Banach fixed point and flow approach for rough analysis

Banach fix ed point and flo w approac h f or rough anal y sis February 20, 2026 Y v ain Bruned 1 , Yingtong Hou 1 , Paul Laubie 1 , Zhicheng Zhu 2 1 U niv ersite de Lor raine, CNRS, IECL, F-54000 Nancy , France 2 School of Mathematics and Statis tics, Lanzhou U niv ersity , Lanzhou, 730000, China Email: yvain.bruned@univ- lorraine.fr yingtong.hou@univ- lorraine.fr paul.laubie@univ- lorraine.fr Zhicheng.Zhu_math@outlook.com . Abstract In this paper , w e sho w that the main alg ebraic assumption required to perform a fix ed point ar gument f or rough differential eq uations implies the alg ebraic assumption for the Bailleul flo w approach. This assumption requires that the rough path associated with the equation is given b y a Hopf algebra whose coproduct admits a cocycle and has a tree-like basis. W e sho w that the Hopf algebra of multi-indices does not satisfy the cocycle condition. This is a r igorous result on the impossibility , observed in practice, of per f or ming a fixed point argument f or multi-indices rough paths and multi-indices in Regularity Structures. Contents 1 Introduction 1 2 Rough paths and Hopf alg ebras 6 3 B-series representation of cocy cle-RDE solutions 12 4 log-ODE solution theory of cocy cle-RDEs 18 5 No alg ebraic fix ed point for multi-indices 22 1 Introduction Decorated trees ha v e become the main combinator ial set f or solving a lar g e class of singular stoc hastic par tial differential equations (SPDEs) via the theory of Regularity Structures in [ 29 , 8 , 17 , 5 ]. The ansatz f or the solution, a local e xpansion in ter ms of recentered stoc hastic iterated integrals follo w s the formalism giv en by B-series (see [ 5 , 11 ]). The B-series provides an efficient parametrisation. These decorated trees come with tw o Hopf alg ebras in cointeraction: One f or the recentering of integ rals Intr oduction 2 and the other for their renormalisation. The Hopf algebra f or the recenter ing can be vie wed as an e xtension of the so-called Butcher -Connes-Kreimer coproduct used f or composing B-series in numer ical analy sis [ 12 ] and encoding the nested subdiv erg ences in QFT [ 14 ]. A f e w y ears later , another combinator ial set called multi-indices emerg ed which is v er y efficient for describing scalar -v alued equations such as quasi-linear SPDEs (see [ 38 ]) in the context of R egularity Structures. Its recentering Hopf algebra is giv en in [ 34 ] and one obtains a recursiv e proof f or the stoc hastic es timates in [ 35 ] ; the proof f or the stochas tic estimates was firs t per f or med in [ 17 ] with a non-recursiv e proof. One of the dra wbacks of the multi-indices approach f or a while w as the lac k of a solution theory which is given in the conte xt of decorated trees via Banach fix ed point theorem. Such a path does not seem possible f or multi-indices. A solution theor y w as unco vered in [ 4 ] f or multi-indices rough paths introduced in [ 33 ]. The main idea is to use the flo w approach from Bailleul [ 1 ] that a v oids the use of a fixed point with the idea of iterating a numer ical scheme of the solution. This idea also appears in Davie ’ s solutions f or rough differential equations [ 19 ] and the log -ODE approach from [ 13 ]. These approaches differ from the fix ed point approach proposed in [ 36 , 26 , 25 ] (see also the book [ 21 ] f or an introduction on rough differential equations). In this w ork, w e consider a rough differential equation (RDE) giv en by d Y t = f ( Y t )d X t = X α ∈ [ d ] f α ( Y t ) dX α t (1.1) where α ∈ [ d ] = { 1 , ..., d } ( X is d -dimensional), Y is m -dimensional, and t ∈ [ 0 , T ] . The paths X α f or α ∈ { 1 , ..., d } are γ -H ¨ older f or γ ∈ ( 0 , 1 ) . W e assume that X can be lifted to a rough path X s,t described by a graded Hopf alg ebra ( H , µ, ∆ ) , meaning that one has X s,t ∈ H ∗ . W e consider the dual of the pre vious Hopf alg ebra denoted by ( H ∗ , ⋆, ∆ µ ) and w e suppose giv en a pair ing ⟨· , ·⟩ H on H . This Hopf alg ebra giv es Chen ’ s relation X s,t = X s,u ⋆ X u,t with ⋆ the graded dual of ∆ . One well-kno w e xample is the Butcher -Connes-Kreimer Hopf alg ebra ( H BCK , ⊙ , ∆ BCK ) f or branched rough paths [ 26 ], where ⊙ is the f orest product and ∆ BCK is the Butcher -Connes-Kreimer coproduct. The dual Hopf alg ebra is denoted b y ( H ∗ BCK , ⋆ BCK , ∆  ) where ⋆ BCK is the Grossman-Larson coproduct and ∆  is the unshuffle coproduct. The inner product f or H BCK is denoted b y ⟨· , ·⟩ . W e denote Υ f : H ∗ BCK → C ∞ , the usual elementary differential. Then, one can apply the flo w approach from [ 1 ] if the Hopf algebra satisfies some alg ebraic assumptions needed f or getting elementary differentials and tw o ke y identities giv en in [ 31 , Definition 1.1] that w e recall belo w Assump tion 1 (Algebraic flo w condition) W e suppose that either • Ther e exists a mor phism Λ ∗ : H ∗ → H ∗ BCK . Then, one can define elementar y differ entials as ¯ Υ f [ h ] = Υ f [ Λ ∗ ( h )] . (1.2) Intr oduction 3 • Ther e exists a morphism Φ : H ∗ BCK → H ∗ and elementar y differ entials ¯ Υ f suc h that f or every τ ∈ T ¯ Υ f [ Φ ( τ )] = Υ f [ τ ] . (1.3) F or u, v ∈ H , smoot h enough functions φ, ψ , one has ¯ Υ f [ u ⋆ v ] { φ } = ¯ Υ f [ u ] ◦ ¯ Υ f [ v ] { φ } , (1.4) ¯ Υ f [ ∆ µ u ] { φ ⊗ ψ } = ¯ Υ f [ u ] { φψ } , (1.5) wher e ¯ Υ f [ u ] { φ } is the ext ension of the elementar y differ entials to the ones composed with a smoo th enough function φ . These two identities w ere giv en in a g eneral conte xt in [ 31 ]. Let us mention that the y are v ery close in spir it to the Ne wtonian maps introduced in [ 32 ]. These identities are chec ked in [ 4 ] f or multi-indices. Other types of flow methods ha v e been applied in the context of singular SPDEs. Let us summar ise below the ideas behind the different flo w approaches: • Flo w along the time parameter: This is perf or med in [1] and [4]. It is not clear that such an approach will work in the conte xt of singular SPDEs with R egular ity Structures. • Flo w along a scale λ that appears in the kernel K λ in the mild f or mulation of the singular SPDEs. This is the flo w approach of P aw el Duch [ 20 ] inspired b y the P olchinski flo w [ 37 ] used in QFT f or renor malising F eynman diagrams. A solution theor y with multi-indices has been derived f or the generalised KPZ equation in [16]. • Flo w along a small parameter multipl ying the non-linear interaction of the singular SPDEs. This is the strategy used in [10]. Behind these various methods, one expects to find in the proofs, alg ebraic identities similar to Assumption 1 needed f or the solution theor y . In [ 23 ], the authors e xhibit an algebraic assumption that is sufficient f or per - f or ming a fix ed point argument in