On the conditions of topological equivalence of pseudoharmonic functions defined on disk
Let $D^2 \subset C$ be a closed two-dimensional disk and $f:D^2 \to R$ be a continuous function such that a restriction of $f$ to $\partial D^2$ is a continuous function with a finite number of local extrema and $f$ has a finite number of critical po…
Authors: Yevgen Polulyakh, Iryna Yurchuk
On the onditions of top ologial equiv alene of pseudoharmoni funtions dened on disk P oluly akh E., Y ur h uk I. Institute of Mathematis of Ukrainian National A adem y of Sienes, Kyiv Abstrat. Let D 2 ⊂ C b e a losed t w o-dimensional disk and f : D 2 → R b e a on tin uous funtion su h that a restrition of f to ∂ D 2 is a on tin uous funtion with a nite n um b er of lo al extrema and f has a nite n um b er of ritial p oin ts in In t D 2 su h that ea h of them is saddle (i.e., in its neigh- b orho o d the lo al represen tation of f is f = Rez n + const , where z = x + iy , n ≥ 2 ). This lass of funtions oinides with lass of pseudoharmoni fun- tions dened on D 2 [17℄. First, w e will onstrut an in v arian t of su h funtions whi h on tains all information ab out them. Then, in terms of su h in v arian t the neessary and suien t onditions for pseudoharmoni funtions to b e top ologially equiv alen t will b e obtained. Keyw ords . a pseudoharmoni funtion, a om binatorial diagram, a top o- logial onjugany . Intr odution In [817℄ the problems of top ologial lassiation of funtions, v e- tor elds and others strutures on manifold w ere solv ed. In most ases, su h solutions w ere reeiv ed b y the onstrution of om binatorial ob- jets whi h on tain all neessary fats ab out the struture b eing in v es- tigated. F or example, in [15℄ authors onstruted the spin graphs in order to lassify Morse-Smale's elds on losed t w o-dimensional mani- folds. So, an isomorphism of su h graphs is the neessary and suien t ondition of top ologial lassiation of elds. In [8, 14℄ the lassia- tion of M morsiations and bifurations w as obtained in terms of snak es (the sp eial p erm utations). Our goal is to onstrut an in v arian t of pseudoharmoni funtions de- ned on disk and to reeiv e the onditions for them to b e top ologially equiv alen t. In Setion 1 w e in tro due all neessary denitions whi h onnet the nature of pseudoharmoni funtions and theory of graphs. In Setion 2 the in v arian t of su h funtions is onstruted, and also its main prop erties are studied. In partiular, the in v arian t is a nite onneted graph with a partial orien tation and a partial order on its v erties whi h is generated b y a funtion. The main result of this pap er is Theorem 3.1. in Setion 3 whi h form ulates the neessary and suien t onditions for pseudoharmoni funtions to b e top ologially equiv alen t in terms of their om binatorial diagrams. 1 2 1. Preliminaries Let us remind some denitions and results whi h will b e helpful for us. Let T b e a tree with a set of v erties V and a set of edges E . Supp ose that T is non degenerated ( has at least one edge). Denote b y V ter a set of all terminal v erties of T (i.e. v erties of degree one). Supp ose that for some subset V ∗ ⊆ V the follo wing ondition holds true (1) V ter ⊆ V ∗ . Let also ϕ : T → R 2 is an em b edding su h that (2) ϕ ( T ) ⊆ D 2 , ϕ ( T ) ∩ ∂ D 2 = ϕ ( V ∗ ) . Lemma 1.1 (see [18℄) . A set R 2 \ ( ϕ ( T ) ∪ ∂ D 2 ) has a nite numb er of onne te d omp onents U 0 = R 2 \ D 2 , U 1 , . . . , U m , and for every i ∈ { 1 , . . . , m } a set U i is an op en disk and is b ounde d by a simple lose d urve ∂ U i = L i ∪ ϕ ( P ( v i , v ′ i )) , L i ∩ ϕ ( P ( v i , v ′ i )) = { ϕ ( v i ) , ϕ ( v ′ i ) } wher e L i is an ar of ∂ D 2 suh that the verti es ϕ ( v i ) and ϕ ( v ′ i ) ar e its endp oints, and ϕ ( P ( v i , v ′ i )) is an image of the unique p ath P ( v i , v ′ i ) in T whih onne ts v i and v ′ i . It is kno wn [19℄ that if E 1 , E 2 are losed disks and h : ∂ E 1 → ∂ E 2 is a homeomorphism that there exists a homeomorphism H : E 1 → E 2 su h that H | ∂ E 1 = h . Denition 1.1. Two funtions f , g : D 2 → R ar e al le d top ologially equiv alen t if ther e exist orientation pr eserving home omorphisms h 1 : D 2 → D 2 and h 2 : R → R suh that f = h − 1 2 ◦ g ◦ h 1 . W e remind that funtion f ( x, y ) is harmoni at a p oin t ( x 0 , y 0 ) if ∂ 2 f ∂ x 2 ( x 0 , y 0 ) + ∂ 2 f ∂ y 2 ( x 0 , y 0 ) = 0 . Denition 1.2. F untion f ( z ) is pseudoharmoni at a p oint z 0 = ( x 0 , y 0 ) if ther e exist a neighb orho o d U ( z 0 ) and a home omorphism ϕ of U ( z 0 ) onto itself suh that ϕ ( z 0 ) = z 0 and f ( ϕ ( z )) , z = ( x, y ) , is harmoni. F untion f is pseudoharmoni in a domain if it is pseudoharmoni at an y p oin t of it. Denition 1.3. Point z 0 ∈ D 2 is a regular p oint of f if ther e exist its op en neighb orho o d U ⊆ D 2 and a home omorphism ϕ : U → In t D 2 suh that ϕ ( z 0 ) = 0 and f ◦ ϕ − 1 ( z ) = Rez + f ( z 0 ) for al l z ∈ U . The neighb orho o d U wil l b e al le d anonial . 3 Denition 1.4. Point z 0 ∈ ∂ D 2 is a regular b oundary p oin t of f if ther e exist its neighb orho o d U in D 2 and a home omorphism h : U → D 2 + of suh neighb orho o d into upp er half-disk D 2 + suh that h ( z 0 ) = 0 , h ( U ∩ f − 1 ( f ( z 0 ))) = { 0 } × [0 , 1) , h ( U ∩ ∂ D 2 ) = ( − 1 , 1) × { 0 } and a funtion f ◦ h − 1 is stritly monotone on the interval ( − 1 , 1) × { 0 } . The neighb orho o d U wil l b e name d anonial . Remark 1.1. It is e asy to se e that the anoni al neighb orho o ds fr om Denitions 1.3 and 1.4 an b e hosen as smal l as ne e d. If a p oin t z 0 ∈ Int D 2 is not a regular p oin t of f ∈ F ( D 2 ) it will b e alled riti al . By denition all ritial p oin ts of f are saddle. P oin t of ∂ D 2 that is neither a b oundary regular p oin t nor an isolated p oin t of its lev el urv e will b e alled a riti al b oundary p oint . Denition 1.5. Numb er c is a ritial v alue of f if level set f − 1 ( c ) ontain riti al p oints. Numb er c is a regular v alue of f if a level set f − 1 ( c ) do es not ontain riti al p oints and it is home omorphi to a disjoint union of se gments whih interse t with a b oundary ∂ D 2 only in their endp oints. It is kno wn that an y lev el urv e of pseudoharmoni funtion is home- omorphi to a disjoin t union of trees [1, 3℄. Denition 1.6. Numb er c is a semiregular value of f if it is neither r e gular nor riti al. Remark 1.2. F r om Denitions it fol lows that level urves of semir e g- ular value ontain only b oundary riti al p oints and lo al extr ema of f (they b elong to ∂ D 2 and ar e isolate d p oints of level urves of f ). The level urves of the riti al value ontain the riti al p oints and they also an ontain