Singular dynamics of a q -difference Painlevé equation in its initial-value space
aa r X i v : . [ n li n . S I] A ug Singular dynamics of a q -difference Painlev´e equation in itsinitial-value space N. Joshi and S.B. Lobb
Abstract.
We construct the initial-value space of a q -discrete first Painlev´eequation explicitly and describe the behaviours of its solutions w ( n ) in thisspace as n → ∞ , with particular attention paid to neighbourhoods of excep-tional lines and irreducible components of the anti-canonical divisor. Theseresults show that trajectories starting in domains bounded away from the ori-gin in initial value space are repelled away from such singular lines. However,the dynamical behaviours in neighbourhoods containing the origin are compli-cated by the merger of two simple base points at the origin in the limit. Weshow that these lead to a saddle-point-type behaviour in a punctured neigh-bourhood of the origin. Dedicated to Rodney J. Baxter on the occasion of his 75th birthday.
1. Introduction
The discrete Painlev´e equations are known to have a simple, beautiful geometricstructure [ ], but very little is known about their general solutions. Motivated byBoutroux’s famous study [ ] of the first Painlev´e equation P I : y tt = 6 y − t , as | t | → ∞ , we consider the limit | ξ | → ∞ of its q -discrete version q P I : w (cid:0) q ξ (cid:1) w (cid:0) ξ/q (cid:1) = w ( ξ ) − /ξw ( ξ ) , (1.1)where we assume | q | >
2, 18 ] and multi-dimensionally consistent integrable lattice equations[
1, 14 ].Starting at a given point in C , repeated iteration of the equation on complexspirals (for q ∈ C ) leads to a sequence of solution values of a discrete Painlev´eequation, but no information comparable to that found by Boutroux about globalbehaviours of solutions of P I is available in the plane of the independent variable ξ ∈ C . Instead, we consider solution trajectories in the space of initial values,which is compactified by embedding in P × P and regularized by resolving allsingularities . We denote the resulting initial value space by S . Mathematics Subject Classification. Note that P × P is the two-complex-dimensional space parametrized by pairs of homoge-neous coordinates ([ x : x ] , [ y : y ])= ([ λ x : λ x ] , [ µ y : µ y ]) for all non-zero ( x j , y j ) ∈ C , j = 1 ,
2, and non-zero λ , µ ∈ C . More precisely, writing ξ = q n ξ , w n = w ( ξ ), w n +1 = w ( qξ ), w n − = w ( ξ/q ),we study the solutions in S = ∪ n ∈ C S n , where the fibre S n is locally described byaffine coordinates ( w n , w n − ) and resolved by blowing up 8 base points. Equation(1.1) is symmetric under w n +1 ↔ w n − . In the remainder of the paper, we studythe equivalent forward and backward systems (cid:18) uv (cid:19) = qv − tquv u − tu v , (cid:18) uv (cid:19) = v − qtuv u − q tu v (1.2)where t := 1 /ξ , u n := w n , v n := w n − . (Note that bar now denotes t t/q .)To our knowledge, the term “discrete Painlev´e equation” was proposed byFokas, Its and Kitaev [ ] for another discrete version of P I , which arises fromtransformations of the classical fourth Painlev´e equation. The latter equation is anadditive-type discrete equation, where coefficient functions are linear in n insteadof exponential, i.e., ξ = ξ q n , as in Equation (1.1).Many discrete Painlev´e equations have now been identified. Following Okamoto’sconstruction for the classical Painlev´e equations [ ], Sakai [ ] showed how toconstruct initial value spaces of discrete Painlev´e equations as rational surfaces ob-tained by 9-point resolution of P or equivalently, 8-point resolution of P × P . Inparticular, Sakai showed that the initial value space associated with Equation (1.2)is described by the affine Weyl group A (1)7 .Qualitative information about solutions of (1.2) is valuable because Nishioka[ ] showed that they are higher transcendental functions, which cannot be ex-pressed in terms of basic or q -special functions or in terms of solutions of first-orderor linear second-order difference equations. We describe solutions by studying thelimit | ξ | → ∞ , or equivalently | t | → In the limit | t | →
0, the system (1.2) becomes (cid:18) uv (cid:19) = v uv ! , (cid:18) uv (cid:19) = uvu ! (1.3)or, equivalently w w w = 1, for either w = u or v . This map is one of the periodiccases of the QRT mappings studied by Tsuda [ ], which is periodic with period3, for any initial value. The autonomous system has an invariant given by K ( u, v ) = u v + u + vuv (1.4)i.e., K ( u, v ) − K ( u, v ) = 0 = K ( u, v ) − K ( u, v ) when u and v satisfy Equations(1.3). The singularities of the invariant curve (i.e., points where the gradient of K ( u, v ) vanishes) are given by ( u, v ) = ( ω, ω ) where ω = 1, and ( u, v ) = (0 , ∞ ),( u, v ) = ( ∞ , K = 3 ω . Therefore, thereare 3 singular fibres in the pencil of curves (1.4).It is interesting to note that the so called tronqu´ee solutions identified byBoutroux for the first Painlev´e equation also approach double zeroes of the lat-ter’s autonomous invariant. In this sense, the solutions that approach the equilib-rium values ( u, v ) = ( ω, ω ) of the autonomous q P I act as discrete analogues of thetronqu´ee solutions. In this paper, however, we focus on the generic solutions of thedynamical system (1.2). INGULAR DYNAMICS OF A q -DIFFERENCE PAINLEV´E EQUATION 3 Fixed points are important for studying limiting solu-tions. The cases where the solutions of Equation (1.1) approach a steady state as n → + ∞ have been considered in [
12, 16 ]. Joshi [ ] showed that there existsa true solution satisfying | w | → n → + ∞ , which is asymptotic to a diver-gent asymptotic series expansion in powers of 1 /ξ , but is not a singularity of thelimiting invariant K . This vanishing, unstable solution was called the quicksilver solution in [ ] and its trajectory lies in a neighbourhood of the origin in S , whichis punctured at two base points (called b , b below) that approach the origin. (SeeFigure 3.2 for a typical such neighbourhood.) Our results show that the dynamicsof Equations (1.2) in this punctured neighbourhood resemble that near a saddlepoint in the initial value space S . The structure of the paper is as follows. In Sec-tion 2 we construct the initial-value space S and provide explicit coordinate chartsto describe it completely; key points are illustrated here, while full details are givenin Appendix A. We also describe the irreducible components of its anti-canonicaldivisor, with explicit details given in Appendix B. In Section 3.2 we examine theaction of the mapping on exceptional lines, with details given in Appendix C. Wethen describe approximate solutions near exceptional lines, with particular atten-tion paid to a neighbourhood of the origin in Section 3.3. A conclusion and summaryare provided in Section 4.
