J ) = I (see figure 1a).Upper and lower bounds on the Hausdorff dimension of the attractor may be obtained from propertiesof the constituent maps of the IFS by using results described in [11] together with rigorous computations ofbounds on the renormalisation fixed-point, g . The operator R is defined on suitable sets of functions havinga critical point of (even) integer degree d at the origin. Note that the choice of α preserves the normalisation g (0) = 1. We note that both g and α depend on d ; in what follows we suppress this dependence in thenotation, except where needed, for clarity.The initial interval of the IFS is determined by the points along the forward orbit, (cid:0) g k (0) (cid:1) k ≥ = (cid:0) , , α − , g ( α − ) , α − , ... (cid:1) , (4)of the critical point. In fact, the endpoints of the subintervals in successive generations of the IFS comprisethis forward orbit. The functional form (equations 1,2) of the IFS is common across families of maps for anyinteger degree d ≥ H ( A ) ≈ .
538 045 143 580 549 911 671 415 567 , (5)1 a r X i v : . [ m a t h . D S ] F e b . − . . . . x − . − . − . − . . . . . . g ( x ) α − g ( g (1)) 1 α − α − g ( x )region of interest g k (0) for k = 1 , , − . − . . . . . . . x − . − . . . . . . . αxαg ( x ) (a) Fixed point function g ( x ) (b) Maps αx and αg ( x )Figure 1: (a) Feigenbaum’s renormalisation fixed point function (for the case d = 2) near the origin. The restrictionof g to the interval J = [ g ( g (1)) ,
1] where g ( g (1)) ≈ .
76 is plotted as a thick line (red in colour copy), and (b) αx (thin line, blue in colour copy) and αg ( x ) (thick line, red in colour copy) where αg ( x ) is plotted over [ g ( g (1)) , I = [ α − , α ≈ − .
5. The inverses of these maps form the IFS. where A d denotes the attractor corresponding to the universality class of maps with degree d critical point.The approach that we take, making use of the IFS along with rigorous bounds on the relevant renormalisationfixed points, delivers rigorous bounds on the dimension but, due to computational cost (and inevitablelooseness of bounds), is unable to match the precision of those numerical techniques.Figure 2 shows the subintervals corresponding to the first three generations of the IFS for the Feigenbaumattractor in the case d = 2. Following the method described in [11] we use bounds on the derivatives of the IFS functions to bound theHausdorff dimension of the attractor. At generation n of the IFS, we have 2 n maps, Ψ σ indexed by symbolsequences σ = a n − . . . a a ∈ { , } n , as shown in figure 2 (for n ≤ σ = Ψ a n − ...a a = Ψ a n − ◦ · · · ◦ Ψ a ◦ Ψ a . (6)The principal ingredient of the IFS approach to calculating dimensions is knowledge of the contractivity and‘coercivity’ factors of each map. For each Ψ σ , uniform contractivity and coercivity constants, c σ and d σ respectively, satisfy 0 < d σ ≤ | Ψ σ ( x ) − Ψ σ ( y ) || x − y | ≤ c σ < ∀ x, y ∈ I with x (cid:54) = y . (7)We calculate bounds on the derivatives of each map. Smoothness of the Ψ σ givesinf z ∈ I | Ψ (cid:48) σ ( z ) | ≤ | Ψ( x ) − Ψ( y ) || x − y | ≤ sup z ∈ I | Ψ (cid:48) σ ( z ) | ∀ x, y ∈ I with x (cid:54) = y , (8)and thus suitable constants, c σ and d σ , may be found by bounding the suprema and infima of the absolutevalues of the derivatives of the maps. Solving the corresponding partition function equations for s n and r n , (cid:88) σ c s n σ = 1 , (cid:88) σ d r n σ = 1 , (9)2 − . − . . . . . . . x I I I I I I I I I I I I I I I Figure 2: The first three generations of the IFS in the case d = 2, illustrating the iterative construction ofthe attractor. To aid understanding we emphasise the parts of the construction corresponding to the symbolsequence { } (blue/green in colour version) and sequence { } (red/orange) with solid lines (see section2). The dashed lines indicate a change of symbol after the first generation and the dotted lines indicate achange after the second generation. The map Ψ (blue/green), is order-reversing, and Ψ (red/orange) isorder-preserving.at each generation n then gives bounds r n ≤ dim H ( A ) ≤ s n , (10)on the Hausdorff dimension of the attractor, A , of the IFS.The derivatives of the IFS functions are given by:Ψ (cid:48) : x (cid:55)→ α − , (11)Ψ (cid:48) : x (cid:55)→ α − ( g − ) (cid:48) ( α − x ) , (12)defined on