the context of rough differential equations. It is mainl y giv en by a cocy cle condition on the Hopf algebra H . W e recall it in the next assumption Assump tion 2 (Algebraic fix ed point) The Hopf alg ebra H is eq uipped wit h L α : H → H , a f amily of linear maps indexed by α ∈ [ d ] , homog eneous of degr ee one, satisfying the cocy cle condition ∆ L α ( u ) = (id ⊗ L α ) ∆ ( u ) + L α ( u ) ⊗ 1 . (1.6) Her e, 1 is the unit of H . Mor eov er , one has a pairing ⟨· , ·⟩ H : H ⊗ H → suc h that • L α ( 1 ) is a basis of H 1 . • F or every x, y ∈ H , one has ⟨ L α ( x ) , L β ( y ) ⟩ H = δ β α ⟨ x, y ⟩ H . (1.7) Intr oduction 4 From [ 14 , Theorem 2] on the univ ersal proper ty of H BCK , the Butcher -Connes- Kreimer Hopf algebra, identity (1.6), the e xistence of 1 -cocy cle implies the exis tence of a unique morphism Λ : H BCK → H sending B α + to L α . W e denote the dual map b y Λ ∗ : H ∗ → H ∗ BCK . Here, B α + takes a f orest and connects all the roots to a ne w root labelled b y α . No w , w e can state one of the main results of the present paper: Theorem 1.1 Assump tion 2 implies Assumption 1. The f act that some combinator ial sets possess tw o solution theor ies, fix ed point and flo w , has been kno wn since [ 1 ] (Shuffle Hopf Alg ebra) and [ 2 ] (Butc her -Connes- Kreimer Hopf alg ebra). Then, a natural q uestion is to find a counter -e xample to the opposite direction proposed by the previous theorem. W e are able to pro vide multi-indices as a counter -ex ample via the second main result of the present paper . Indeed, one can ha v e a mor phism betw een H ∗ BCK and H ∗ without (1.6). This fact has been observ ed f or multi-indices and it is explained in the proof of Proposition 5.2. Theorem 1.2 The multi-indices Hopf alg ebra does no t satisfy Assumption 2. This theorem is just a conseq uence of Theorem 5.1 whic h sho ws that the multi-indices do not satisfy (1.6) meaning that one cannot find a 1 -cocy cle L such that L ( 1 )  = 0 . The tw o abov e-mentioned theorems can be summarised in the f ollo wing w a y: Decor ated trees can be used for fixed point and flow approac hes whereas multi- indices wor ks only f or flo w approac hes. The second part of this claim is not r igorous as the Assumption 2 is a sufficient but not a necessar y assumption f or getting the fix ed point. But w e still g et a mathematical theorem on what has been obser v ed in practice (Only flo w approaches ha v e been applied to multi-indices), that is showing that the alg ebraic assumption f or the fix ed point is not satisfied. W e conclude this introduction with a couple of remarks bef ore summarising the content of the paper . Remar k 1.3 One can g et an analogue of Assumption 2 within the conte xt of R egular ity Structures. One e xpects to get a def ormed cocycle property in the sense that there e xists L α : H → H , a famil y of linear maps inde xed b y α ∈ N d +1 , homog eneous of degree one, satisfying the deformed cocy cle condition ∆ L α ( u ) = ( L α ⊗ id) ∆ ( u ) + X ℓ ∈ N d +1 X ℓ ℓ ! ⊗ L α + ℓ ( u ) . (1.8) where no w monomials of the f orm X ℓ are par t of H . A similar e xpression is giv en in [ 8 , Proposition 4.17] but used f or the recenter ing and it appears f or the first time in [ 29 , Section 8] e xpressed f or a different decorated trees basis. In the identity (1.8), one has an infinite sum but, in practice, it is finite as the decorations α + ℓ cannot be arbitrar il y larg e. One can also make sense of this infinite sum b y using a bigrading introduced in [ 8 , Section 2.3]. Then, one has to use this condition to sho w Intr oduction 5 the fix ed point via R egular ity Structures f ollo wing [ 5 ] where a B-ser ies f or malism is used f or the solution expansion. The identity (1.4) can be pro ved the same w ay as in [ 3 , Proposition 2.2] which giv es a mor phism proper ty f or ⋆ dual of ∆ . One has to use the explicit f or mula f or the product ⋆ giv en in [ 9 , Proposition 3.17]. The other identity (1.5) which corresponds to a Leibniz rule should be more straightf or w ard. The f act that R egular ity S tr uctures multi-indices will not satisfy Assumption 2 will follo w the same argument as f or rough paths multi-indices. Theref ore, one g ets a theoretical argument regarding the f ailure of finding a fixed point f or the multi-indices f or singular SPDEs. Remar k 1.4 The paper [ 23 ] only cov ers the case where the equation (1.1) takes v alues in R m . In [ 15 ], the authors consider equations taking v alues in homog e- neous space which requires non-commutative Hopf alg ebras. W e e xpect similar assumptions to work in this case but one has to chec k analyticall y that a variant of Assumption 2 pro vides a fix ed point in this context. The same is true f or the Assumption 1 as in [31] the manif old case is treated by using c har ts. Let us outline the paper by summarising the content of its sections. In Section 2, w e recall the der iv ation of the notion “rough paths”, and motivate the Hopf algebra and the one-cocy cle condition (1.6). Moreov er , w e br iefly recall impor tant concepts in Butcher -Connes-Kreimer Hopf algebra since the proof of our main theorem is based on the mor phism sending H ∗ to H ∗ BCK . In Section 3, w e use the univ ersal result from Theorem 3.1 to get the e xistence of a mor phism Λ : H BCK → H from the 1 -cocy cle of H . Then, w e are able in Proposition 3.2 to define a T -B-series from the B-ser ies on the trees. W e f ollo w b y pro ving a f or mula f or the composition law of T -B-series in Corollar y 3.6 that proceeds from the one on the trees in Proposition 3.4. W e finish the section by introducing controlled rough paths (see Definition 3.7) and w e sho w that the B-ser ies based on the rough path X is a controlled rough path in Corollary 3.9. In Section 4, w e pro v e that the elementar y differentials of T -B-series obtained in Section 3 satisfy the tw o ke y identities in Assumption 1. This is a consequence of Proposition 4.5 that uses deepl y the results on H BCK (see Propositions 4.2 and 4.3). W e conclude this section by the proof of Theorem 1.1. In Section 5, w e discuss the solution theor y of RDEs driven b y multi-indices rough paths. W e sho w in Theorem 5.1 that the multi-indices Hopf algebra does not satisfy the one-cocy cle condition which is a natural sufficient condition in implementing the fix ed point argument f or RDEs solutions. This someho w answers the question in the literature that why it seems to be impossible to find the fixed point f or multi-indices RDEs. W e finish the section b y sho wing that multi-indices satisfy Assumption 1 in Proposition 5.2. A ckno wledg ements Y . B., Y . H., P . L. g ratefull y ackno w ledge funding suppor t from the European Research Council (ER C) through the ER C Starting Grant Lo w R egular ity Dynamics via Decorated T rees (LoRDeT), g rant agreement No. 101075208. Vie ws and opinions e xpressed are R ough p aths and Hopf algebras 6 ho w ev er those of the author(s) only and do not necessarily reflect those of the European U nion or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. This project was star ted the week of the 6th October 2025 when Zhic heng Zhu visited the U niversit ´ e de Lorraine in Nancy . This visit was funded b y the ER C LoRDeT . 2 R ough paths and Hopf algebras 2.1 R ough paths Let us briefly recall the derivation the notion rough paths and their associated Hopf-alg ebraic definition from the RDEs’ point of vie w . Recall that the targ et RDE (1.1) is d Y t = X α ∈ [ d ] f α ( Y t )d X α t where [ d ] := { 1 , . . . , d } and t ∈ [ 0 , T ] . Suppose the integ ration with respect to X t is w ell-defined. Then, the equation can be re wr itten in the integral form d Y t = Y s + X α ∈ [ d ] Z t s f α ( Y r 1 )d X α r 1 f or any 0 ≤ s ≤ t ≤ T . T a ylor e xpansion of f a ( Y r 1 ) around Y s yields d Y t = Y s + X α ∈ [ d ] X n ∈ N X b ∈ [ m ] n 1 n ! Z t s n Y i =1 ∂ b i f α ( Y s )( Y b i r 1 − Y b i s )d X α r 1 where ∂ b i is the partial der ivativ e in the direction of the b i -th v ar iable. N otice that w e can repeat the “integration- T ay lor procedure” f or Y r 1 and g et d Y t = Y s + X α ∈ [ d ] f α ( Y s ) Z t s d X a r 1 (2.1) + X α ∈ [ d ] X β ∈ [ d ] X b ∈ [ m ] f b β ∂ b f a ( Y s ) Z t s Z r 1 s d X β r 2 d X α r 1 + X α ∈ [ d ] X β ∈ [ d ] 2 X b ∈ [ m ] 2 f b 1 β 1 f b 2 β 2 ∂ b 1 b 2 f a ( Y s ) Z t s Z r 1 s d X β 1 r 2 Z r 1 s d X β 2 r 3 d X α r 1 + R where f b β is the b -th entr y of f β , ∂ b 1 b 2 is a shor thand notation of ∂ b 1 ∂ b 2 , and the remainder R can be made e xplicit b y iterating the integration- T a y lor procedure. One observes that if we want to f ormally and algebraicall y encode such expansion of the solution via a Hopf alg ebra H , there are three main ingredients that need to be defined: • the multiplication of iterated integrals (from monomials in the T a y lor expansion), R ough p aths and Hopf algebras 7 • the composition of integrals, which satisfies the Chen ’ s relation [18], • the iteration that preserves necessary proper ties of iterated integrals. The first tw o notions lead to Definition 2.1 of rough paths, and the third proper ty is ensured b y the one-cocy cle condition (1.6) which links elements in H to tree structures. Since classical iterated integrals of X is ill-defined, one needs to find some s tochastic integ ration such as It ˆ o or Stratono vich integrals to mak e sense of the rough integrals. Ho we ver , the pre viously mentioned properties of iterated integrals are needed to be preserved, whic h leads to the notion of rough paths. Suppose H ( µ, ∆ ) = L n ∈ N H n and H ∗ ( ⋆, ∆ µ ) = L n ∈ N H ∗ n are a graded Hopf alg ebra and its graded dual Hopf algebra. Further denote | h | the grade of h ∈ H , i.e., h ∈ H | h | . Definition 2.1 (R ough paths) A H -rough path of regularity γ is a map X : [ 0 , T ] × [ 0 , T ] → H ∗ satisfying the f ollowing three conditions 1. For e very h 1 , h 2 ∈ H ⟨ X s,t , µ ( h 1 , h 2 ) ⟩ := ⟨ X s,t , h 1 ⟩ · ⟨ X s,t , h 2 ⟩ . (2.2) 2. (Chen ’ s relation) For an y 0 ≤ s ≤ r ≤ t ≤ T , X s,t = X s,r ⋆ X r,t . (2.3) 3. For an y h ∈ H | h | sup s  = t |⟨ X s,t , h ⟩| | t − s | γ | h | < ∞ . Remar k 2.2 In practice, µ is deter mined by the proper ty of the chosen rough integration. F or e xample, if the rough integral satisfies integration b y parts, the shuffle product is chosen. The Chen ’ s relation can be e xpressed equiv alently in the dual w ay ⟨ X s,t , h ⟩ = ⟨ X s,r ⊗ X r,t , ∆ h ⟩ = ⟨ X s,r , h ( 1 ) ⟩⟨ X s,r , h ( 2 ) ⟩ where w e use the Sw eedler notation f or ∆ given b y ∆ h = X ( h ) h ( 1 ) ⊗ h ( 2 ) . 2.2 Butcher -Connes-Kreimer Hopf alg ebra and branched r ough paths The Butcher -Connes-Kreimer Hopf algebra H BCK (Span R ( F ) , ⊙ , ∆ BCK ) , is defined on the linear span of f orests of non-planar rooted trees. Here, ⊙ is the commutativ e f orest product which represents the juxtaposition of rooted trees. W e denote the set of f orests b y F and the set of rooted trees by T . H BCK is equipped with 1 -cocy cle ( B α + ) α ∈ [ d ] which is a famil y linear maps. Consider a f orest Q ⊙ n i τ i , B α +  Q ⊙ n i τ i  amounts to g rafting τ i to a common root meaning that all the roots of the τ i are connected via ne w edges to a ne w root decorated b y α . R ough p aths and Hopf algebras 8 Example 1 W e pro vide belo w an e xample of computation with the map B α + B α + β ⊙ γ δ ! = α γ δ β . W e identify H BCK with its dual H ∗ BCK via the basis of f orest from H BCK . Definition 2.3 ( H BCK perfect pairing) The per f ect pair ing of H BCK is ⟨· , ·⟩ : H ∗ BCK × H BCK → R (2.4) ⟨ f ∗ 1 , f 2 ⟩ := δ f 1 f 2 S ( f 2 ) , f or any f ∗ 1 ∈ F ∗ and an y f 2 ∈ F where δ is the Kroneck er delta and S is the symmetry factor in Definition 2.7. Due to this identification, w e will make the f ollowing abuse of notations in the sequel τ , f f or τ ∗ , f ∗ . Branched rough paths introduced in [ 26 ] liv e in H ∗ BCK (Span R ( F ) , ⋆ BCK , ∆  ) , the dual of H BCK . The unshuffle coproduct runs o ver the par tition of f orest into tw o, and trees are pr imitiv e elements of ∆  , i.e., f or an y τ , σ ∈ T ∆  τ = 1 ⊗ τ + τ ⊗ 1 , ∆  ( τ ⊙ σ ) = ∆  τ ⊙ ∆  σ, where 1 is the tr ivial (empty) f orest, and ( h 1 ⊗ h 2 ) ⊙ ( h 3 ⊗ h 4 ) := ( h 1 ⊙ h 3 ) ⊗ ( h 2 ⊙ h 4 ) . The ⋆ BCK product is the Grossman-Larson product [ 22 ] which can be also obtained through the Guin-Oudom construction [ 27 , 28 ] on the pre-Lie g rafting product ▷ BCK of rooted trees. W e recall the definitions of ▷ BCK and ⋆ BCK . Definition 2.4 (Grafting product ▷ BCK ) For any σ, τ ∈ T , the grafting product σ ▷ BCK τ amounts to adding an edge connecting the root of σ to one v er te x of τ and summing o v er all vertices of τ . σ ▷ BCK τ = X v ∈ V τ σ ▷ v τ , (2.5) where V τ is the v er tex set of τ and σ ▷ v τ adds an edg e betw een the root of σ and the v ertex v . Example 2 (Grafting pr oduct ▷ BCK ) Here, we giv e an ex ample for the g rafting product. W e omit the decorations on the nodes. • ▷ BCK = + 2 , ▷ BCK = + 2 where the coefficient 2 appears since the trees are non-planar . R ough p aths and Hopf algebras 9 With the product ⋆ BCK , H ∗ BCK f or ms the universal en v elop of the Lie algebra obtained from the antisymmetrisation of the pre-Lie product ▷ BCK . No w , w e introduce ⋆ BCK through the Guin-Oudom procedure, which constructs an algebra isomor phic to the Grossman-Larson alg ebra. Let us first define the simultaneous grafting product ¯ ⋆ BCK , the reduced v ersion of ⋆ BCK . Definition 2.5 (Simultaneous grafting ¯ ⋆ BCK ) For an y Q ⊙ n i =1 σ i ∈ F and τ ∈ T , the simultaneous grafting reads ⊙ n Y i =1 σ i ¯ ⋆ BCK τ = X v 1 ,...,v n ∈ V τ σ 1 ▷ v 1 . . . σ n ▷ v n τ which means choose n v ertices of τ and add an edge connecting the root of σ i to v i . The nodes v i are not necessarily distinct. Moreo v er , for τ i ∈ T σ ¯ ⋆ BCK ⊙ m Y j =1 τ i = n X i =1 σ ¯ ⋆ BCK τ i ⊙ ⊙ Y j  = i τ j which is analogue to the Leibniz rule. Example 3 (Simultaneous grafting ¯ ⋆ BCK ) One has  • ⊙  ¯ ⋆ BCK  • ⊙  = ⊙ + ⊙ + 2 ⊙ + ⊙ + 2 ⊙ + • ⊙ + 2 • ⊙ + 2 • ⊙ + 2 • ⊙ . Through the Guin-Oudom constr uction, the ⋆ BCK product is obtained by partitioning the fores t Q ⊙ n i =1 σ i into two par ts and simultaneousl y grating one par t to Q ⊙ m i =1 τ i while putting the other par t aside in the f orest. Definition 2.6 ( ⋆ BCK ) For any f 1 , f 2 ∈ F . The product ⋆ BCK is defined as the f ollowing. f 1 ⋆ BCK 1 = 1 ⋆ BCK f 1 = f 1 , f 1 ⋆ BCK f 2 = M ⊙ ◦ (id ⊗ · ¯ ⋆ BCK f 2 ) ◦ ∆  f 1 , where M ⊙ ( f 1 ⊗ f 2 ) = f 1 ⊙ f 2 . Example 4 ( ⋆ BCK )  • ⊙  ⋆ BCK  • ⊙  = • ⊙  ¯ ⋆ BCK  • ⊙  + ⊙  • ¯ ⋆ BCK  • ⊙  +  • ⊙  ¯ ⋆ BCK  • ⊙  R ough p aths and Hopf algebras 10 where  • ⊙  ¯ ⋆ BCK  • ⊙  is sho wn in Example 3. Since H BCK is based on non-planar trees, w e hav e to introduce the symmetry f actors of trees and f orests, which play impor tant roles in both the pair ing (inner product) and B-series for solutions. Definition 2.7 (Symmetry f actor) The symmetr y factor of a f orest f is the num- ber of per mutations of vertices of f letting f globall y in variant. For an y τ = B α +  Q ⊙ n τ i τ i  ∈ T S ( τ ) = c Y j =1 r j ! S ( τ j ) r j where c is the number of isomor phic classes among τ i ∈ T , and r j is the cardinal of the j -th class in which elements are isomor phic to τ j . For any f = Q ⊙ Card( f ) i =1 σ i ∈ F with Card( f ) the number of trees in f , one has S ( f ) = c f Y j =1 r ( f ) j ! S ( σ j ) r ( f ) j , where c f is the number of isomorphic classes among σ i ∈ T , and r ( f ) j is the cardinal of the j -th class in which elements are isomorphic to σ j . Example 5 Belo w , w e compute the symmetry factors f or trees where w e ha v e assumed that all the nodes decorations are the same. S     = 1 , S   = 2! ( 3! ) 2 = 72 , S  ⊙  = 2! · 2 2 = 8 . The coproduct ∆ BCK and the product ⋆ BCK are adjoint under the pair ing (2.4), i.e, f or an y h 1 , h 2 ∈ H ∗ BCK and h ∈ H BCK . ⟨ h 1 ⋆ BCK h 2 , h ⟩ = ⟨ h 1 ⊗ h 2 , ∆ BCK h ⟩ . Remar k 2.8 There is another pairing in the literature (see [30]), which is ( h ∗ 1 , h 2 ) = δ h 1 h 2 . One can notice that the duality ( h 1 ⋆ BCK h 2 , h ) = ( h 1 ⊗ h 2 , ∆ BCK h ) f orces the coefficients resulting from the product ⋆ BCK to be equal to those from ∆ BCK . Consequently , w e lose the relativ ely natural f or mula of ⋆ BCK where the identity defining the grafting (2.5) has to be chang ed. R ough p aths and Hopf algebras 11 Due to the definition of the inner product (2.4), an element in X ∈ H ∗ BCK has the g eneral f orm X = X f ∈ F ⟨ X , f ⟩ S ( f ) f ∗ . No w , to construct Branc hed rough paths that sol v e RDEs (1.1), w e define recursivel y X to satisfy simultaneously , for h, h 1 , h 2 ∈ H BCK ⟨ X s,t , • α ⟩ := Z t s d X α r = X α t − X α s , ⟨ X s,r , h 1 ⊙ h 2 ⟩ := ⟨ X s,r , h 1 ⟩ · ⟨ X s,r , h 1 ⟩ , ⟨ X s,t , B α + ( h ) ⟩ := Z t s ⟨ X s,r , h ⟩ d X α r . (2.6) Here, the integral is in the rough sense (It ˆ o, Stratono vich, etc.). One can check that this definition tog ether with ⋆ BCK satisfy Definition 2.1 and f or m a rough path. From the e xpansion (2.1) of solutions of RDEs, one notices that there are tw o main ingredients: Iterated integrals of the signals X , and the monomials of derivativ es of the vector field f . The iterated integrals are represented by branc hed rough paths through the map X (2.6). Theref ore, it is left to find the algebraic e xpression of the monomials, which are called elementar y differentials and are defined as f ollowing. Definition 2.9 (Elementary differentials) Elementary differentials are maps Υ f : H ∗ BCK × R m → R m , which are linear in T ∗ , and ha v e the f ollowing e xpression. Υ f [ • α ] ( y ) = f α ( y ) , Υ f " B α + ⊙ n Y i =1 τ i !# ( y ) := X b ∈ [ m ] n n Y i =1 ∂ b i f α ( y ) ! n Y i =1 Υ b i f [ τ i ]( y ) , where Υ b f is the b -th coordinate of Υ f . Outside the pre vious cases, the elementar y differential is zero. A ccording to [ 26 , Theorem 5.1], given the signal X and its associated branched rough path X ∈ H ∗ BCK , the solution of equation (1.1) admits the B-ser ies e xpression Y t = B T ( Y s , X s,t ) (2.7) where one has f or Z ∈ H ∗ BCK B T ( y , Z ) = y + X τ ∈ T Υ f [ τ ]( y ) S ( τ ) ⟨ Z , τ ⟩ . (2.8) B-series representa tion of cocy cle-RDE solutions 12 3 B-series representation of cocy cle-RDE solutions It is sho wn in [ 23 ] that Assumption 2 tog ether with the smoothness assumption on the vector field f is sufficient to perf or m a fix ed point argument for proving the e xistence and uniqueness of solutions of RDEs (1.1). In the sequel, we call the RDEs driven b y rough paths associated to a one-cocycle-Hopf alg ebra cocycle-RDEs. In this section, w e will sho w that the solution of cocy cle-RDEs admits a B-series representation and the composition law of B-series ensures that their coefficients f or m controlled rough paths. Notabl y , according to the univ ersal proper ty of H BCK (Theorem 3.1), the one-cocycle condition leads to a morphism Λ : H ∗ → H ∗ BCK . Theref ore, proper ties of H BCK elementary differentials are passed to those of H . The alg ebraic proper ties recalled in this section cor respond to Assumption 2. 3.1 1-cocy cle Hopf algebra and H BCK W e hav e seen the need of iterations preser ving proper ties of iterated integ rals f or sol ving the RDE (1.1). This motivates the properties of the Hopf algebra H which pro vides a sufficient condition in pro ving the fixed point ar gument. Let the graded reduced Hopf alg ebra H ( µ, ∆ ) equipped with L α : H → H , a f amily of linear maps index ed by α ∈ [ d ] , homog eneous of degree one, satisfying the cocy cle condition f or h ∈ H ∆ L α ( h ) = (id ⊗ L α ) ∆ ( h ) + L α ( h ) ⊗ 1 . (3.1) By reduced, we mean that H 0 = R 1 where 1 is the unit for H . Let us recall [ 14 , Theorem 2] on the univ ersal property of H BCK . Theorem 3.1 The Hopf alg ebra H BCK endow ed with the cocycle B + is initial in the cat egor y of Hopf alg ebras endo wed with a 1 -cocycle. In plain wor ds, any Hopf alg ebra H endow ed with a 1 -cocycle L admits a unique morphism Λ : H BCK → H sending B + to L . It is naturally g eneralised to a collection of cocycles L α by decor ating the v ertices. By Theorem 3.1, w e ha v e a unique morphism of Hopf algebra: Λ : ( H BCK , ⊙ , ∆ BCK ) → ( H , µ, ∆ ) such that Λ ◦ B α + = L α ◦ Λ . This mor phism induces a dual mor phism: Λ ∗ : ( H ∗ , ⋆, ∆ · ) → ( H ∗ BCK , ⋆ BCK , ∆  ) . The space H split as H = I ⊕ K , with I = Im ( Λ ) , and K = Im ( Λ ) ⊥ the or thogonal of Im ( Λ ) . Moreov er , I is closed b y µ , and ( L α ) α ∈ [ d ] , to be more precise, this is actuall y the closure of H 0 b y µ , and ( L α ) α ∈ [ d ] . W e denote b y T an orthogonal basis of I , we naturall y identify I and I ∗ via this basis. B-series representa tion of cocy cle-RDE solutions 13 Proposition 3.2 F or ev er y h ∈ T , w e define the norm ∥ · ∥ H by ∥ h ∥ 2 H := ⟨ h, h ⟩ H . (3.2) F or every X ∈ I ∗ , one has X = X h ∈ T ⟨ X , h ⟩ H ∥ h ∥ 2 H h. (3.3) Mor eov er , one can define the elementary differential of h ∈ T by ¯ Υ f ( h ) := Υ f ( Λ ∗ ( h )) (3.4) and the f ollowing T -Butcher series by B T ( y , Z ) := B T ( y , Λ ∗ ( Z )) . (3.5) Then one has B T ( y , Z ) = y + X h ∈ T ¯ Υ f [ h ]( y ) ∥ h ∥ 2 H ⟨ Z , h ⟩ H . (3.6) Pr oof. The map ( B T ( y , · ) − y ) , and ( B T ( y , · ) − y ) are linear , moreo v er, the f ormer is e xactl y the elementar y differential Υ f as sho wn b y the f or mula: y + Υ f ( X ) = y + X τ ∈ T Υ f [ τ ] S ( τ ) ⟨ X , τ ⟩ . = B T ( y , X ) . Let us write B T ( y , Z ) − y as the composition ( B T ( y , · ) − y ) ◦ Λ ∗ e valuated at Z . In the basis T , one has B T ( y , Z ) = y + B T y , X h ∈ T ⟨ Z , h ⟩ H ∥ h ∥ 2 H h ! = y + X h ∈ T ( B T ( y , · ) − y )( h ) ∥ h ∥ 2 H ⟨ Z , h ⟩ H . = y + X h ∈ T ( B T ( y , · ) − y ) ◦ Λ ∗ ( h ) ∥ h ∥ 2 H ⟨ Z , h ⟩ H . = y + X h ∈ T Υ f [ Λ ∗ ( h )]( y ) ∥ h ∥ 2 H ⟨ Z , h ⟩ H . = y + X h ∈ T ¯ Υ f [ h ]( y ) ∥ h ∥ 2 H ⟨ Z , h ⟩ H . B-series representa tion of cocy cle-RDE solutions 14 3.2 B-series solutions and controlled r ough paths The follo wing mor phism proper ty of elementar y differentials ensures that the composition of T ay lor coefficients is algebraicall y compatible with the composition of rough paths (Chen ’ s relation), which fur ther yields the composition of solutions and guarantees B-series as solutions. Lemma 3.3 Elementar y differ entials are momor phisms with respect to the ¯ ⋆ BCK pr oduct, i.e., for any σ j , τ ∈ T Υ f     ⊙ n σ Y j =1 σ j   ¯ ⋆ BCK τ   ( y ) = X b ∈ [ m ] n σ   n σ Y j =1 Υ b j f [ σ j ]( y )   n σ Y j =1 ∂ b j Υ f [ τ ]( y ) . Pr oof. W e first prov e the case when n σ = 1 b y induction. The proof of the case when n σ > 1 f ollow s from the univ ersal proper ty from the Guin-Oudom cons truction [ 27 , 28 ] that builds ¯ ⋆ BCK out of the grafting product. Let τ = B α +  Q ⊙ n τ i =1 τ i  and suppose that, f or any τ i , Υ[ σ ¯ ⋆ BCK τ i ] ( y ) = X b ∈ [ m ] Υ b f [ σ ]( y ) ∂ b Υ f [ τ i ]( y ) . By the definition of elementary differentials, Υ f " B α + ⊙ n τ Y i =1 τ i !# ( y ) = X c ∈ [ m ] n τ n τ Y i =1 ∂ c i f α ( y ) ! n τ Y i =1 Υ c i f [ τ i ]( y ) . Then, the Leibniz rule implies X b ∈ [ m ] Υ b f [ σ ] ∂ b Υ f [ τ ] = X b ∈ [ m ] Υ b f [ σ ]( y ) ∂ b   X c ∈ [ m ] n τ n τ Y i =1 ∂ c i f α ( y ) ! n τ Y i =1 Υ c i f [ τ i ]( y ) .   = X b ∈ [ m ] X c ∈ [ m ] n τ Υ b f [ σ ]( y ) ∂ b n τ Y i =1 ∂ c i f α ( y ) ! n τ Y i =1 Υ c i f [ τ i ]( y ) ! + X b ∈ [ m ] X c ∈ [ m ] n τ n τ Y i =1 ∂ c i f α ( y ) ! Υ b f [ σ ]( y ) ∂ b n τ Y i =1 Υ c i f [ τ i ]( y ) ! . Notice, from the definition of elementary differentials, that X b ∈ [ m ] X c ∈ [ m ] n τ Υ b f [ σ ]( y ) ∂ b n τ Y i =1 ∂ c i f α ( y ) ! n τ Y i =1 Υ c i f [ τ i ]( y ) ! =Υ f " B α + σ ⊙ ⊙ n τ Y i =1 τ i !# . B-series representa tion of cocy cle-RDE solutions 15 Moreo v er, b y the induction hypothesis and b y the Leibniz r ule on ∂ and on ¯ ⋆ BCK , X b ∈ [ m ] X c ∈ [ m ] n τ n τ Y i =1 ∂ c i f α ( y ) ! Υ b f [ σ ]( y ) ∂ b n τ Y i =1 Υ c i f [ τ i ]( y ) ! = Υ f " B α + σ ¯ ⋆ BCK ⊙ n τ Y i =1 τ i !# ( y ) . W e conclude from the fact that σ ¯ ⋆ BCK τ = B α + σ ⊙ ⊙ n τ Y i =1 τ i ! + B α + σ ¯ ⋆ BCK ⊙ n τ Y i =1 τ i ! . Follo wing from this mor phism proper ty of elementar y differentials, B-ser ies ha v e the composition law . This ensures that the ansatz is preserved while composing local solutions to a “flow”, whic h is cr ucial in the solution theor y and is at the hear t of the concept of controlled rough paths introduced in [25, 26]. Proposition 3.4 The Butcher -Connes-Kreimer T -B-series satisfies the follo wing composition law . B T ( B T ( y , Z 1 ) , Z 2 ) = B T ( y , Z 1 ⋆ BCK Z 2 ) f or any Z 1 , Z 2 ∈ H ∗ BCK . Pr oof. For the simplicity in notations, w e identify the spaces H BCK and H ∗ BCK since the y are isomor phic. By the definition of the T -Butcher ser ies, B T ( B T ( y , Z 1 ) , Z 2 ) = B T ( y , Z 1 ) + X τ ∈ T Υ f [ τ ]( B T ( y , Z 1 )) S ( τ ) ⟨ Z 2 , τ ⟩ . The T ay lor e xpansion of the elementar y differentials around y yields Υ f [ τ ]( B T ( y , Z 1 )) = X β ∈ N m 1 β ! ∂ β Υ f [ τ ]( y ) m Y i =1  B i T ( y , Z 1 ) − y i  β i = X β ∈ N m 1 β ! ∂ β Υ f [ τ ]( y ) m Y i =1   X σ i ∈ T Υ i f [ σ i ]( y ) S ( σ i ) ⟨ Z 1 , σ i ⟩   β i = X β ∈ N m 1 β ! ∂ β Υ f [ τ ]( y ) X σ k i i ∈ T m Y i =1 β i Y k i =1 Υ i f [ σ k i i ]( y ) S ( σ k i i ) ⟨ Z 1 , σ k i i ⟩ B-series representa tion of cocy cle-RDE solutions 16 = X f = Q ⊙ n i =1 σ ⊙ r i i Q n i =1 S ( σ i ) r i S ( f ) X b i j ∈ [ m ] r i n Y i =1 r i Y j =1   n Y i =1 Υ b j f [ σ i ]( y ) S ( σ i ) ⟨ Z 1 , σ i ⟩   n Y i =1 r i Y j =1 ∂ b i j Υ f [ τ ]( y ) where β is a multi-index with m entries, the multi-index f actorial β ! = Q m i =1 β i ! , and ∂ β is the the abbreviation of the differential operator Q m i =1 ∂ β i i . In the last identity , we assume that the σ i are pairwise disjoint trees and w e ha v e used the notation σ ⊙ r i i = Q ⊙ r i j =1 σ i . Using the fact that Z 1 is a character combined with Lemma 3.3, one g ets Υ f [ τ ]( B T ( y , Z 1 )) = B T ( y , Z 1 ⋆ BCK τ ) . W e hav e also used the fact that Υ f is zero when ev aluated on a f orest with more than one tree. Theref ore, B T ( B T ( y , Z 1 ) , Z 2 ) = B T ( y , Z 1 ) + X τ ∈ T B T ( y , Z 1 ⋆ BCK τ ) S ( τ ) ⟨ Z 2 , τ ⟩ = X f ∈ F B T ( y , Z 1 ⋆ BCK f ) S ( f ) ⟨ Z 2 , f ⟩ = B T ( y , Z 1 ⋆ BCK Z 2 ) . Remar k 3.5 The proof abo ve follo ws the proof of [ 11 , Theorem 4.6] whic h com- poses R egular ity Structures B-series. Corollary 3.6 The T -Butcher series satisfies the f ollowing composition law B T ( B T ( y , Z 1 ) , Z 2 ) = B T ( y , Z 1 ⋆ Z 2 ) f or any Z 1 , Z 2 ∈ H ∗ . Pr oof. One has from Proposition 3.2 B T ( y , Z 1 ) = B T ( y , Λ ∗ ( Z 1 )) . Then, from Proposition 3.4, one has B T ( B T ( y , Λ ∗ ( Z 1 )) , Λ ∗ ( Z 2 )) = B T ( y , Λ ∗ ( Z 1 ) ⋆ BCK Λ ∗ ( Z 2 )) W e conclude from the mor phism proper ty of Λ ∗ f or the product ⋆ that implies B T ( y , Λ ∗ ( Z 1 ) ⋆ BCK Λ ∗ ( Z 2 )) = B T ( y , Λ ∗ ( Z 1 ⋆ Z 2 )) = B T ( y , Z 1 ⋆ Z 2 ) . B-series representa tion of cocy cle-RDE solutions 17 It is sho wn in [ 26 , Theorem 5.2] that the coefficients of branched rough paths B-series f or m controlled rough paths. This enables the application of the Se wing Lemma [ 