2. The Initial Value Space
In this section, we describe the major features of the initial value space S ,obtained by resolution of singularities of the dynamical system (1.2). The detailsof the calculations are provided in Appendices A and B. Consider the dynamical sys-tem (1.2) in P × P , which is covered by four charts described in Figure 2.1.Singularities occur where the numerator and denominator vanish simultaneously inthe definition of at least one of u , v , u , or v . We call these base points , following [ ].We resolve these in the standard way to obtain successive charts ( u ij , v ij ), where i corresponds to the number of blow-ups and j corresponds to a chart indicated inFigure 2.1.The definition of v and u in Equations (1.2) show that there are two base pointsin Chart 1: b : (cid:18) u v (cid:19) = (cid:18) t (cid:19) , b : (cid:18) u v (cid:19) = (cid:18) qt (cid:19) , (2.1)while there is one base point in each of Charts 2 and 3 respectively: b : (cid:18) u v (cid:19) = (cid:18) (cid:19) , b : (cid:18) u v (cid:19) = (cid:18) (cid:19) . (2.2)The explicit resolutions of these base points are provided in Appendix A. We recall here that the standard operation of resolution or blowing up a base point at( α, β ) means replacing ( u, v ) by two new coordinate charts ˜ u = ( u − α ) / ( v − β ) , ˜ v = v − β , orˆ u = ( u − α ) , ˆ v = ( v − β ) / ( u − α ). This has the effect of replacing the base point by an exceptional line. Each resolution lowers the self-intersection number of the line containing the base point byunity. In the chart (˜ u, ˜ v ), this line is defined by ˜ v = 0 and parametrized by ˜ u , whilst in chart (ˆ u, ˆ v ),the line is defined by ˆ u = 0 and parameterized by ˆ v . For this and other standard constructions ofalgebraic geometry, the reader is referred to [ ]. N. JOSHI AND S.B. LOBB
Chart 1 : (cid:18) u v (cid:19) = (cid:18) uv (cid:19) v u /v u Chart 3 : (cid:18) u v (cid:19) = (cid:18) u /v (cid:19) /v /u Chart 4 : (cid:18) u v (cid:19) = (cid:18) /u /v (cid:19) v /u Chart 2 : (cid:18) u v (cid:19) = (cid:18) /uv (cid:19) Figure 2.1.
The four coordinate charts of P × P The resolution of b and b lead to no further singularities (see § A.1-A.2).However, the resolutions of b and b lead to four further base points, which aregiven in terms of coordinate charts defined in § A.3-A.4 by b : (cid:18) u v (cid:19) = (cid:18) (cid:19) , b : (cid:18) u v (cid:19) = (cid:18) (cid:19) , (2.3) b : (cid:18) u v (cid:19) = (cid:18) − q (cid:19) , b : (cid:18) u v (cid:19) = (cid:18) − q (cid:19) . (2.4)The resolution of b leads to b , b leads to b , while those of b and b lead to b and b respectively. The details can be found in § A.3-A.8. The exceptional lines E j replacing each base point b j , j = 1 , . . . , H u denotes a line where u is constant and H v denotes a line where v isconstant. The equivalence classes of lines H u , H v , and E i for i = 1 , .., S ) [ ]. This is equipped with asymmetric bilinear form ( · , · ) on elements of Pic( S ) given by(1) ( H u , H u ) = ( H v , H v ) = ( H u , E i ) = ( H v , E i ) = 0,(2) ( H u , H v ) = 1,(3) ( E i , E j ) = − δ ij . INGULAR DYNAMICS OF A q -DIFFERENCE PAINLEV´E EQUATION 5 H u − E − E H u − E − E H v − E − E H v − E − E E E E − E E − E E E − E E − E E Figure 2.2.
