Definition 2.6 .

A control-Lyapunov function (CLF) for

x Λ™ = h ⁒ ( x , u ) Λ™ π‘₯ β„Ž π‘₯ 𝑒 \dot{x}=h(x,u) Λ™ start_ARG italic_x end_ARG = italic_h ( italic_x , italic_u ) (10)

is a continuous, positive definite, proper function V : ℝ n β†’ ℝ normal-: 𝑉 normal-β†’ superscript ℝ 𝑛 ℝ V:{\mathbb{R}}^{n}\to{\mathbb{R}} italic_V : blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT β†’ blackboard_R for which there exist a continuous, positive definite function W : ℝ n β†’ ℝ normal-: π‘Š normal-β†’ superscript ℝ 𝑛 ℝ W:{\mathbb{R}}^{n}\to{\mathbb{R}} italic_W : blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT β†’ blackboard_R , and a nondecreasing function Ξ± : [ 0 , ∞ ) β†’ [ 0 , ∞ ) normal-: 𝛼 normal-β†’ 0 0 \alpha:[0,\infty)\to[0,\infty) italic_Ξ± : [ 0 , ∞ ) β†’ [ 0 , ∞ ) , satisfying

βˆ€ ΞΆ ∈ βˆ‚ P ⁑ V ⁒ ( x ) , inf | u | ≀ Ξ± ⁒ ( | x | ) ⁑ ⟨ ΞΆ , h ⁒ ( x , u ) ⟩ ≀ - W ⁒ ( x ) formulae-sequence for-all 𝜁 subscript 𝑃 𝑉 π‘₯ subscript infimum 𝑒 𝛼 π‘₯ 𝜁 β„Ž π‘₯ 𝑒 π‘Š π‘₯ \displaystyle\forall\zeta\in\partial_{P}V(x),\;\;\inf_{|u|\leq\alpha(|x|)}% \langle\zeta,h(x,u)\rangle\leq-W(x) βˆ€ italic_ΞΆ ∈ βˆ‚ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT italic_V ( italic_x ) , roman_inf start_POSTSUBSCRIPT | italic_u | ≀ italic_Ξ± ( | italic_x | ) end_POSTSUBSCRIPT ⟨ italic_ΞΆ , italic_h ( italic_x , italic_u ) ⟩ ≀ - italic_W ( italic_x )

for all x ∈ ℝ n π‘₯ superscript ℝ 𝑛 x\in{\mathbb{R}}^{n} italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT . In this case, we call ( V , W ) 𝑉 π‘Š (V,W) ( italic_V , italic_W ) a Lyapunov pair for ( 10 ) .

Definition 8

A crossed module is a homomorphism of groups βˆ‚ : E β†’ G normal-: normal-β†’ 𝐸 𝐺 \partial:E\rightarrow G βˆ‚ : italic_E β†’ italic_G together with an action β–· normal-β–· \triangleright β–· of G 𝐺 G italic_G on E 𝐸 E italic_E by automorphisms, such that

βˆ‚ ⁑ ( g β–· e ) = g ⁒ ( βˆ‚ ⁑ e ) ⁒ g - 1 β–· 𝑔 𝑒 𝑔 𝑒 superscript 𝑔 1 \partial(g\triangleright e)=g(\partial e)g^{-1} βˆ‚ ( italic_g β–· italic_e ) = italic_g ( βˆ‚ italic_e ) italic_g start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT
( βˆ‚ ⁑ e ) β–· Ξ΅ = e ⁒ Ξ΅ ⁒ e - 1 β–· 𝑒 πœ€ 𝑒 πœ€ superscript 𝑒 1 (\partial e)\triangleright\varepsilon=e\varepsilon e^{-1} ( βˆ‚ italic_e ) β–· italic_Ξ΅ = italic_e italic_Ξ΅ italic_e start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT
Definition 4.1 .