the interval I = [ α − , α = g (1) − .In order to form the partition function equations for the IFS we require bounds on contractivity overthe fundamental interval of the IFS. Clearly Ψ is monotonically decreasing, with Ψ (cid:48) constant. However,Ψ (cid:48) presents some challenges. We need to bound ( g − ) (cid:48) on the interval α − I . It might at first seem thata covering by sub-intervals is required, however we are able to establish rigorously for d = 2 , , (cid:48) is monotonic on I (illustrated in figure 3 for the case d = 2) and therefore we need only evaluate it at theendpoints of the IFS interval. We can therefore bound the contractivities and coercivities by evaluatingderivatives only at the endpoints of IFS subintervals.We prove the monotonicity of Ψ (cid:48) by reference to g (cid:48)(cid:48) (illustrated in figure 3c for the case d = 2).Since the second derivative of g is proven to be strictly negative, g (cid:48) must be strictly monotonicallydecreasing on the relevant interval [ g ( g (1)) ,
1] = Ψ ( I ) (note also that g (cid:48) is bounded away from 0 on thisinterval, establishing monotonicity of g itself). To calculate the derivative of the inverse map we use thestandard result that ( g − ) (cid:48) ( y ) = g (cid:48) ( x ) − with y = g ( x ). Thus we may calculate g (cid:48) ( x ) − instead of ( g − ) (cid:48) ( y ),provided that we have a means to solve y = g ( x ) for x in a rigorous manner. We thus prove that ( g − ) (cid:48) ismonotonically increasing on the interval α − I and thus Ψ (cid:48) is monotonically decreasing on I .We use a ball B , in a suitable space of analytic functions, previously proven to contain the renormalisationfixed point, g , as described in detail in [14]. The ball B is centred on a high-degree polynomial approximationto g . For the implementation of this method we need to address a technical issue when using bounds on g (in the space of functions) to compute bounds on its inverse g − . For example, given a ball of functions B .
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555 rigorous dimension boundsestimated dimension value (c) Rigorous bounds on g (cid:48)(cid:48) (f) Rigorous bounds on dimensionFigure 3: Construction of the IFS maps and resulting bounds on dimension for the case d = 2. The rectanglesprovide a coarse rigorous covering of the graphs of the corresponding functions: (a) the fixed point function, g ; (b) the first derivative of g , with the vertical dashed lines indicating the interval [ g ( g (1)) , g , demonstrating that g (cid:48)(cid:48) ( x ) < g ( g (1)) , andΨ ; and (e) rigorous bounds on the derivatives of Ψ and Ψ . (f) shows convergence of rigorous upper andlower bounds on dimension with increasing IFS generation.4ontaining an invertible function f , it is not sufficient, in order to bound f − , to find another ball C suchthat computing bounds on B ◦ C := { f ◦ h : f ∈ B, h ∈ C } results in a ball D ⊇ B ◦ C containing theidentity function. Looseness of the bounds computed for the composition means that this does not implythat f − ∈ C in general. To circumvent this, we instead make use of the function ball B (cid:51) g directly, togetherwith a rigorous root-finding method to bound an interval X = [ a, b ] (cid:51) x = g − ( y ) such that g ( X ) (cid:51) y forgiven y .As described earlier we need rigorous bounds on Ψ (cid:48) at the endpoints of IFS intervals, which raises a furtherobstacle: given a ball B = B ( f, r ) it is not possible, in general, to find a ball B (cid:48) such that B (cid:48) ⊇ { h (cid:48) : h ∈ B } ,as the corresponding derivative operator is unbounded. We note, however, that we require bounds valid onlyfor the renormalisation fixed point g itself. Thus, to overcome this problem, we may make use of the fixedpoint equation, g ( x ) = αg ( g ( α − x )) , (13)differentiating both sides to give g (cid:48) ( x ) = g (cid:48) ( g ( α − x )) g (cid:48) ( α − x ) , (14)and note that the composition of a derivative with a function, f (cid:48) ◦ g , is indeed bounded within a ball offunctions for suitably-bounded functions f, g , as shown in [8].When bounding the contractivities and coercivities of the IFS maps in successive generations, we use thechain rule: at generation n + 1 we have I σ n +1 σ =(Ψ σ n +1 ◦ Ψ σ )( I ) = Ψ σ n +1 ( I σ ) , (15)Ψ (cid:48) σ n +1 σ =(Ψ σ n +1 ◦ Ψ σ ) (cid:48) = (Ψ (cid:48) σ n +1 ◦ Ψ σ ) · Ψ (cid:48) σ . (16)For instance, at generation 2 we require the derivatives Ψ (cid:48) ij evaluated at the endpoints of I for i, j ∈ { , } .We have Ψ (cid:48) ij = (Ψ i ◦ Ψ j ) (cid:48) = (Ψ (cid:48) i ◦ Ψ j ) · Ψ (cid:48) j , (17)where Ψ (cid:48) j has already been computed at the endpoints of I at generation n = 1 and Ψ (cid:48) i must then be computedat the endpoints of the subinterval I j = Ψ j ( I ).The partition function equations are solved using rigorous interval arithmetic to give new upper and lowerbounds on the Hausdorff dimension of the attractor at each generation.Using a ball of functions, B , with (cid:96) radius 10 − centred on a polynomial of degree 40 that we previouslyproved contains the fixed point g [14], using the techniques of [8], after 20 generations of the IFS we obtainrigorous bounds on the Hausdorff dimension of the Feigenbaum attractor A :0 . < dim H ( A ) < . . (18)Figure 3f shows the convergence of the upper and lower bounds of the dimension. We adapt the procedure above to the attractors corresponding to the universality class of maps with odddegree d critical point exemplified by the prototypical family x n +1 = 1 − µ | x n | d . (19)In order to encode the absolute value, we modify the renormalisation operator, equation (3), defining separateanalytic functions for use with positive and negative operands: g ( x ) = (cid:40) g + ( x ) , x ≥ g − ( x ) , x < , (20)5 . − .
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64 rigorous dimension boundsestimated dimension value (c) Rigorous bounds on g (cid:48)(cid:48) (f) Rigorous bounds on dimensionFigure 4: Construction of the IFS maps and resulting bounds on dimension for the case d = 3. The rectanglesprovide a coarse rigorous covering of the graphs of the corresponding functions: (a) the fixed point function, g , for which g ( x ) = g + ( | x | ); (b) the first derivative of g , with the vertical dashed lines indicating the interval[ g ( g (1)) , g , demonstrating that g (cid:48)(cid:48) ( x ) < g ( g (1)) , and Ψ ; and (e) rigorous bounds on the derivatives of Ψ and Ψ . (f) shows convergenceof rigorous upper and lower bounds on dimension with increasing IFS generation.6here g − ( x ) = g + ( − x ) for x <
0. The renormalisation operator R becomes: R : (cid:40) g + ( x ) (cid:55)→ αg + ( g − ( α − x )) ,g − ( x ) (cid:55)→ αg + ( g + ( α − x )) , (21)where α = g (1) − = g + (1) − < . It suffices to work with just g + on a carefully chosen domain using amodified operator which with a slight abuse of notation we also refer to as R : R ( g + )( x ) : g + ( x ) (cid:55)→ αg + ( g + ( − α − x )) (22)where α = g + (1) − (note the presence of − α − , rather than α − in the inner bracket). The techniques usedfor d = 2 above may now be used for maps with odd degree critical point, for which the corresponding map g ( x ) = g + ( | x | ), is not itself analytic at the origin.The maps defining the corresponding IFS have the same functional form as for d = 2, equations (1) and(2), with α = α d being the relevant universal constant. We work with a ball of functions proven to contain thecorresponding fixed point, g = g d (in fact, g + ), of the renormalisation operator using the method describedin detail in [15].Figure 4 illustrates the results in the case d = 3. We establish rigorously that g (cid:48)(cid:48) is strictly negativeon the domain [ g ( g (1)) ,
1] (illustrated in figure 4c), thus proving that g (cid:48) is strictly monotonically decreasing(note that g (cid:48) is itself bounded away from zero, establishing monotonicity of g ). Thus, Ψ (cid:48) is monotonicallydecreasing on the domain [ α − ,
1] as required. Again, the significance of this is that we need only bound thederivatives at the endpoints of the intervals in order to bound contractivity of the IFS maps.Figure 4f shows the convergence of the rigorous upper and lower bounds of the Hausdorff dimension witheach step. The bounds produced verify the previously published numerical estimate for the dimension of theattractor for maps with degree 3 critical point [9]:dim H ( A ) ≈ . . Using a ball of functions, B , with (cid:96) radius 10 − , centred on a polynomial of degree 120 that we previouslyproved contains the fixed point, g [15], after 20 generations of the IFS we obtain rigorous bounds on theHausdorff dimension of the attractor for the case d = 3:0 . < dim H ( A ) < . . (23) Following the procedure set out for the dimension of attractors of maps with degree 2 critical point, we areable to calculate rigorous bounds on the degree 4 equivalent.We establish that g (cid:48) is strictly negative on the interval [ g ( g (1)) ,