The rational surface S obtained from the resolutionof singularities in Equation (1.2) indicating proper transforms ofrelevant curves.From the resolution described above (see Figure 2.2), it follows that the nodalcurves with self-intersection -2 are D = H v − E − E , D = H v − E − E ,D = E − E , D = E − E ,D = E − E , D = E − E ,D = H u − E − E , D = H u − E − E . These form irreducible components of the anti-canonical divisor, defined by − K S = 2 H u +2 H v − P i =1 E i . By the definitions above, we have − K S = P j =1 D j .From the intersection rules on the Picard group given above, it follows that( D , D ) = 1 = ( D , D ), while ( D , D j ) = 0 for j = 3 , , , ,
7. Further detailsabout the intersections of D j and the corresponding generalized Cartan matrix areprovided in Appendix B. By representing each D j as a node and connecting twonodes D i , D j by a simple edge if ( D i , D j ) = 0, we obtain the Dynkin diagramgiven in Figure 2.3, which represents the affine Weyl group A (1)7 . By consideringits orthogonal complement in Pic( S ), we also show explicitly in Appendix B thatEquation (1.2) has a symmetry group given by A (1)1 .
3. Dynamics
Let the forward iteration given in Equation (1.2) on S be denoted by ϕ . Westudy iterations of points near the components D j of the anti-canonical divisor in § E , E , E , E in § E and E in § N. JOSHI AND S.B. LOBB D D D D D D D D Figure 2.3.
Intersection diagram of { D j } j =1 , which is equivalentto the Dynkin diagram of A (1)7 . D i . The iteration ofthese components under ϕ is computed in Appendix B. For example, consider D = H u − E − E . Given t = t , a finite initial point near D lies near the line v = 0 (with u = t ), so that v ( t ) = δv ′ ( t ) + O ( δ ) for some finite v ′ ( t ),and small 0 < δ ≪
1. The results of Appendix B show that u ( t /q ) ∼ − u ( t ) t q ,v ( t /q ) ∼ δ qv ′ ( t ) t (cid:0) q + u ( t ) t (cid:1) ( u ( t ) − t ) , (3.1)which again lies near D after iterating forward 4 steps. Arguments similar to thesein Appendix B show that the set D , . . . , D is closed under the action of ϕ , thetime evolution of D j is periodic and, moreover, their iteration occurs in two orbitsof period four each, as shown in Figure 3.1. D D D D D D D D Figure 3.1.
Action of ϕ on divisors. E , E , E , E . As shown in Ap-pendix C, we have the following behaviour under forward iteration E → E → H u +2 H v − E − E − E − E − E , and E → E → H v − E , (3.2) INGULAR DYNAMICS OF A q -DIFFERENCE PAINLEV´E EQUATION 7 and under backward iteration we have H u − E ← E ← E , and 2 H u + H v − E − E − E − E − E ← E ← E . (3.3)This means that under forward and backward iterations of these exceptional lines,we are sent eventually to regular points. There are certain special cases: for ex-ample ( u , v ) = (0 ,
0) on E is mapped under forward iteration to the point( u , v )= ( − q ξ,
0) on E . This point is mapped through the chain (3.2) oncemore, before emerging into regular space. Here we consider dynamics in a neigh-bourhood of the origin in initial value space. Assume d r (0 ,
0) is a disk of radius r around the origin in S for sufficiently small r >
0. We assume that t lies in a smalldomain d r ( t ), around a given point t ≪
1, for sufficiently small r > U ⊂ d r (0 ,
0) lying in S , which is punctured at ( t,
0) and (0 , qt ) . Wedenote the small punctured domain around ( t,
0) by σ and that around (0 , qt ) by σ . An example of such a domain is drawn in Figure 3.2. σ σ r S Figure 3.2.
Punctured domain U The mapping of σ j , j = 1 , ϕ are obtained by the results in AppendixC. These show that the exceptional lines in each region act as repellers, and thatiterates move away from each region.In the domain U , for sufficiently small ( u, v ), we find u ∼ − u vt u ∼ − qtuv (3.4) v ∼ − tu v v ∼ − u vq t (3.5)These show that U is mapped to a domain with large values of v (i.e., towards theorigin in chart 3, which corresponds to ( u, v ) = (0 , ∞ )) by the forward map, whileit is mapped to a domain with large values of u (i.e., towards the origin in chart2, which corresponds to ( u, v ) = ( ∞ , v and finite u is attracted tothe origin, while values of u and v lying in U are repelled away to a domain withlarge values of u and finite v . That is, the origin is a saddle-point of the dynamicalsystem. This result is consistent with the origin being a repelling fixed point of theautonomous map, and with the quicksilver solution being an unstable solution, asshown in [ ]. N. JOSHI AND S.B. LOBB
Appendix B contains explicit calculationsof the mappings of the irreducible components D j of the anti-canonical divisor,and Appendix C contains explicit calculations of the mappings of the remainingexceptional lines.Based on these results, we can express the action of ϕ on elements of thestandard basis of the Picard group in matrix form: ϕ H u H v E E E E E E E E = − − − − − − − − − − − − − − − − − − − − − − − − − − − − H u H v E E E E E E E E (3.6)where ϕ : Pic( S ) → Pic( S ) is the action of forward iteration in n . Because allthe eigenvalues of this matrix are roots of unity, its algebraic entropy (see Bellonand Viallet [ ]) vanishes. This shows that the complexity of the dynamics of thissystem does not grow exponentially in the time step n .