b oundary riti al p oints and lo al extr ema. F rom Theorem 4.1 [6℄, see also [1℄, it follo ws that for an y ritial b oundary p oin t there exists an homeomorphism of its anonial neigh- b orho o d on to half-disk whi h maps that p oin t to origin and an image of its lev el set onsists of nite n um b er of ra ys outgoing from it. It is easy to pro v e that the lev el urv es of a semiregular v alue of pseu- doharmoni funtion are isomorphi to trees, in general disonneted. F or an y b oundary ritial p oin t z 0 the n um b er of domains on whi h half-disk is divided b y a set f − 1 ( f ( z 0 )) is greater than 2. Ev ery su h domain an b e asso iated with a sign either + or − dep ending on a sign of the dierene f ( z ) − f ( z 0 ) in it. W e should remark that if a n um b er of the domains is ev en then the domains adjoining to ∂ D 2 ha v e the same sign. Therefore a p oin t z 0 is lo al extrem um of f | ∂ D 2 . It is ob vious that in the ase when this sign is min us it is a lo al maxim um, otherwise it is minim um. Let n > 1 and z 1 , . . . , z 2 n is a sequene of p oin ts on ∂ D 2 ∼ = S 1 and they are passed in this order in the p ositiv e diretion on ∂ D 2 . Let γ k 4 a) b) − + − + − + − + − Figure 1. In ase a ) p oin t is regular but in ase b ) it is lo al maxim um of f | ∂ D 2 . is a p ositiv ely orien ted lose ar of ∂ D 2 from z k to z k +1 or z 1 when k = 2 n . Denote b y ˚ γ k an op en ar γ k without its endp oin ts. Denition 1.7 (see [20℄) . Supp ose that for a ontinuous funtion f : D → R ther e exists n = N ( f ) ≥ 2 and a se quen e of p oints z 1 , . . . , z 2 n − 1 , z 2 n ∈ F r D (in this or der these p oints ar e p asse d in the p ositive dir e tion on F r D ) suh that the fol lowing onditions ar e satise d: 1) every p oint of the domain In t D = D \ F r D is a r e gular p oint of f ; 2) ˚ γ 2 k − 1 6 = ∅ for k ∈ { 1 , . . . , n } , and any p oint of the ar ˚ γ 2 k − 1 is a r e gular b oundary p oint of f (in p artiular, the r estrition of f to γ 2 k − 1 is stritly monotone); 3) the ar s γ 2 k , k ∈ { 1 , . . . , n } , ar e onne te d omp onents of the level sets of f . Suh funtion wil l b e al le d a w eakly regular on D . Lemma 1.2 (see [20℄) . L et f b e a we akly r e gular funtion on D . Then N ( f ) = 2 . Denition 1.8 (see [20℄) . Supp ose that for some n ≥ 2 and a se quen e of p oints z 1 , . . . , z 2 n ∈ F r D a funtion f satises al l onditions of Def- inition 1.7 ex ept 3, but inste ad, the fol lowing ondition holds true 3 ′ ) for j = 2 k , k ∈ { 1 , . . . , n } , an ar γ j b elongs to some level set of f . Suh funtion wil l b e name d an almost w eakly regular on D . Let f b e an almost w eakly regular funtion on D . Denote b y 2 · N ( f ) a minimal n um b er of p oin ts and ars satisfying Denition 1.8. Ob viously , 2 · N ( f ) only dep ends on f . Prop osition 1.1 (see [20℄) . If for some n ≥ 2 and a se quen e of p oints z 1 , . . . , z 2 n ∈ F r D a funtion f satises al l onditions of Denition 1.8 and n = N ( f ) , then a ol le tion of sets { ˚ γ 2 k − 1 } n k =1 oinides with a ol le tion of onne te d omp onents of the set of r e gular b oundary p oints of f . 