A periodic partially ordered set (ppset for short), of degree k > 0 π‘˜ 0 k>0 italic_k > 0 , is a partially ordered set P 𝑃 P italic_P together with an action of β„€ k superscript β„€ π‘˜ {\mathbb{Z}}^{k} blackboard_Z start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT by order preserving transformations. In other words, to give a ppset of degree k π‘˜ k italic_k is to give the poset P 𝑃 P italic_P together with a collection of commuting order preserving automorphisms T 1 , … , T k subscript 𝑇 1 … subscript 𝑇 π‘˜ T_{1},\ldots,T_{k} italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT (We will call the automorphisms obtained by the action of β„€ k superscript β„€ π‘˜ {\mathbb{Z}}^{k} blackboard_Z start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT the shifts of P 𝑃 P italic_P ). A map f : P β†’ Q : 𝑓 β†’ 𝑃 𝑄 f:P\rightarrow Q italic_f : italic_P β†’ italic_Q between two ppsets is a function f 𝑓 f italic_f which is order preserving and satisfies the relation

f ∘ ( 1 , 1 , … , 1 ) = ( 1 , 1 , … , 1 ) ∘ f 𝑓 1 1 … 1 1 1 … 1 𝑓 f\circ(1,1,\ldots,1)=(1,1,\ldots,1)\circ f italic_f ∘ ( 1 , 1 , … , 1 ) = ( 1 , 1 , … , 1 ) ∘ italic_f

where the symbol ( 1 , 1 , … , 1 ) 1 1 … 1 (1,1,\ldots,1) ( 1 , 1 , … , 1 ) represents the action of the appropriate group element (which could of course be different in P 𝑃 P italic_P and Q 𝑄 Q italic_Q ). We write Map ⁑ ( P , Q ) Map 𝑃 𝑄 {\operatorname{Map}}(P,Q) roman_Map ( italic_P , italic_Q ) for the set of maps between ppsets P 𝑃 P italic_P and Q 𝑄 Q italic_Q . A morphism between two ppsets P 𝑃 P italic_P and Q 𝑄 Q italic_Q is an equivalence class of maps. The equivalence is given by precomposition with the shifts of P 𝑃 P italic_P and postcomposition with the shifts of Q 𝑄 Q italic_Q . The set of morphisms between P 𝑃 P italic_P and Q 𝑄 Q italic_Q is denoted Hom ⁑ ( P , Q ) Hom 𝑃 𝑄 {\operatorname{Hom}}(P,Q) roman_Hom ( italic_P , italic_Q ) .

Definition 1

Let A 𝐴 A italic_A be an abelian variety over a field k π‘˜ k italic_k . A point p ∈ A ⁒ ( k Β― ) 𝑝 𝐴 normal-Β― π‘˜ p\in A(\bar{k}) italic_p ∈ italic_A ( Β― start_ARG italic_k end_ARG ) is called almost rational (a.r.) if

Οƒ ⁒ ( p ) - p = p - Ο„ ⁒ ( p ) β‡’ p = Οƒ ⁒ ( p ) = Ο„ ⁒ ( p ) 𝜎 𝑝 𝑝 𝑝 𝜏 𝑝 β‡’ 𝑝 𝜎 𝑝 𝜏 𝑝 \sigma(p)-p=p-\tau(p)\Rightarrow p=\sigma(p)=\tau(p) italic_Οƒ ( italic_p ) - italic_p = italic_p - italic_Ο„ ( italic_p ) β‡’ italic_p = italic_Οƒ ( italic_p ) = italic_Ο„ ( italic_p )

for all Οƒ , Ο„ ∈ πΊπ‘Žπ‘™ ⁒ ( k Β― / k ) 𝜎 𝜏 πΊπ‘Žπ‘™ normal-Β― π‘˜ π‘˜ \sigma,\tau\in{\mbox{Gal}}(\bar{k}/k) italic_Οƒ , italic_Ο„ ∈ Gal ( Β― start_ARG italic_k end_ARG / italic_k ) .

Definition 2.9 .