1] (see figure 5b). However, it is necessaryto compute rigorous bounds for g (cid:48)(cid:48) on much smaller subintervals to establish that g (cid:48)(cid:48) is strictly negative onthe required interval, and therefore that g (cid:48) is strictly monotonically decreasing (see figure 5c). The argumentsthen follow as before to show that Ψ (cid:48) is monotonically decreasing on [ α − ,
1] and we are again able to usethe short cut of evaluating derivatives only at the endpoints of the interval. The bounds produced verify thepreviously published estimated value for the dimension for maps with a quartic critical point [12]:dim H ( A ) ≈ . . Using a ball of functions, B , with (cid:96) radius 10 − , centred on a polynomial of degree 160 that we previouslyproved contains the fixed point g [13], after 20 generations of the IFS we obtain rigorous bounds on theHausdorff dimension of the attractor in the case d = 4:0 . < dim H ( A ) < . . (24)Figure 5f shows the convergence of the rigorous upper and lower bounds of the Hausdorff dimension.7 . . . . x − . − . . . . . . g ( x ) g ( x ) − . . . . x − . − . − . . . . . . . Ψ (a) Rigorous bounds on g (d) Rigorous bounds on Ψ and Ψ − . . . . x − − − − − − g ( g (1)) 1 g ( x ) − . . . . x − . − . − . − . − . − . . . . Ψ (b) Rigorous bounds on g (cid:48) (e) Rigorous bounds on Ψ (cid:48) and Ψ (cid:48) − . . . . x − . − . − . − . − . . . . g ( g (1)) 1 g ( x )0 .
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70 rigorous dimension boundsestimated dimension value (c) Rigorous bounds on g (cid:48)(cid:48) (f) Rigorous bounds on dimensionFigure 5: Construction of the IFS maps and resulting bounds on dimension for the case d = 4. The rectanglesprovide a coarse rigorous covering of the graphs of the corresponding functions: (a) the fixed point function g ; (b) the first derivative of g , with the vertical dashed lines indicating the interval [ g ( g (1)) , g ; from the (coarse) covering shown it is not clear that g (cid:48)(cid:48) ( x ) < g ( g (1)) , g (cid:48)(cid:48) ( x ) <
0; (d) rigorous bounds on Ψ and Ψ ; (e)rigorous bounds on the derivatives of Ψ and Ψ ; (f) shows convergence of rigorous upper and lower boundson dimension with increasing IFS generation. 8 Conclusion
Using balls of functions previously proven to contain the renormalisation fixed points for universality classesof maps with degree 2, 3 and 4 critical points, we have calculated rigorous bounds on the Hausdorff dimensionsof the corresponding Feigenbaum attractors at the accumulations of period-doubling.In order to extend this calculation to maps with critical points of degrees higher than 4 a modified versionof the method presented above is needed. The short cut described in section 2, in which we evaluate thederivatives of Ψ and Ψ only at the endpoints of IFS sub-intervals, cannot be used in the case where thesecond derivative of g has mixed sign. Instead, one would either need to partition subintervals into monotonicsegments or to bound the suprema and infima on a rigorous covering of the fundamental interval by smallsub-intervals. As observed in section 4, obtaining tight bounds demands finer subdivisions, increasing thecomputational cost significantly.It is also more difficult computationally to bound the renormalisation fixed-point functions themselvesas the degree of the critical point is increased: obtaining sufficiently tight bounds requires working with ahigh truncation degree in a suitable space of analytic functions, while maintaining control over all higher-order terms. The domains of the corresponding power series need to be chosen carefully to ensure that therenormalisation operator is well-defined and differentiable (with compact derivative) on a suitable ball inorder that a contraction mapping argument may be used to establish that the ball contains a fixed point. References [1] P. Cvitanovic, A. Pikovsky,
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