5 Denition 1.9 (see [20℄) . A simple ontinuous urve γ : [0 , 1] → W is al le d an U -tra jetory if f ◦ γ is str ongly monotone on the se gment [0 , 1 ] . Denition 1.10 (see [20℄) . L et f b e a we akly r e gular funtion on the disk D , let γ 1 , . . . , γ 4 b e ar s fr om Denition 1.7 . If thr ough every p oint of a set Γ whih is dense in ˚ γ 2 ∪ ˚ γ 4 p asses a U -tr aje tory, then the funtion f is al le d regular on D . Theorem 1.1 (see [20℄) . L et f b e a r e gular funtion on D , γ 1 , . . . , γ 4 b e the ar s fr om Denition 1.7 . L et D ′ = I 2 , if ˚ γ 2 6 = ∅ and ˚ γ 4 6 = ∅ ; D ′ = D 2 , if ˚ γ 2 ∪ ˚ γ 4 = ∅ ; D ′ = D 2 + , if exatly one of sets either ˚ γ 2 or ˚ γ 4 is empty. Supp ose that φ : F r D → F r D ′ is a home omorphism suh that φ ( K ) = K ′ , wher e K = f − 1 min z ∈ D ( f ( z )) ∪ max z ∈ D ( f ( z )) , K ′ = n ( x, y ) ∈ D ′ y ∈ min ( x,y ) ∈ D ′ ( y ) , max ( x,y ) ∈ D ′ ( y ) o . Ther e exists a home omorphism H f of disk D onto D ′ suh that H f | K = φ and f ◦ H − 1 f ( x, y ) = ay + b , ( x, y ) ∈ D ′ , for some a, b ∈ R , a 6 = 0 . 2. Combina torial inv ariant of pseudoharmoni funtions A t rst w e should remind the term of Reeb's graph. Let M b e a smo oth ompat manifold. Supp ose that f : M → R is a smo oth funtion with a nite n um b er of ritial p oin ts. Let us dene onneted omp onen t of lev el urv es of f − 1 ( a ) , where a ∈ R , as la y er. Then M is the union of all la y ers of f . Also w e an dene the relation of equiv alene as the prop ert y of p oin ts to b elong to a same la y er and onsider the quotien t spae b y this relation. It is homeomorphi to a nite graph named Reeb's graph and let us denote it b y Γ K − R ( f ) . Its v erties are omp onen ts of lev el urv es su h that they on tain the ritial p oin ts. Let D 2 b e a losed orien ted disk and f : D 2 → R b e a pseudohar- moni funtion. W e should remark that for a manifold with b oundary the onstrution of Reeb's graph is an op en problem therefore there is a reason to obtain another in v arian t for su h funtions. Constrution of invariant for pseudoharmoni funtion named as om binatorial diagram: 1) W e onstrut Reeb's graph Γ K − R ( f | ∂ D 2 ) of the restrition of f to ∂ D 2 . It is isomorphi to irle with ev en n um b er of v erties of degree 2 (v erties are lo al extrema of the restrition of f to ∂ D 2 ) and x an orien tation on Γ K − R ( f | ∂ D 2 ) whi h is generated b y the orien tation of D 2 . 6 2) Let a i b e the ritial v alues of f and c j b e the semiregular v alues. W e add to Γ K − R ( f | ∂ D 2 ) those onneted omp onen ts of sets f − 1 ( a 1 ) [ . . . [ f − 1 ( a k ) [ f − 1 ( c 1 ) [ f − 1 ( c 2 ) [ . . . [ f − 1 ( c l ) , of lev el urv es that on tain ritial and b oundary ritial p oin ts. It is ob vious that new v erties app ear on Γ K − R ( f | ∂ D 2 ) . W e set P ( f ) = Γ K − R ( f | ∂ D 2 ) ∪ [ i b f − 1 ( a i ) ∪ [ j b f − 1 ( c j ) , where b f − 1 ( a i ) ⊂ f − 1 ( a i ) , b f − 1 ( c j ) ⊂ f − 1 ( c j ) are those onneted omp onen ts of lev el sets that on tain ritial and b oundary rit- ial p oin ts. 3) W e put a partial order on v erties of P ( f ) b y using the v al- ues of f : v 1 < v 2 ⇐ ⇒ f ( x 1 ) < f ( x 2 ) , where v 1 , v 2 ∈ P ( f ) , x 1 , x 2 are p oin ts orresp onding to v erties v 1 , v 2 , resp etiv ely . In ase of the same v alues of funtion on v erties they will b e non omparable. This partial order is strit [21℄ sine the relation is an tireexiv e, an ti- symmetri and transitiv e. P ( f ) will b e alled ombinatorial diagr am of pseudoharmoni funtion f . By the onstrution P ( f ) is a nite partially orien ted graph with a strit partial order on v erties. 