30 , Lemma 3.1] to the solution ansatz, guaranteeing its conv erg ence when composing local solutions to a flo w , which was firs t constructed in [ 25 ] and adapted f or branched rough paths in [ 26 ]. [ 30 , Proposition 3.8] also sho w s the equiv alence betw een the unique controlled rough path solution and the B-series solution. No w w e will sho w that the result of [ 26 , Theorem 5.2] is actuall y a consequence of the composition la w of B-series (Proposition 3.4). In fact, the proofs of [ 26 , Theorem 5.2] and [30, Lemma 3.10] cor respond to composition of B-series. There are tw o equiv alent definitions f or controlled rough paths in the literature. The first one is [ 26 , Definition 8.1], which is more in the B-series style, and the other one is a definition with a more Hopf algebraic flav our [ 30 , Definition 3.2]. Here, w e will stic k to the latter one, since we will use the main result of [ 23 ] which is based on it. Let T ≤ N := T ∩ ( L n ≤ N H n ) . Definition 3.7 (Controlled r ough path) Let X be a γ -H ¨ older H -rough path, and let N be the larg est integ er such that N γ ≤ 1 . An X -controlled rough path is a path W : [ 0 , T ] → H satisfying: • For an y s ≤ t ∈ [ 0 , T ] , ⟨ 1 , W t ⟩ = W t = W s + X h ∈ T ≤ N − 1 ⟨ h, W s ⟩⟨ X s,t , h ⟩ + R N s,t where R N s,t ≤ C 1 | t − s | N γ f or some constant C 1 . • For an y h ∈ T ≤ N − 1 , ⟨ h, W t ⟩ = X g ∈ T ≤ N − 1 ⟨ g , W s ⟩⟨ X s,t ⋆ h/ ∥ h ∥ 2 H , g ⟩ + R τ s,t , where the remainder R h s,t ≤ C 2 | t − s | ( N −| h | ) γ f or some constant C 2 . • For an y h ∈ H \ T , ⟨ h, W ⟩ = 0 . Remar k 3.8 Here we hav e a ∥ h ∥ 2 H difference from [ 30 , Definition 3.2] and [ 23 , Definition 5.4] due to the different definition of the inner product (see Remark 2.8 f or details). Corollary 3.9 The elementar y differ entials of T -Butc her series f orm a controlled r ough path Y : [ 0 , T ] → T through ⟨ h, Y t ⟩ := ¯ Υ f [ h ]( Y t ) ∥ h ∥ 2 H with h ∈ T ∗ and ⟨ 1 , Y t ⟩ := Y t = Y s + X h ∈ T ≤ N − 1 ⟨ h, Y s ⟩⟨ X s,t , h ⟩ + R N s,t . (3.7) log-ODE solution theor y of cocy cle-RDEs 18 Pr oof. By Corollar y 3.6 B T ( Y t , Z ) = B T ( B T ( Y s , X s,t ) , Z ) = B T ( Y s , X s,t ⋆ Z ) , which implies that X h ∈ T ≤ N − 1 ⟨ h, Y t ⟩⟨ Z , h ⟩ = X g ∈ T ≤ N − 1 ⟨ g , Y s ⟩⟨ X s,t ⋆ Z , g ⟩ + R N s,t = X g ∈ T ≤ N − 1 X k ∈ T ≤ N −| h |− 1 X h ∈ T ≤ N − 1 ⟨ g , Y s ⟩⟨ k ⋆ h, g ⟩ ⟨ X s,t , k ⟩ ∥ k ∥ 2 H ⟨ Z , h ⟩ ∥ h ∥ 2 H + R N s,t . Thus, let us match the coefficients in front of ⟨ Z , h ⟩ and get ⟨ h, Y t ⟩ = X g ∈ T ≤ N − 1 X k ∈ T ≤ N −| h |− 1 ⟨ g , Y s ⟩⟨ k ⋆ h, g ⟩ ⟨ X s,t , k ⟩ ∥ k ∥ 2 H ∥ h ∥ 2 H + R h s,t = X g ∈ T ≤ N − 1 ⟨ g , Y s ⟩⟨ X s,t ⋆ h/ ∥ h ∥ 2 H , g ⟩ + R h s,t , where R h s,t ≤ C 2 | t − s | ( N −| h | ) γ f or some constant C 2 . Indeed, in the first line, w e sum up k ∈ T ≤ N −| h |− 1 , and thus the iterated integ rals ⟨ X s,t , k ⟩ are N − | h | − 1 - H ¨ older continuous f or all k . In f act, the first line reco v ers the conditions in [ 26 , Definition 8.1] of controlled rough paths. 4 log-ODE solution theory of cocy cle-RDEs In the log-ODE method, one needs the composition of elementar y differentials with smooth functions. Theref ore, we introduce the f ollo wing definition. Definition 4.1 (elementary v ector fields) For an y τ ∈ T and any y ∈ R m , we define the elementary v ector field Υ f [ τ ] {·} ( y ) : C ∞ ( R m , R m ) → C ∞ ( R m , R m ) as the functional Υ f [ τ ] { ψ } ( y ) := m X b =1 Υ b f [ τ ]( y ) ∂ b ψ ( y ) . (4.1) For g ener ic element Q ⊙ n i =1 τ i ∈ F we set Υ f [ 1 ] { ψ } ( y ) := ψ ( y ) , (4.2) Υ f [ ⊙ n Y i =1 τ i ] { ψ } ( y ) := m X b 1 ,...,b n =1 n Y i =1 Υ b i f [ τ i ]( y ) n Y i =1 ∂ b i ψ ( y ) . For h ∈ H , we set ¯ Υ f [ h ] { ψ } ( y ) := Υ f [ Λ ∗ ( h )] { ψ } ( y ) . (4.3) log-ODE solution theor y of cocy cle-RDEs 19 One can immediatel y remark that the identity (4.1) becomes elementary differentials in Definition (2.9) when ψ = I m y , where I m is the m -dimensional identity matrix. One can obser v e that the elementar y differential has the f ollo wing mor phism proper ty , which connects the e xpansions of solutions to the combinatorial objects. Proposition 4.2 The elementar y differ ential is a homomorphism when one defines the pr oduct betw een elementar y differ entials as the composition, whic h means for any u, v ∈ H ∗ BCK and y ∈ R Υ f [ u ⋆ BCK v ] { ψ } ( y ) = Υ f [ u ] {·} ( y ) ◦ Υ f [ v ] { ψ } ( y ) wher e ◦ stands f or the composition of elementar y v ector fields in the second input, i.e., Υ f [ u ] {·} ( y ) ◦ Υ f [ v ] { ψ } ( y ) = Υ f [ u ] { Υ f [ v ] { ψ }} ( y ) . Pr oof. Let us wr ite u = Q ⊙ n u j =1 σ j and v = Q ⊙ n v i =1 τ i where σ j , τ i ∈ T . By the definition of ⋆ , Υ f [ u ⋆ BCK v ] { ψ } ( y ) =Υ f [ ( ∆  u )(id ⊗ ¯ ⋆ BCK v ) ] { ψ } ( y ) = X K ⊔ L = { 1 ,...,n u } Υ f [ µ ⊙ Y k ∈ K σ k , ⊙ Y l ∈ L σ l ¯ ⋆ BCK v ! ] { ψ } ( y ) = X K ⊔ L = { 1 ,...,n u } m X { b k } k ∈ K =1 m X c 1 ,...,c n v =1 X L 1 ⊔ ... ⊔ L n v = L Y k ∈ K Υ b k f [ σ k ]( y ) n v Y i =1 Υ c i f [ ⊙ Y l ∈ L i σ l ¯ ⋆ BCK τ i ]( y ) n v Y i =1 ∂ c i Y k ∈ K ∂ b k ψ ( y ) . On the other hand, by the definition of elementar y v ector fields and the Leibniz r ule Υ f [ u ] { Υ f [ v ] { ψ }} ( y ) = m X b 1 ,...,b n u =1 n u Y j =1 Υ b j f [ σ j ]( y ) n u Y j =1 ∂ b j Υ f [ v ] { ψ } ( y ) = m X b 1 ,...,b n u =1 m X c 1 ,...,c n v =1 X K ⊔ L = { 1 ,...,n u } n u Y j =1 Υ b j f [ σ j ]( y ) Y l ∈ L ∂ b l n v Y i =1 Υ c i f [ τ i ]( y ) ! Y k ∈ K ∂ b k n v Y i =1 ∂ c i ψ ( y ) . A ccording to Lemma 3.3 m X { b l } l ∈ L =1 Y l ∈ L Υ b l f [ σ l ]( y ) Y l ∈ L ∂ b l n v Y i =1 Υ c i f [ τ i ]( y ) ! log-ODE solution theor y of cocy cle-RDEs 20 = m X { b l } l ∈ L =1 X L 1 ⊔ ... ⊔ L n v = L n v Y i =1   Y l ∈ L i Υ b l f [ σ l ]( y ) Y l ∈ L i ∂ b l Υ c i f [ τ i ]( y )   = X L 1 ⊔ ... ⊔ L n v = L n v Y i =1 Υ c i f [ ⊙ Y l ∈ L i σ l ¯ ⋆ BCK τ i ]( y ) . Finall y , we ha ve Υ f [ u ] { Υ f [ v ] { ψ }} ( y ) = m X c 1 ,...,c n v =1 X K ⊔ L = { 1 ,...,n u } m X { b k } k ∈ K =1 X L 1 ⊔ ... ⊔ L n v = L Y k ∈ K Υ b k f [ σ k ]( y ) n v Y i =1   Υ c i f [ ⊙ Y l ∈ L i σ l ¯ ⋆ BCK τ i ]( y )   n v Y i =1 ∂ c i Y k ∈ K ∂ b k ψ ( y ) ! =Υ f [ u ⋆ BCK v ] { ψ } ( y ) . Proposition 4.3 F or any u ∈ H ∗ BCK , y ∈ R m and φ, ψ ∈ C ∞ , w e hav e Υ f [ ∆  u ] { φ ⊗ ψ } ( y ) = Υ f [ u ] { φψ } ( y ) (4.4) wher e f or any u, v ∈ H ∗ BCK , Υ f [ u ⊗ v ] { φ ⊗ ψ } ( y ) := Υ f [ u ] { φ } ( y ) Υ f [ v ] { ψ } ( y ) . Pr oof. W e only ha v e to pro v e the identity for the primitive elements of ∆  that are e xactl y T since the primitive elements are “g enerators” and both ∆  and elementary v ector fields are linear in F ∗ . One can g enerate the proof from the pr imitive elements to the symmetr ic alg ebra by induction on the cardinal of u ∈ F ∗ that is the number of trees in the f orest