4. Conclusions
In this paper, we explicitly constructed the initial value space of the q -discretefirst Painlev´e system (1.2) as t →
0, and for that of the scalar equation (1.1), in thelimit ξ → ∞ . We provided the coordinate charts of the space of initial values indetail, identifying exceptional lines and irreducible components of the anti-canonicaldivisor and, based on these, showed how to deduce qualitative information aboutthe dynamics in the limit.We found that the union of exceptional lines is a repeller for the flow; i.e.,if a solution is near an exceptional line at any given time, after one time step itis immediately mapped to a different region. This is analogous to the results forPainlev´e I found in [ ]. The set of the irreducible components of the anti-canonicaldivisor is invariant under the time flow and the dynamical system is periodic onthe level of each such component.Nevertheless, the solutions are neither simple or periodic. In particular, weshowed that the origin acts as a saddle point for the generic flow. The remainingfixed point solutions, which approach the singularities on three fibres of the au-tonomous pencil (1.4) (see § tronqu´ee solutions identified by Boutroux for the first Painlev´e equation, which also approachdouble points of the latter’s autonomous Hamiltonian.The dynamics of most other q -discrete Painlev´e equations in limits when t → t → ∞ remain unknown. In particular, it would be useful to study the q -discretePainlev´e equations known as q P II and q P III , which have q P I as a degenerate limit.Finally, since these q -discrete Painlev´e equations arise as reductions of Yang-Baxter maps, it would be interesting to relate their limiting behaviours to thephysical setting in which such maps play a role. INGULAR DYNAMICS OF A q -DIFFERENCE PAINLEV´E EQUATION 9 Appendix A. Blow up of base points
Here we provide explicit details of the process of resolution of each of the 8base points of q P I . A.1. Blow up of base point b . This base point arises in Chart 1 when weiterate the map forward.Define new coordinates (cid:18) u v (cid:19) = (cid:18) ( u − t ) /v v (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u v + tv (cid:19) . (A.1)In these coordinates, Equations (1.2) become u = ( u v + t ) (cid:0) qu − u v t − u v t − t (cid:1) qu v = u ( u v + t ) (A.2a)and u = v − qtv ( u v + t ) v = v ( u v + t ) (cid:0) v − q u v t − qt − q v t (cid:1) ( v − qt ) (A.2b)where the base point b is replaced by the exceptional line E defined by v = 0.There are no new base points in this chart.To look in the other chart, define new coordinates (cid:18) u v (cid:19) = (cid:18) u − tv / ( u − t ) (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u + tu v (cid:19) . (A.3)The equations in these coordinates become u = ( u + t ) v (cid:0) q − u v t − u v t − v t (cid:1) qv = 1( u + t ) v (A.4a)and u = u v − qtu ( u + t ) v v = u ( u + t ) v (cid:0) u v − q u v t − qt − q u v t (cid:1) ( u v − qt ) where the base point b is replaced by the exceptional line E defined by u = 0.There are no new base points in this chart. A.2. Blow up of base point b . Define new coordinates (cid:18) u v (cid:19) = (cid:18) u / ( v − qt ) v − qt (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u v v + qt (cid:19) . (A.5) The equations in these coordinates become u = u v ( v + qt ) (cid:0) qu v − qt − u v t − qu v t (cid:1) q ( u v − t ) v = u v − tu v ( v + qt ) (A.6a)and u = 1 u ( v + qt ) v = u ( v + qt ) (cid:0) − q u v t − q u v t − q u t (cid:1) (A.6b)where the base point b is replaced by the exceptional line E defined by v = 0.There are no new base points in this chart.To look in the other chart, define new coordinates (cid:18) u v (cid:19) = (cid:18) u ( v − qt ) /u (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u u v + qt (cid:19) . (A.7)The equations in these coordinates become u = u ( u v + qt ) (cid:0) qu − qt − u v t − qu t (cid:1) q ( u − t ) v = u − tu ( u v + qt ) (A.8a)and u = v ( u v + qt ) v = ( u v + qt ) (cid:0) v − q u v t − q u v t − q t (cid:1) v (A.8b)where E is defined by u = 0. There are no new base points in this chart. A.3. Blow up of base point b . Define new coordinates (cid:18) u v (cid:19) = (cid:18) u /v v (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u v v (cid:19) . (A.9)In terms of the original variables, this is (cid:18) u v (cid:19) = (cid:18) / ( uv ) v (cid:19) ⇒ (cid:18) uv (cid:19) = (cid:18) / ( u v ) v (cid:19) . (A.10)The equations in these coordinates become u = v (cid:0) qu − qu v t − t (cid:1) qu (1 − u v t ) v = u (1 − u v