1 1 2 0 1 2 1 1 2 0 1 2 1 3 1 0 1 2 1 2 Figure 2. Example of a diagram of some pseudohar- moni funtion. W e onstruted the om binatorial in v arian t of f as subset of D 2 . W e will onsider it as the abstrat partially orien ted graph with xed relation of partial order on the set of v erties V ( P ( f )) . Denition 2.1. Two ombinatorial diagr ams P ( f ) and P ( g ) ar e iso- morphi if ther e exists an isomorphism φ : P ( f ) → P ( g ) b etwe en them whih pr eserves a strit p artial or der given on their verti es (maps φ V ( P ( f )) and φ − 1 V ( P ( g )) ar e monotone) and the orientation. 7 W e put the natural top ology on the diagram P ( f ) . F or example, it an b e in tro dued b y struture of one-dimensional CW-omplex on P ( f ) . All v erties of the graph P ( f ) an b e onsidered as 0-dimensional ells, similarly , all edges an b e onsidered as 1-dimensional ells. P ( f ) also an b e regarded as a subset of R 3 and all edges are straigh t seg- men ts. Denition 2.2. Home omorphism ϕ : P ( f ) → P ( g ) is said to real- ize an isomorphism φ : P ( f ) → P ( g ) of om binatorial diagrams if ϕ V ( P ( f )) = φ V ( P ( f )) and fr om φ ( e ) = e ′ it fol lows that ϕ ( e ) = e ′ for any e dge e ∈ E ( P ( f )) . Remark 2.1. It is le ar that every isomorphism φ of ombinatorial diagr ams is r e alize d by some home omorphism but it is not uniquely dene d: for every e dge e ∈ E ( P ( f )) we an arbitr arily ho ose a home- omorphism ϕ e : e → φ ( e ) suh that maps ϕ e and φ ar e the same on e ∩ V ( P ( f )) . W e onstruted the om binatorial diagram P ( f ) as a subset of D 2 therefore supp ort of diagram in D 2 is orretly dened sine it is the set (3) P f = Γ K − R ( f | ∂ D 2 ) ∪ [ i b f − 1 ( a i ) ∪ [ j b f − 1 ( c j ) , where b f − 1 ( a i ) and b f − 1 ( c j ) are onneted omp onen ts of lev el urv es of f whi h on tain the ritial and b oundary ritial p oin ts. Similarly , to the v erties of P ( f ) orresp onds the set V f whi h is the supp ort of the set of its v erties in D 2 . F untion f indues a strit partial order on it using the follo wing orrelations x 1 < x 2 ⇔ f ( x 1 ) < f ( x 2 ) . Denote b y M ( f ) ⊂ ∂ D 2 the set of lo al extr ema of f on D 2 . By the onstrution ev ery p oin t of this set orresp onds to some v ertex of P ( f ) , th us M ( f ) ⊂ V f . Other v erties of P ( f ) are haraterized b y the prop ert y that ea h of them is a ommon endp oin t of at least three edges, therefore it has no neigh b orho o d that is homeomorphi to segmen t in the spae P ( f ) . Denition 2.3. C r -subgraph of P ( f ) is a sub gr aph q ( f ) suh that: • q ( f ) is a simple oriente d yle; • arbitr ary p air of adja ent verti es v i , v i +1 ∈ q ( f ) is omp ar able. Let ϕ : P ( f ) → D 2 b e an arbitrary em b edding of top ologial spae P ( f ) in to D 2 su h that ϕ ( P ( f )) = P f . Gran ting what w e said ab o v e it is ob vious that an inlusion M ( f ) ⊆ ϕ ( V ( P ( f ))) is equiv alen t to ϕ ( V ( P ( f ))) = V f . In what fol lows unless otherwise stipulate d we assume that for any emb e dding of P ( f ) into D 2 the orientation of C r - sub gr aph oinides with the orientation of ∂ D 2 . 