A left-symmetric algebra (LSA) structure on a Lie algebra π”₯ π”₯ \mathfrak{h} fraktur_h is a bilinear product π”₯ Γ— π”₯ ⟢ π”₯ , ( x , y ) ↦ x β‹… y formulae-sequence ⟢ π”₯ π”₯ π”₯ maps-to π‘₯ 𝑦 β‹… π‘₯ 𝑦 \mathfrak{h}\times\mathfrak{h}\longrightarrow\mathfrak{h},\,(x,y)\mapsto x\cdot y fraktur_h Γ— fraktur_h ⟢ fraktur_h , ( italic_x , italic_y ) ↦ italic_x β‹… italic_y , which satisfies

(9) x β‹… ( y β‹… z ) - ( x β‹… y ) β‹… z = y β‹… ( x β‹… z ) - ( y β‹… x ) β‹… z β‹… π‘₯ β‹… 𝑦 𝑧 β‹… β‹… π‘₯ 𝑦 𝑧 β‹… 𝑦 β‹… π‘₯ 𝑧 β‹… β‹… 𝑦 π‘₯ 𝑧 x\cdot(y\cdot z)-(x\cdot y)\cdot z=y\cdot(x\cdot z)-(y\cdot x)\cdot z italic_x β‹… ( italic_y β‹… italic_z ) - ( italic_x β‹… italic_y ) β‹… italic_z = italic_y β‹… ( italic_x β‹… italic_z ) - ( italic_y β‹… italic_x ) β‹… italic_z

and

(10) [ x , y ] = x β‹… y - y β‹… x . π‘₯ 𝑦 β‹… π‘₯ 𝑦 β‹… 𝑦 π‘₯ [x,y]=x\cdot y-y\cdot x. [ italic_x , italic_y ] = italic_x β‹… italic_y - italic_y β‹… italic_x .
Definition 1.6 .

A skew-symmetric non-degenerate bilinear form Ο‰ πœ” \omega italic_Ο‰ on the Lie algebra 𝔀 𝔀 \mathfrak{g} fraktur_g is called symplectic if it closed, i.e.

Ο‰ ⁒ ( [ x , y ] , z ) + Ο‰ ⁒ ( [ y , z ] , x ) + Ο‰ ⁒ ( [ z , y ] , x ) = 0 , βˆ€ x , y , z ∈ 𝔀 . formulae-sequence πœ” π‘₯ 𝑦 𝑧 πœ” 𝑦 𝑧 π‘₯ πœ” 𝑧 𝑦 π‘₯ 0 for-all π‘₯ 𝑦 𝑧 𝔀 \omega([x,y],z)+\omega([y,z],x)+\omega([z,y],x)=0,\;\forall x,y,z\in\mathfrak{% g}. italic_Ο‰ ( [ italic_x , italic_y ] , italic_z ) + italic_Ο‰ ( [ italic_y , italic_z ] , italic_x ) + italic_Ο‰ ( [ italic_z , italic_y ] , italic_x ) = 0 , βˆ€ italic_x , italic_y , italic_z ∈ fraktur_g .
Definition 10.1 .

Let ( C ⁒ [ 1 ] , π”ͺ ) 𝐢 delimited-[] 1 π”ͺ (C[1],\mathfrak{m}) ( italic_C [ 1 ] , fraktur_m ) , ( C β€² ⁒ [ 1 ] , π”ͺ β€² ) superscript 𝐢 β€² delimited-[] 1 superscript π”ͺ β€² (C^{\prime}[1],\mathfrak{m}^{\prime}) ( italic_C start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT [ 1 ] , fraktur_m start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ) be L ∞ subscript 𝐿 L_{\infty} italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT algebras and Ξ΄ , Ξ΄ β€² 𝛿 superscript 𝛿 β€² \delta,\,\delta^{\prime} italic_Ξ΄ , italic_Ξ΄ start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT be the associated coderivation. A sequence Ο† = { Ο† k } k = 1 ∞ πœ‘ superscript subscript subscript πœ‘ π‘˜ π‘˜ 1 \varphi=\{\varphi_{k}\}_{k=1}^{\infty} italic_Ο† = { italic_Ο† start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT with Ο† k : E k ⁒ C ⁒ [ 1 ] β†’ C β€² ⁒ [ 1 ] : subscript πœ‘ π‘˜ β†’ subscript 𝐸 π‘˜ 𝐢 delimited-[] 1 superscript 𝐢 β€² delimited-[] 1 \varphi_{k}:E_{k}C[1]\to C^{\prime}[1] italic_Ο† start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT : italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_C [ 1 ] β†’ italic_C start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT [ 1 ] is said to be an L ∞ subscript 𝐿 L_{\infty} italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT homomorphism if the corresponding coalgebra homomorphism Ο† ^ : E ⁒ C ⁒ [ 1 ] β†’ E ⁒ C β€² ⁒ [ 1 ] : ^ πœ‘ β†’ 𝐸 𝐢 delimited-[] 1 𝐸 superscript 𝐢 β€² delimited-[] 1 \widehat{\varphi}:EC[1]\to EC^{\prime}[1] ^ start_ARG italic_Ο† end_ARG : italic_E italic_C [ 1 ] β†’ italic_E italic_C start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT [ 1 ] satisfies