u . On the left-hand side of the equality Υ f [ ∆  τ ] { φ ⊗ ψ } ( y ) = Υ f [ τ ⊗ 1 ] { φ ⊗ ψ } ( y ) + Υ f [ 1 ⊗ τ ] { φ ⊗ ψ } ( y ) = ψ ( y ) m X b =1 Υ b f [ τ ]( y ) ∂ b φ ( y ) + φ ( y ) m X b =1 Υ b f [ τ ]( y ) ∂ b ψ ( y ) = m X b =1 Υ b f [ τ ]( y ) ∂ b ( φ ( y ) ψ ( y )) = Υ f [ τ ] { φψ } ( y ) since Υ f [ 1 ]( φ ) = φ and we ha ve applied the Leibniz rule. Remar k 4.4 The proof f or Proposition 4.2 and Proposition 4.3 f or the case of planar trees are per f or med in [ 31 ]. Since those f or H BCK are needed in this paper and to be self-contained, w e ha v e decided to pro vide their proofs. log-ODE solution theor y of cocy cle-RDEs 21 Proposition 4.5 W e recall t he morphism of Hopf alg ebras giv en below Λ ∗ : ( H ∗ , ⋆, ∆ µ ) → ( H ∗ BCK , ⋆ BCK , ∆  ) . Then ¯ Υ f [ u ⋆ v ] { φ } = ¯ Υ f [ u ] ◦ ¯ Υ f [ v ] { φ } , ¯ Υ f [ ∆ µ u ] { φ ⊗ ψ } = ¯ Υ f [ u ] { φψ } . Pr oof. In this proof, w e will use the f act that Λ ∗ is a Hopf algebra morphism betw een ( H ∗ , ⋆, ∆ µ ) and ( H ∗ BCK , µ, ∆  ) . This implies Λ ∗ ( u ⋆ v ) = Λ ∗ ( u ) ⋆ BCK Λ ∗ ( v ) (4.5) (Λ ∗ ⊗ Λ ∗ )∆ µ = ∆  Λ ∗ . (4.6) For the firs t identity , one has ¯ Υ f [ u ⋆ v ] { φ } = Υ f [ Λ ∗ ( u ⋆ v )] { φ } = Υ f [ Λ ∗ ( u ) ⋆ BCK Λ ∗ ( v )] { φ } = Υ f [ Λ ∗ ( u )] ◦ Υ f [ Λ ∗ ( v )] { φ } = ¯ Υ f [ u ] ◦ ¯ Υ f [ v ] { φ } . where from the first to the second line, w e hav e used (4.5). From the second to the third line, w e ha v e applied Proposition 4.2. For the second identity , one has ¯ Υ f [ ∆ µ u ] { φ ⊗ ψ } ( y ) = Υ f [ (Λ ∗ ⊗ Λ ∗ )∆ µ u ] { φ ⊗ ψ } ( y ) = Υ f [ ∆  Λ ∗ ( u )] { φ ⊗ ψ } ( y ) = Υ f [ Λ ∗ ( u )] { φψ } ( y ) = ¯ Υ f [ u ] { φψ } ( y ) where from the first to the second line, w e hav e used (4.6). From the second to the third line, w e ha v e applied Proposition 4.3. Pr oof of Theorem1.1. Firs tly , Theorem 3.1 states that the cocycle condition (1.6) in Assumption 2 ensures the e xistence of the mor phism Λ ∗ and the e xistence of its dual map Λ . Recall that the space H splits as H = I ⊕ K , with I = Im ( Λ ) , and K = Im ( Λ ) ⊥ the or thogonal of Im ( Λ ) . W e denote b y T an or thogonal basis of I , we naturally identify I and I ∗ via this basis. Proposition 3.2 show s that the solution of cocy cle-RDE admits a T -B-series e xpression whose elementar y differentials are defined via (1.2). Theref ore, it is left to chec k that elementar y differentials of T -B-series satisfy the tw o k ey identities in Assumption 2. For H ∗ BCK , Proposition 4.2 sho ws that (1.4) holds while (1.5) is pro v ed in Proposition 4.3. W e can pass those tw o identities to H ∗ b y using the f act that Λ is a morphism of Hopf algebras (see Proposition 4.5). No algebraic fixed point for mul ti-indices 22 Remar k 4.6 One obser v es that Assumption 1 requires the e xistence of the morphism (f or defining elementar y differentials) but not necessar ily the one-cocy cle condition. R oughl y speaking, the one-cocy cle condition implies the exis tence of the mor phism. That is wh y log-ODE method is more general than the fix ed point argument in this sense. Ho w ev er , one has to keep in mind that the one-cocycle condition is sufficient f or showing the fix ed-point. It is not prov ed to be necessary . 5 No alg ebraic fixed point f or multi-indices Another combinatorial set coming from singular SPDEs has been used for defining rough paths. They come eq uipped with a Hopf algebra str ucture necessary f or the Chen ’ s relation. Let us introduce this Hopf alg ebra H M (Span R ( F M ) , · , ∆ M ) . W e are giv en abstract variables ( z k ) k ∈ N , a multi-inde x z β with β : N → N ha ving a finite suppor t giv en by z β := Y k ∈ N z β ( k ) k which is a monomial in the variables z k . In practice, one has to work with more v ar iables of the f or m z ( i,k ) with i ∈ [ d ] but w e pref er to look at simple multi-indices f or having lighter notations as the proof is ex actly the same. W e are interested in specific multi-indices called populated multi-indices. They satisfy the population condition: [ β ] := X k ∈ N ( 1 − k ) β ( k ) = 1 . (5.1) One has a surjectiv e map from the rooted trees to the populated multi-indices that w e denote b y Φ and giv en b y Φ ( B + ( τ 1 , ..., τ n )) = z n n Y i =1 Φ ( τ i ) . Denote the set of populated multi-indices by M . Then, w e consider the symmetr ic space ov er populated multi-indices. This v ector space is formed of f orests of multi-indices which are collections of multi-indices without an y order among them. The f orest product ˜ Q m i =1 z β i (or z β · z α ) is the juxtaposition of multi-indices. The identity element of the fores t product is the empty multi-index z 0 which is the multi-indice with β ( k ) = 0 f or ev er y k ∈ N . W e denote this space by F M . Norms and symmetry factors are tw o impor tant quantities of multi-indices. W e w ould use the f ollowing notations and f ormulae for them. • The nor m of a multi-inde x z β is the sum of each element of β | z β | = X k ∈ N β ( k ) . No algebraic fixed point for mul ti-indices 23 • The symmetry factor of a multi-indice counts the total number of permuting k children of eac h node z k S M ( z β ) = Y k ∈ N ( k ! ) β ( k ) . (5.2) The abo v e-mentioned quantities can be e xtended and applied to a f orest of multi- indices. Then the norms are     ˜ Y m i =1 z β i     = m X i =1 | z β i | . The symmetry factor f or a f orest of multi-indices ˜ z ˜ β = ˜ Q m i =1  z β i  r i with distinct β i is S M  ˜ z ˜ β  = m Y i =1 r i !  S M ( z β i )  r i . (5.3) In the sequel, w e will also use the follo wing symmetr y f actor: S ext ( ˜ Y n i =1 z β i ) = S M ( ˜ Q n i =1 z β i ) S M ( Q n i =1 z β i ) = m Y i =1 r i ! For tw o f orests of multi-indices ˜ z ˜ α and ˜ z ˜ β , the pair ing is defined via the inner product ⟨ ˜ z ˜ α , ˜ z ˜ β ⟩ := S M ( ˜ z ˜ α ) δ ˜ α ˜ β , (5.4) where δ ˜ α ˜ β = 1 , if ˜ α = ˜ β , otherwise it is equal to 0 . Equipped with these notations, w e are able to pro vide an e xplicit f or mula f or the coproduct denoted b y ∆ M from [7, Theorem 3.5]: ∆ M z β = z 0 ⊗ z β + z β ⊗ z 0 + X β = β 1 + ··· + β n + ˆ β n ∈ N ∗ 1 S ext ( ˜ Q n i =1 z β i ) ˜ Y n i =1 z β i ⊗ ¯ D n z ˆ β , (5.5) where the β i satisfy the population condition and the image of the map ¯ D n liv es in Span R ( M ) . The sum P β = β 1 + ··· + β n + ˆ β does not count the order among β i , which means β = β 1 + · · · + β n + ˆ β is a par tition ov er β . The map ¯ D is giv en by ¯ D z ˆ β = X k ∈ N ∗ k ˆ β ( k − 1 ) + 1 ˆ β ( k ) z k − 1 ∂ z k z ˆ β . (5.6) No algebraic fixed point for mul ti-indices 24 Partition ( j ) n ˜ Q n i =1 z β i z ˆ β j ¯ D n z ˆ β 1 1 z 0 z 0 z 1 z 2 2 z 2 0 z 2 + 4 z 0 z 2 1 2 1 z 0 z 1 z 0 z 2 2 z 0 z 1 3 1 z 2 0 z 2 z 1 z 0 4 2 z 0 · z 0 z 1 z 2 6 z 0 z 1 5 2 z 0 · z 0 z 1 z 2 2 z 0 The f ormula w as first