t ) (A.11a)and u = u ( v − qt ) v v = u v − q v t − qu tu ( v − qt ) (A.11b) INGULAR DYNAMICS OF A q -DIFFERENCE PAINLEV´E EQUATION 11 where the base point b is replaced by the exceptional line E defined by v = 0.There is a new base point at b : (cid:18) u v (cid:19) = (cid:18) (cid:19) . (A.12)To look in the other chart, define new coordinates (cid:18) u v (cid:19) = (cid:18) u v /u (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u u v (cid:19) . (A.13)In terms of the original variables, this is (cid:18) u v (cid:19) = (cid:18) /uuv (cid:19) ⇒ (cid:18) uv (cid:19) = (cid:18) /u u v (cid:19) . (A.14)The equations now become u = u v ( q − qu t − v t ) q (1 − u t ) ,v = 1 − u tv , (A.15a)and u = u v − qtu v ,v = v (cid:0) u v − q u v t − qt (cid:1) ( u v − qt ) . (A.15b)where E is now given by u = 0. There are no further base points in this chart. A.4. Blow up of base point b . Define new coordinates (cid:18) u v (cid:19) = (cid:18) u /v v (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u v v (cid:19) . (A.16)In terms of the original variables, this is (cid:18) u v (cid:19) = (cid:18) uv /v (cid:19) ⇒ (cid:18) uv (cid:19) = (cid:18) u v /v (cid:19) . (A.17)The equations in these coordinates become u = u (cid:0) qu v − qt − u v t (cid:1) q ( u v − t ) v = u v − tu v (A.18a)and u = 1 − qv tu v = u v (cid:0) − q u t − qv t (cid:1) (1 − qv t ) (A.18b)where b is replaced by the exceptional line E , which is defined by v = 0. Thereare no new base points in this chart.To look in the other chart, define new coordinates (cid:18) u v (cid:19) = (cid:18) u v /u (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u u v (cid:19) . (A.19) In terms of the original variables, this is (cid:18) u v (cid:19) = (cid:18) u / ( uv ) (cid:19) ⇒ (cid:18) uv (cid:19) = (cid:18) u / ( u v ) (cid:19) . (A.20)The equations now become u = qu v − qv t − u tqv ( u − t ) v = v ( u − t ) u (A.21a)and u = v (1 − qu v t ) v = u (cid:0) v − q t − qu v t (cid:1) v (1 − qu v t ) (A.21b)where E is defined by u = 0. There is a new base point in this chart at b : (cid:18) u v (cid:19) = (cid:18) (cid:19) . (A.22) A.5. Blow up of base point b . Define new coordinates (cid:18) u v (cid:19) = (cid:18) u /v v (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u v v (cid:19) . (A.23)In terms of the original variables, this is (cid:18) u v (cid:19) = (cid:18) / ( uv ) v (cid:19) ⇒ (cid:18) uv (cid:19) = (cid:18) / ( u v ) v (cid:19) . (A.24)The equations in these coordinates become u = qu v − qu v t − tqu (1 − u v t ) v = u v (cid:0) − u v t (cid:1) (A.25a)and u = u ( v − qt ) v = u v − q t − qu tu v ( v − qt ) (A.25b)where b is replaced by the exceptional line E defined by v = 0. There is a newbase point at b : (cid:18) u v (cid:19) = (cid:18) − q (cid:19) . (A.26)To look in the other chart, define new coordinates (cid:18) u v (cid:19) = (cid:18) u v /u (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u u v (cid:19) . (A.27)In terms of the original variables, this is (cid:18) u v (cid:19) = (cid:18) / ( uv ) uv (cid:19) ⇒ (cid:18) uv (cid:19) = (cid:18) / ( u v ) u v (cid:19) . (A.28) INGULAR DYNAMICS OF A q -DIFFERENCE PAINLEV´E EQUATION 13 The equations in these coordinates become u = v (cid:0) qu − qu v t − t (cid:1) q (1 − u v t ) v = u (cid:0) − u v t (cid:1) (A.29a)and u = u v − qtv v = u v − q v t − qtu ( u v − qt ) (A.29b)whereas E is now u = 0. There are no further base points appearing here. A.6. Blow up of base point b . Define new coordinates (cid:18) u v (cid:19) = (cid:18) u /v v (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u v v (cid:19) . (A.30)In terms of the original variables, this is (cid:18) u v (cid:19) = (cid:18) u v / ( uv ) (cid:19) ⇒ (cid:18) uv (cid:19) = (cid:18) u v / ( u v ) (cid:19) . (A.31)The equations in these new coordinates become u = qu v − qt − u tqv ( u v − t ) v = u v − tu (A.32a)and u = v (cid:0) − qu v t (cid:1) v = u (cid:0) v − q t − qu v t (cid:1) (1 − qu v t ) (A.32b)where the base point b is replaced by the exceptional line E defined by v = 0.There is a new base point at b : (cid:18) u v (cid:19) = (cid:18) − q (cid:19) . (A.33)To look in the other chart, define new coordinates (cid:18) u v (cid:19) = (cid:18) u v /u (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u u v (cid:19) . (A.34)In terms of the original variables, this is (cid:18) u v (cid:19) = (cid:18) u / ( u v ) (cid:19) ⇒ (cid:18) uv (cid:19) = (cid:18) u / ( u v ) (cid:19) . (A.35)The equations in these new coordinates become u = qu v − qv t − tqu v ( u − t ) v = v ( u − t ) (A.36a) and u = u v (cid:0) − qu v t (cid:1) v = (cid:0) u v − q t − qu v t (cid:1) v (1 − qu v t ) (A.36b)where E is now defined by u = 0. There are no further base points in this chart. A.7. Blow up of base point b . Define new coordinates (cid:18) u v (cid:19) = (cid:18) ( u + q ) /v v (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u v − qv (cid:19) . (A.37)In terms of the original variables, this is (cid:18) u v (cid:19) = (cid:18) (1 + quv ) / ( uv ) v (cid:19) ⇒ (cid:18) uv (cid:19) = (cid:18) / [( u v − q ) v ] v (cid:19) . (A.38)The equations now become u = q v − qu v + q v t − q u v t + qu v t + tq ( q − u v ) (1 + qv t − u v t ) v = v ( u v − q ) (cid:0) qv t − u v t (cid:1) , (A.39)and u = − ( q − u v ) ( v − qt ) v = − q + u v − qu t ( u v − q ) ( v − qt ) (A.40)where the base point b is now replaced by the exceptional line E defined by v = 0. There are no further base points in this chart.To look in the other chart, define new coordinates (cid:18) u v (cid:19) = (cid:18) u + qv / ( u + q ) (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u − qu v (cid:19) . (A.41)In terms of the original variables, this is (cid:18) u v (cid:19) = (cid:18) (1 + quv ) / ( uv ) uv / (1 + quv ) (cid:19) ⇒ (cid:18) uv (cid:19) = (cid:18) / [( u − q ) u v ] u v (cid:19) . (A.42)The equations in these coordinates become u = q u v − qu v + q u v t − q u v t + qu v t + tq ( q − u ) (1 + qu v t − u v t ) v = u v ( u − q ) (cid:0) qu v t − u v t (cid:1) (A.43)and u = − ( q − u ) ( u v − qt ) v = − qv + u v − qtv ( u − q ) ( u v − qt ) (A.44)where now E is defined by u = 0. There are no further base points appearinghere. INGULAR DYNAMICS OF A q -DIFFERENCE PAINLEV´E EQUATION 15 A.8. Blow up of base point b . Define new coordinates (cid:18) u v (cid:19) = (cid:18) ( u + q ) /v v (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u v − qv (cid:19) . (A.45)In terms of the original variables, this is (cid:18) u v (cid:19) = (cid:18) ( u v + q ) uv / ( uv ) (cid:19) ⇒ (cid:18) uv (cid:19) = (cid:18) ( u v − q ) v / [( u v − q ) v ] (cid:19) . (A.46)The equations become u = qu v − q − u tq ( qv − u v + t ) v = qv − u v + tq − u v t (A.47)and u = v (cid:0) q v t − qu v t (cid:1) v = ( u v − q ) (cid:0) v + q v t − q t − qu v t (cid:1) (1 + q v t − qu v t ) (A.48)where the base point b is now replaced by the exceptional line E defined by v = 0. There are no further base points in this chart.To look in the other chart, define new coordinates (cid:18) u v (cid:19) = (cid:18) u + qv / ( u + q ) (cid:19) ⇒ (cid:18) u v (cid:19) = (cid:18) u − qu v (cid:19) . (A.49)In terms of the original variables, this is (cid:18) u v (cid:19) = (cid:18) u v + q / [( u v + q ) uv ] (cid:19) ⇒ (cid:18) uv (cid:19) = (cid:18) ( u − q ) u v / [( u − q ) u v ] (cid:19) . (A.50)The equations in these coordinates become u = qu v − q v − tqv ( qu v − u v + t ) v = qu v − u v + tq − u (A.51)and u = u v (cid:0) q u v t − qu v t (cid:1) v = ( u − q ) (cid:0) u v + q u v t − q t − qu v t (cid:1) (1 + q u v t − qu v t ) (A.52)where E is now defined by u = 0. There are no further base points in this chart. Appendix B. Dynamics of solutions near divisors D i Defining A ij := 2( D i , D j ) / ( D j , D j ), we can express the intersection informa-tion between D i and D j in a generalised Cartan matrix A := ( A ij ) i,j =1 . For { D j } j =1 defined in § A = − − − − − − − − − − − − − − − − . B.1. Symmetries of the system.
We find the symmetries of the systemby constructing vectors orthogonal to the components D , . . . , D , and definingcorresponding actions which leave this set invariant. A vector α ∈ Pic( S ) is givenby the linear combination α := α u H u + α v H v + X i =1 α i E i (B.1)and its intersection with each D j is given by( α, D ) = α u + α + α , ( α, D ) = − α + α , ( α, D ) = − α + α , ( α, D ) = α v + α + α , ( α, D ) = α u + α + α , ( α, D ) = − α + α , ( α, D ) = − α + α , ( α, D ) = α v + α + α . For orthogonality to be satisfied, it follows that α = α = α =: a, (B.2) α = α = α =: b, (B.3)where a and b are arbitrary, and hence α u = − b, (B.4) α v = − a, (B.5) α = 2 b − a, (B.6) α = 2 a − b. (B.7)Thus α becomes α = a ( − H v − E + 2 E + E + E + E ) + b ( − H u + 2 E − E + E + E + E )=: aF + bF (B.8)where, if we define B i,j := 2( F i , F j ) / ( F j , F j ), we find the generalised Cartan matrix B := ( B ij ) i,j =1 given by B = (cid:18) − − (cid:19) . (B.9) INGULAR DYNAMICS OF A q -DIFFERENCE PAINLEV´E EQUATION 17 This leads to the Dynkin diagram shown in Figure B.1, which corresponds to theroot lattice A (1)1 . F F ∞ Figure B.1.