Ο† ^ ∘ Ξ΄ = Ξ΄ β€² ∘ Ο† ^ . ^ πœ‘ 𝛿 superscript 𝛿 β€² ^ πœ‘ \widehat{\varphi}\circ\delta=\delta^{\prime}\circ\widehat{\varphi}. ^ start_ARG italic_Ο† end_ARG ∘ italic_Ξ΄ = italic_Ξ΄ start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ∘ ^ start_ARG italic_Ο† end_ARG .

We say that Ο† πœ‘ \varphi italic_Ο† is an L ∞ subscript 𝐿 L_{\infty} italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT isomorphism , if there exists a sequence of homomorphisms ψ = { ψ k } k = 1 ∞ πœ“ superscript subscript subscript πœ“ π‘˜ π‘˜ 1 \psi=\{\psi_{k}\}_{k=1}^{\infty} italic_ψ = { italic_ψ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT , ψ : E k ⁒ C β€² ⁒ [ 1 ] β†’ C β€² ⁒ [ 1 ] : πœ“ β†’ subscript 𝐸 π‘˜ superscript 𝐢 β€² delimited-[] 1 superscript 𝐢 β€² delimited-[] 1 \psi:E_{k}C^{\prime}[1]\to C^{\prime}[1] italic_ψ : italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_C start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT [ 1 ] β†’ italic_C start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT [ 1 ] such that its associated coalgebra homomorphism ψ ^ : E ⁒ C β€² ⁒ [ 1 ] β†’ E ⁒ C ⁒ [ 1 ] : ^ πœ“ β†’ 𝐸 superscript 𝐢 β€² delimited-[] 1 𝐸 𝐢 delimited-[] 1 \widehat{\psi}:EC^{\prime}[1]\to EC[1] ^ start_ARG italic_ψ end_ARG : italic_E italic_C start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT [ 1 ] β†’ italic_E italic_C [ 1 ] satisfies

ψ ^ ∘ Ο† ^ = i ⁒ d E ⁒ C ⁒ [ 1 ] , Ο† ^ ∘ ψ ^ = i ⁒ d E ⁒ C β€² ⁒ [ 1 ] . formulae-sequence ^ πœ“ ^ πœ‘ 𝑖 subscript 𝑑 𝐸 𝐢 delimited-[] 1 ^ πœ‘ ^ πœ“ 𝑖 subscript 𝑑 𝐸 superscript 𝐢 β€² delimited-[] 1 \widehat{\psi}\circ\widehat{\varphi}=id_{EC[1]},\quad\widehat{\varphi}\circ% \widehat{\psi}=id_{EC^{\prime}[1]}. ^ start_ARG italic_ψ end_ARG ∘ ^ start_ARG italic_Ο† end_ARG = italic_i italic_d start_POSTSUBSCRIPT italic_E italic_C [ 1 ] end_POSTSUBSCRIPT , ^ start_ARG italic_Ο† end_ARG ∘ ^ start_ARG italic_ψ end_ARG = italic_i italic_d start_POSTSUBSCRIPT italic_E italic_C start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT [ 1 ] end_POSTSUBSCRIPT .