introduced in [ 24 , Theorem 13]. W e illustrate the pre vious definitions recalling an e xample giv en in [ 7 , Example 3.6] W e onl y giv e the full computation f or z ˆ β 1 = z 0 z 1 z 2 ¯ D z ˆ β 1 = 1 1 + 1 1 z 2 0 z 2 + 2 1 + 1 1 z 0 z 2 1 = 2 z 2 0 z 2 + 4 z 0 z 2 1 . Other terms are listed in the f ollo wing tables. Finally w e hav e ∆ ( z β ) = z 0 ⊗ z 2 0 z 1 z 2 + z 2 0 z 1 z 2 ⊗ z 0 + 2 z 0 ⊗ z 2 0 z 2 + 4 z 0 ⊗ z 0 z 2 1 + 2 z 0 z 1 ⊗ z 0 z 1 + z 2 0 z 2 ⊗ z 0 + 3 z 0 ⊙ z 0 ⊗ z 0 z 1 + 2 z 0 ⊙ z 0 z 1 ⊗ z 0 . Let ¯ ∆ M the reduced coproduct, f or x ∈ F M , w e ha v e ¯ ∆ M ( x ) = ∆ M ( x ) − z 0 ⊗ x − x ⊗ z 0 . Belo w , we present e xamples of computations with this reduced coproduct which are needed f or the main theorem of the section. W e use the notation ¯ ∆ M m,n which means that w e consider onl y the ter ms of the f orm ˜ Q m i =1 ˜ z ˜ β i ⊗ ˜ Q n j =1 ˜ z ˜ α j with the ˜ β i and ˜ α j being non-zero, when one computes ¯ ∆ M m,n x with x ∈ F M . One also has the identity ¯ ∆ M = X m,n ∈ N ¯ ∆ M m,n . In the computations below , we will use a shor t hand notation f or fores t of multi- indices. All multi-indices are wr itten with the elements in decreasing order , f or e xample: z 1 z 0 and ne v er z 0 z 1 . This notation allo ws us to omit the · . Indeed, with this conv ention, z 0 z 1 z 0 can only be parsed as z 0 · z 1 z 0 . Thus z 0 z 0 z 0 z 0 is indeed z 0 · z 0 · z 0 · z 0 . Belo w , we wr ite the computations f or fores ts of multi-indices ˜ z ˜ β with | ˜ z ˜ β | ∈ { 1 , 2 , 3 } and w e giv e the e xpression of each non-zero ¯ ∆ M m,n component. ¯ ∆ M 1 , 1 z 0 z 0 2 z 0 ⊗ z 0 z 1 z 0 z 0 ⊗ z 0 ¯ ∆ M 1 , 2 ¯ ∆ M 2 , 1 z 0 z 0 z 0 3 z 0 ⊗ z 0 z 0 3 z 0 z 0 ⊗ z 0 z 0 z 1 z 0 z 0 ⊗ z 0 z 0 + z 0 ⊗ z 1 z 0 z 0 z 0 ⊗ z 0 + z 1 z 0 ⊗ z 0 z 1 z 1 z 0 2 z 0 ⊗ z 1 z 0 z 1 z 0 ⊗ z 0 z 2 z 0 z 0 2 z 0 ⊗ z 1 z 0 z 0 z 0 ⊗ z 0 No algebraic fixed point for mul ti-indices 25 ¯ ∆ M 1 , 3 ¯ ∆ M 2 , 2 z 0 z 0 z 0 z 0 4 z 0 ⊗ z 0 z 0 z 0 6 z 0 z 0 ⊗ z 0 z 0 z 0 z 0 z 1 z 0 z 0 ⊗ z 0 z 0 z 0 + 2 z 0 ⊗ z 0 z 1 z 0 2 z 0 z 0 ⊗ z 0 z 0 + z 1 z 0 ⊗ z 0 z 0 z 0 z 1 z 1 z 0 2 z 0 ⊗ z 0 z 1 z 0 + z 0 ⊗ z 1 z 1 z 0 2 z 0 z 0 ⊗ z 1 z 0 + 2 z 1 z 0 ⊗ z 0 z 0 z 0 z 2 z 0 z 0 2 z 0 ⊗ z 0 z 1 z 0 + z 0 ⊗ z 2 z 0 z 0 2 z 0 z 0 ⊗ z 1 z 0 + z 0 z 0 ⊗ z 0 z 0 z 1 z 0 z 1 z 0 2 z 0 ⊗ z 0 z 1 z 0 z 0 z 0 ⊗ z 0 z 0 + 2 z 1 z 0 ⊗ z 1 z 0 z 1 z 1 z 1 z 0 3 z 0 ⊗ z 1 z 1 z 0 6 z 1 z 0 ⊗ z 1 z 0 z 2 z 1 z 0 z 0 4 z 0 ⊗ z 1 z 1 z 0 + 2 z 0 ⊗ z 2 z 0 z 0 3 z 0 z 0 ⊗ z 1 z 0 + 2 z 1 z 0 ⊗ z 1 z 0 z 3 z 0 z 0 z 0 3 z 0 ⊗ z 2 z 0 z 0 3 z 0 z 0 ⊗ z 1 z 0 ¯ ∆ M 3 , 1 z 0 z 0 z 0 z 0 4 z 0 z 0 z 0 ⊗ z 0 z 0 z 0 z 1 z 0 z 0 z 0 z 0 ⊗ z 0 + 2 z 0 z 1 z 0 ⊗ z 0 z 0 z 1 z 1 z 0 2 z 0 z 1 z 0 ⊗ z 0 + z 1 z 1 z 0 ⊗ z 0 z 0 z 2 z 0 z 0 z 0 z 0 z 0 ⊗ z 0 + z 2 z 0 z 0 ⊗ z 0 z 1 z 0 z 1 z 0 2 z 0 z 1 z 0 ⊗ z 0 z 1 z 1 z 1 z 0 3 z 1 z 1 z 0 ⊗ z 0 z 2 z 1 z 0 z 0 2 z 0 z 1 z 0 ⊗ z 0 + z 2 z 0 z 0 ⊗ z 0 z 3 z 0 z 0 z 0 z 0 z 0 z 0 ⊗ z 0 Theorem 5.1 Let L a cocycle of degr ee 1 of the Hopf alg ebr a of multi-indices, then L ( z 0 ) = 0 . Moreo v er the obstruction appear in degr ee 4 , which is the minimal degr ee for suc h an obstruction to appear . Pr oof. The idea of the proof is to use the defining proper ty of a cocy cle to compute the v alue of L f or z 0 , z 0 , z 1 z 0 , z 1 z 1 z 0 , z 2 z 0 z 0 , and to notice that it f orces L ( z 0 ) = 0 . W e need to go up to deg ree 4 because the multi-inde x of lo wes t deg ree representing se v eral trees is z 2 z 1 z 0 z 0 , thus one cannot hope to find an obstr uction in degree less than 4 . The proof no w reduces to a tedious ex ercise of linear algebra in v ol ving a matrix 8 × 12 . W e recall that a cocycle L is a map such that ∆ M ( L ( x )) = x ( 1 ) ⊗ L ( x ( 2 ) ) + L ( x ) ⊗ z 0 where w e use the Sw eedler notation ∆ M ( x ) = x ( 1 ) ⊗ x ( 2 ) . Since z 0 is the onl y multi-inde x with one v er te x, w e ha v e λ ∈ R such that: L ( z 0 ) = 6 λz 0 . W e hav e ¯ ∆ M ( z 0 ) = 0 , and ¯ ∆ M ( L ( z 0 )) = z 0 ⊗ L ( z 0 ) . Thus w e ha v e µ ∈ R such that L ( z 0 ) = 6 λz 1 z 0 + 6 µ ( z 0 z 0 − 2 z 1 z 0 ) . No algebraic fixed point for mul ti-indices 26 W e hav e ¯ ∆ M ( z 1 z 0 ) = z 0 ⊗ z 0 , and ¯ ∆ M ( L ( z 1 z 0 )) = z 0 ⊗ L ( z 0 ) + z 1 z 0 ⊗ L ( z 0 ) . Thus w e ha v e γ ∈ R such that L ( z 1 z 0 ) = 3 λz 1 z 1 z 0 + 2 µ ( z 0 z 0 z 0 − 3 z 2 z 0 z 0 ) + γ ( 2 z 0 z 0 z 0 − 6 z 0 z 1 z 0 + 3 z 1 z 1 z 0 ) . Bef ore beginning the computations in degree 4 , let us compute ker( ¯ ∆ M 2 , 2 ) ∩ ker( ¯ ∆ M 3 , 1 ) . One ma y chec k that ker( ¯ ∆ M 3 , 1 ) = Span( v 1 , v 2 , v 3 , v 4 ) with v 1 = z 0 z 0 z 0 z 0 − 4 z 3 z 0 z 0 z 0 v 2 = z 0 z 0 z 1 z 0 − z 1 z 0 z 1 z 0 − z 3 z 0 z 0 z 0 v 3 = 3 z 0 z 1 z 1 z 0 − 3 z 1 z 0 z 1 z 0 − z 1 z 1 z 1 z 0 v 4 = z 0 z 2 z 0 z 0 + z 1 z 0 z 1 z 0 − z 2 z 1 z 0 z 0 − z 3 z 0 z 0 z 0 . W e may c heck that k er( ¯ ∆ M 2 , 2 ) ∩ ker( ¯ ∆ M 3 , 1 ) = Span( w 1 , w 2 ) with w 1 = z 0 z 0 z 0 z 0 − 3 z 0 z 2 z 0 z 0 − 3 z 1 z 0 z 1 z 0 + 3 z 2 z 1 z 0 z 0 − z 3 z 0 z 0 z 0 w 2 = 3 z 0 z 0 z 0 z 0 − 12 z 0 z 0 z 1 z 0 + 6 z 0 z 1 z 1 z 0 + 6 z 1 z 0 z 1 z 0 − 2 z 1 z 1 z 1 z 0 ¯ ∆ M ( w 1 ) = 4 z 0 ⊗ z 0 z 0 z 0 − 12 z 0 ⊗ z 0 z 1 z 0 + 12 z 0 ⊗ z 1 z 1 z 0 ¯ ∆ M ( w 2 ) = 0 . Let us compute L ( z 1 z 1 z 0 ) . W e set ν µ = − z 0 z 0 z 0 z 0 + 6 z 0 z 0 z 1 z 0 − 6 z 1 z 0 z 1 z 0 − 2 z 3 z 0 z 0 z 0 ¯ ∆ M ( ν µ ) = 2 z 0 ⊗ z 0 z 0 z 0 − 6 z 0 ⊗ z 2 z 0 z 0 + 6 z 1 z 0 ⊗ z 0 z 0 − 12 z 1 z 0 ⊗ z 1 z 0 . Since ¯ ∆ M ( L ( z 1 z 1 z 0 )) = 2 z 0 ⊗ L ( z 1 z 0 ) + 2 z 1 z 0 ⊗ L ( z 0 ) + z 1 z 1 z 0 ⊗ L ( z 0 ) w e ha v e ¯ ∆ M ( L ( z 1 z 1 z 0 ) − λz 1 z 1 z 1 z 0 − 2 µν µ ) = γ ( 2 z 0 ⊗ z 0 z 0 z 0 − 6 z 0 ⊗ z 0 z 1 z 0 + 3 z 0 ⊗ z 1 z 1 z 0 ) . Thus, L ( z 1 z 1 z 0 ) − λz 1 z 1 z 1 z 0 − 2 µν µ ∈ ker( ¯ ∆ M 2 , 2 ) ∩ ker( ¯ ∆ M 3 , 1 ) , and γ = 0 . Let us compute L ( z 2 z 0 z 0 ) . W e set ν λ = − z 0 z 0 z 0 z 0 + 6 z 0 z 2 z 0 z 0 − 2 z 3 z 0 z 0 z 0 ¯ ∆ M ( ν λ ) = − 4 z 0 ⊗ z 0 z 0 z 0 + 12 z 0 ⊗ z 0 z 1 z 0 + 6 z 0 z 0 ⊗ z 1 z 0 + 6 z 2 z 0 z 0 ⊗ z 0 . Since ¯ ∆ M ( L ( z 2 z 0 z 0 )) = 2 z 0 ⊗ L ( z 1 z 0 ) + z 0 z 0 ⊗ L ( z 0 ) + z 2 z 0 z 0 ⊗ L ( 1 ) No algebraic fixed point for mul ti-indices 27 w e ha v e ¯ ∆ M ( L ( z 1 z 1 z 0 ) − λν λ − µ ( z 0 z 0 z 0 z 0 − 4 z 3 z 0 z 0 z 0 )) = λ ( − 4 z 0 ⊗ z 0 z 0 z 0 + 12 z 0 ⊗ z 0 z 1 z 0 − 6 z 0 ⊗ z 2 z 0 z 0 ) . Thus L ( z 1 z 1 z 0 ) − λν λ − µ ( z 0 z 0 z 0 z 0 − 4 z 3 z 0 z 0 z 0 ) ∈ ker( ¯ ∆ M 2 , 2 ) ∩ ker( ¯ ∆ M 3 , 1 ) , and λ = 0 which implies L ( z 0 ) = 0 . Proposition 5.2 The muti-indices Hopf alg ebr a satisfies Assumption 1. Pr oof. The map Φ is a mor phism from ( H ∗ BCK , ⋆ ) into ( H ∗ M , ⋆ M ) where ⋆ M is the graded dual of ∆ M . This f ollo w s from the mor phism proper ty given in [ 34 , Section 6.5]. 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