Dynkin diagram corresponding to the symmetry group.For each vector F i , we define an action on an element x ∈ Pic( X ) as follows: w F i ( x ) := x − x, F i )( F i , F i ) F i = x + 14 ( x, F i ) F i . (B.10)The action of w F on Pic( X ) is w F H u H v E E E E E E E E = / − − / − / − / − / / / / / / / − / − / − / / / − / / − / − / / / − / − / / − / / / − / − / − / / H u H v E E E E E E E E , (B.11)and the action of w F on Pic( X ) is w F H u H v E E E E E E E E = − / − / − / − /
21 0 0 1 / − / − / − / − / / / / / /
40 0 0 0 1 0 0 0 0 01 / − / / / − / − /
40 0 0 0 0 0 1 0 0 01 / − / / − / / − /
40 0 0 0 0 0 0 0 1 01 / − / / − / − / / H u H v E E E E E E E E . (B.12)Note that the action of w F i on each of the D i is the identity, i.e., w F i ( D j ) = D j ∀ i = 1 , j = 1 , .., , (B.13)and taking either of these actions to the second power gives the identity, so the w F i are indeed reflections; the span of these reflections forms the affine Weyl group W .Including a Dynkin automorphism σ : F ↔ F , we obtain the extended affine Weylgroup f W . Note that the action ( w F w F ) has the same effect on the elements ofthe Picard group as does the 4th power of the mapping (1.1). That is, if we denotethe operation of forward shift in the discrete time variable by ϕ , we have ϕ ( K ) = ( w F w F ) K, (B.14) for K an element of the Picard group.In the following, we examine behaviour of solutions near each irreducible com-ponent D , . . . , D of the anti-canonical divisor. We focus on the forward iterationhere, as the case of backward iteration is entirely analogous. B.2. Behaviour near D := H v =0 − E − E . The component D is essentiallythe coordinate axis v = v = 0, where u = t, u = 0. Suppose that at atime t we are near D , i.e., v ( t ) = v ( t ) is close to zero. Expanding theequations (1.2) for forward iteration in the chart ( u , v ), we find to leading orderfor v ( t ) = ǫ ≪ u ( t ) = 1 u ( t ) + O ( ǫ ) v ( t ) ∼ u ( t ) v ( t ) u ( t ) − t = O ( ǫ ) (B.15)while for v ( t ) = ǫ we have u ( t ) = u ( t ) + O ( ǫ ) v ( t ) ∼ v ( t ) u ( t ) (1 − u ( t ) t ) = O ( ǫ ) . (B.16)The image lies near the line v = 0, that is, near the component D . B.3. Behaviour near D . The component D is defined in local coordinatesby v = 0, u = 0, where u = 0. Suppose that at a time t we are near thisdivisor; i.e., v ( t ) , u ( t ) are close to zero. Expanding the equations (1.2) forforward iteration in the chart ( u , v ), we find to leading order for v ( t ) = ǫ ≪ u ( t ) ∼ v ( t )( qu ( t ) − t ) qu ( t ) = O ( ε ) v ( t ) = u ( t ) + O ( ε ) (B.17)and for u ( t ) = ε u ( t ) ∼ v ( t ) u ( t )( q − v ( t ) t ) q = O ( ε ) v ( t ) = 1 v ( t ) + O ( ε ) . (B.18)We see that the image near the line u = 0, near the component D . B.4. Behaviour near D . The component D is the line E − E is definedin local coordinates by v = 0 , u = 0, where u = − q . Suppose that at atime t we are near this component; i.e., v ( t ) = ε , or u ( t ) = ε where ε ≪ u ( t ) = − qt u ( t ) + O ( ε ) ,v ( t ) ∼ u ( t ) v ( t ) = O ( ε ) , (B.19)while, in the coordinates of the second chart we obtain u ( t ) = − v ( t ) qt + O ( ε ) ,v ( t ) ∼ u ( t ) = O ( ε ) . (B.20)The image lies near the line v = 0, or rather the component D . INGULAR DYNAMICS OF A q -DIFFERENCE PAINLEV´E EQUATION 19 B.5. Behaviour near D . The component D is given by u = u = 0,where v = 0. Suppose that at a time t we are near this component; i.e., u ( t ) = u ( t ) = ε ≪
1. Expanding the equations for u , v , we find inthe first chart u ( t ) = − qt v ( t ) + O ( ε ) v ( t ) ∼ u ( t ) v ( t ) = O ( ε ) (B.21)while in the coordinates of the second chart we obtain (cid:26) u ( t ) = − qt v ( t ) + O ( ε ) v ( t ) ∼ v ( t ) u ( t ) = O ( ε ) . (B.22)The image lies near the line v = 0, or rather the component D . B.6. Behaviour near D . The divisor D is essentially the coordinate axis v = v = 0, where u = 0. Suppose that at a time t we are near this divisor;i.e., v ( t ) = v ( t ) = ε is close to zero. Expanding the equations for u , v , wefind for | v | ≪ u ( t ) = − qt u ( t ) + O ( ε ) v ( t ) ∼ v ( t )( u ( t ) − t ) u ( t ) = O ( ε ) (B.23)while for | v ≪
1, we find u ( t ) = − qt u ( t ) + O ( ε ) v ( t ) ∼ u ( t ) v ( t )(1 − u ( t ) t ) + O ( ε ) (B.24)The image lies near the line v = 0, or rather the component D B.7. Behaviour near D . The component D is defined in local coordinatesby v = 0 , u = 0, where u = − q . Suppose that at a time t we are near thiscomponent; i.e., v ( t ), u ( t ) are close to zero. Expanding the equations forforward iteration in chart 2, we see that for | v | = ε ≪ u ( t ) ∼ − qv ( t ) t ( q + u ( t )) = O ( ε ) ,v ( t ) = − t u ( t ) + O ( ε ) , (B.25)and for | u | = ε ≪ u ( t ) ∼ − qv ( t ) u ( t ) t ( qv ( t ) + 1) = O ( ε ) ,v ( t ) = − v ( t ) t + O ( ε ) . (B.26)We see that this is near the line u = 0, or rather the component D . B.8. Behaviour near D . The component D is the line E − E that arisesunder the first blow-up at the point b in the region where u is finite, v is infinite.It is defined in local coordinates by v = 0, u = 0, where v = 0. Suppose thatat a time t we are near this component; i.e., v ( t ) = ε , u ( t ) = ε , where ε ≪ (cid:26) u ( t ) ∼ − t u ( t ) + O ( ε ) v ( t ) = − t u ( t ) v ( t ) = O ( ε ) (B.27)and in the coordinates of the second chart u ( t ) = − t v ( t ) + O ( ε ) ,v ( t ) ∼ − t u ( t ) v ( t ) = O ( ε ) . (B.28)The image lies near the line v = 0, or rather the component D . B.9. Behaviour near D . The component D is essentially the coordinateaxis u = u = 0 where v = qt, v = 0. Suppose that at a time t we arenear this component; i.e., u ( t ) = u ( t ) = ε ≪
1. Expanding the equations forforward iteration in ( u , v ), we see that in the coordinates of the first chart (cid:26) u ( t ) = − t v ( t ) + O ( ε ) ,v ( t ) ∼ u ( t ) = O ( ε ) (B.29)and in the coordinates of the second chart u ( t ) = − t v ( t ) + O ( ε ) v ( t ) ∼ u ( t ) = O ( ε ) . (B.30)The image lies near the line v = 0, or rather the component D . Appendix C. Mappings of remaining exceptional lines
In this appendix, we analyse the dynamics starting with initial values near theexceptional lines E , E , E , E . For simplicity and conciseness, we focus on thelocal results of iterating a neighbourhood near each exceptional line and, to do so,we assume that initial values in a neighbourhood of E i are analytic functions of t close to a point t i , for each i = 1 , , ,
8. Being analytic away from singularities,the birational map (1.2) maps a disk near an exceptional line to another disk ofnon-zero size.