In this case, we say that two L ∞ subscript 𝐿 L_{\infty} italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT algebras, ( C ⁒ [ 1 ] , π”ͺ ) 𝐢 delimited-[] 1 π”ͺ (C[1],\mathfrak{m}) ( italic_C [ 1 ] , fraktur_m ) and ( C β€² ⁒ [ 1 ] , π”ͺ β€² ) superscript 𝐢 β€² delimited-[] 1 superscript π”ͺ β€² (C^{\prime}[1],\mathfrak{m}^{\prime}) ( italic_C start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT [ 1 ] , fraktur_m start_POSTSUPERSCRIPT β€² end_POSTSUPERSCRIPT ) are L ∞ subscript 𝐿 L_{\infty} italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT isomorphic.

Definition 3.1

Let Ξ” k superscript Ξ” π‘˜ \Delta^{k} roman_Ξ” start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT be the standard k π‘˜ k italic_k -simplex. Let Ο† : Ξ” k β†’ ℍ Β― n : πœ‘ β†’ superscript Ξ” π‘˜ superscript Β― ℍ 𝑛 \varphi:\Delta^{k}\to\overline{\mathbb{H}}^{n} italic_Ο† : roman_Ξ” start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT β†’ Β― start_ARG blackboard_H end_ARG start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT be a continuous map that maps the 0 0 -skeleton of Ξ” k superscript Ξ” π‘˜ \Delta^{k} roman_Ξ” start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT to βˆ‚ ⁑ ℍ n superscript ℍ 𝑛 \partial\mathbb{H}^{n} βˆ‚ blackboard_H start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT . Let Q 𝑄 Q italic_Q be the Euclidean convex hull of the Ο† πœ‘ \varphi italic_Ο† -image of the vertices of Ξ” k superscript Ξ” π‘˜ \Delta^{k} roman_Ξ” start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , made in a disc model of ℍ n superscript ℍ 𝑛 \mathbb{H}^{n} blackboard_H start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT . Let ψ : Ξ” k β†’ Q : πœ“ β†’ superscript Ξ” π‘˜ 𝑄 \psi:\Delta^{k}\to Q italic_ψ : roman_Ξ” start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT β†’ italic_Q be the only simplicial map that agrees with Ο† πœ‘ \varphi italic_Ο† on the 0 0 -skeleton.

We say that the map Ο† πœ‘ \varphi italic_Ο† is standard if there exist two homeomorphisms Ξ· : Im ⁒ ( Ο† ) β†’ Q : πœ‚ β†’ Im πœ‘ 𝑄 \eta:{\rm Im}(\varphi)\to Q italic_Ξ· : roman_Im ( italic_Ο† ) β†’ italic_Q and Ξ² : Ξ” k β†’ Ξ” k : 𝛽 β†’ superscript Ξ” π‘˜ superscript Ξ” π‘˜ \beta:\Delta^{k}\to\Delta^{k} italic_Ξ² : roman_Ξ” start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT β†’ roman_Ξ” start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT such that

Ξ· ∘ Ο† ∘ Ξ² = ψ . πœ‚ πœ‘ 𝛽 πœ“ \eta\circ\varphi\circ\beta=\psi. italic_Ξ· ∘ italic_Ο† ∘ italic_Ξ² = italic_ψ .

We say that a foliation β„± β„± \cal F caligraphic_F of Ξ” k superscript Ξ” π‘˜ \Delta^{k} roman_Ξ” start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT is standard if there exists a standard map Ο† : Ξ” k β†’ ℍ Β― n : πœ‘ β†’ superscript Ξ” π‘˜ superscript Β― ℍ 𝑛 \varphi:\Delta^{k}\to\overline{\mathbb{H}}^{n} italic_Ο† : roman_Ξ” start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT β†’ Β― start_ARG blackboard_H end_ARG start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT such that β„± = { Ο† - 1 ⁒ ( x ) } β„± superscript πœ‘ 1 π‘₯ {\cal F}=\{\varphi^{-1}(x)\} caligraphic_F = { italic_Ο† start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_x ) } .