C.1. Behaviour near E . The exceptional line E arises under blow-up ofthe point b in the region where u, v are finite; it is the line v = 0, or equivalently, u = 0. Expanding the equations (1.2) for forward and backward iteration, wefind in the coordinates of the first chart: u ( t ) ∼ t qu ( t ) − t qu ( t ) ,v ( t ) ∼ u ( t ) t , u ( t ) ∼ − qt (1 + qu ( t )) ,v ( t ) ∼ − v ( t ) q , (C.1) INGULAR DYNAMICS OF A q -DIFFERENCE PAINLEV´E EQUATION 21 and in the coordinates of the second chart: u ( t ) ∼ t v ( t )( q − v ( t ) t ) q ,v ( t ) ∼ t v ( t ) , u ( t ) ∼ − q ( q + v ( t )) t v ( t ) ,v ( t ) ∼ − v ( t ) u ( t ) q . (C.2)The image under the forward map is a curve in regular space. Under backwarditeration, we are mapped to near the line v = 0, i.e., to E . C.2. Behaviour near E . The exceptional line E is the line v = 0, u = 0.Suppose that at a time t we are near this exceptional line; i.e., v ( t ), u ( t ) areclose to zero. Expanding the equations for ( u , v ) expressed in terms of thecoordinates ( u , v ), we find u ( t ) ∼ − ( qu ( t ) + 1) t u ( t ) ,v ( t ) ∼ u ( t ) v ( t ) , u ( t ) ∼ t q u ( t ) ,v ( t ) ∼ qt u ( t )(1 − q t u ( t ) , (C.3)while in the coordinates of the second chart, we obtain u ( t ) ∼ − q + v ( t ) t ,v ( t ) ∼ u ( t ) , u ( t ) ∼ v ( t ) q t ,v ( t ) ∼ qt ( v ( t ) − q t ) v ( t ) . (C.4)We see that under forward iteration the image lies near the line v = 0, i.e., near E . Under backward iteration, this is a curve in regular space. C.3. Behaviour near E . E is the line v = 0 or equivalently u = 0.Suppose that at a time t we are near this exceptional line; i.e., v ( t ), u ( t ) areclose to zero. Expanding the equations for ( u , v ) in coordinates ( u , v ), wefind: u ( t ) ∼ − q + u ( t ) t q ,v ( t ) ∼ − qv ( t ) , (C.5)while the backward map for ( u, v ) in ( u , v ) gives u ( t ) ∼ q t ,v ( t ) ∼ − (1 + t u ( t )) q t , (C.6)Now consider these maps in in coordinates ( u , v ): u ( t ) ∼ − q v ( t ) + t q v ( t ) ,v ( t ) ∼ − qv ( t ) u ( t ) , (C.7) u ( t ) ∼ q t ,v ( t ) ∼ − ( t + v ( t )) q t v ( t ) . (C.8)Under forward iteration the image lies near the line v = 0, or rather E shiftedforward. Under backward iteration, we are mapped to near the line u = q t . C.4. Behaviour near E . The exceptional line E is the line v = 0, u = 0.Suppose that at a time t , we are near this exceptional line; i.e., v ( t ), u ( t )are close to zero. Expanding the forward and backward maps in chart 1 for small v ( t ), u ( t ), we find in coordinates ( u , v ) u ( t ) ∼ − ( q + t u ( t )) qt ,v ( t ) ∼ t q , u ( t ) ∼ − q (1 + qt u ) ,v ( t ) ∼ − q (1 + qt u ) v , (C.9)while in the coordinates of the second chart u ( t ) ∼ − q v ( t ) + 1) qt v ( t ) ,v ( t ) ∼ t q , u ( t ) ∼ − v ( t ) q ( q t + v ) ,v ( t ) ∼ q ( q t + v ) u . (C.10)We see that under forward iteration this is near the line v = t/q . Under backwarditeration, we are mapped to near the line v = 0, or rather E shifted backward. Acknowledgements
The authors would like to thank H. Dullin, A. Dzhamay, C. Lustri and T.Takenawa for informative discussions. The research reported in this paper wassupported by Australian Laureate Fellowship Grant
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