8 Denition 2.4. L et ϕ : P ( f ) → D 2 b e an emb e dding of top olo gi al sp a e P ( f ) into D 2 . It is al le d to b e onsisten t with f if the fol lowing orr elations hold true: • ϕ ( P ( f )) = P f ; • M ( f ) ⊆ ϕ ( V ( P ( f ))) ; • a p artial or der on ( V ( P ( f ))) = V f indu e d by a p artial or der on V ( P ( f )) with help of ϕ oinides with a p artial or der indu e d on this set fr om R by f . It is lear that there exist at least one em b edding ϕ : P ( f ) → D 2 whi h is onsisten t with f . If ψ : P ( f ) → P ( f ) is an isomorphism of P ( f ) on to itself (for example, iden tial map) whi h an b e realized b y homeomorphism ˆ ψ : P ( f ) → P ( f ) , then an em b edding ϕ ◦ ˆ ψ is also onsisten t with f . W e should remind that v erties v 1 and v 2 of some graph G are adja- ent if they are endp oin ts of the same edge. Let v b e some v ertex of the diagram P ( f ) and { v i } , i = 1 , k , b e a set of all adjaen t v erties to it. Then there exist p oin ts x and x i of D 2 that orresp ond to v erties v and v i . Denote b y X i ⊆ D 2 the set of p oin ts whi h orresp onds to edge e ( v , v i ) (it is lear that ev ery X i is homeomorphi to segmen t). Let us onsider the follo wing ases: Case 1: x ∈ In t D 2 . Then f ( x ) = f ( x i ) = a , where i = 1 , k and a is a ritial v alue. Therefore v erties v , v 1 , v 2 , . . . , v k are pairwise non omparable. Sine lev el set of the ritial v alue a is a nite tree then all v erties of it are non omparable. Case 2: x ∈ ∂ D 2 . In this ase the p oin t x is either regular or lo al extrem um of f | ∂ D 2 whi h is on tin uous and monotonially inrease (derease) b et w een adjaen t lo al extrema. Therefore, among sets X i there exist su h that funtion monotonially inreases (dereases) on them. Cirle is losed Jordan urv e then there are exatly t w o su h sets X j and X k whose endp oin ts are p oin ts x j and x k . So, it follo ws that among all v erties { v i } adjaen t to v there exist exatly t w o v erties v j and v k whi h are omparable with a v ertex v . F or b oth v j and v k there exist exatly t w o v erties whi h are omparable to it th us these v erties generate a yle (the ase of t w o or more non in terseting yles is imp ossible sine a disk has one b oundary irle). It is ob vious that v together with b oth v erties v j and v k b elong to q ( f ) yle. The fat that the diagram P ( f ) is onstruted b y pseudoharmoni funtion implies sev eral harateristis of it. Main pr op erties of P ( f ) : C1) there exists the unique C r -subgraph q ( f ) ∈ P ( f ) ; 9 C2) P ( f ) \ q ( f ) = S i Ψ i , Ψ j T Ψ i = ∅ , where i 6 = j , and ev ery Ψ i is a tree su h that for an y index i arbitrary t w o v erties v ′ , v ′′ ∈ Ψ i are non omparable; C3) there exists an em b edding ψ : P ( f ) → D 2 su h that ψ ( q ( f )) = ∂ D 2 and ψ ( P ( f ) \ q ( f )) ⊂ In t D 2 ; C4) for ev ery onneted omp onen t Θ of D 2 \ P f the funtion f is regular (see [20℄) on the set Θ . F rom what w as said ab o v e the existene of C r -subgraph and the fairness of C 2 follo w. F rom the existene of C r -subgraph and C 2 it follo ws that q ( f ) is unique. Condition C 3 follo ws from fat that P ( f ) is a diagram of a funtion f , dened on D 2 . C r -subgraph q ( f ) ∈ P ( f ) is unique th us from the denitions it is easy to see that for ev ery em b edding ψ : P ( f ) → D 2 whi h is onsisten t with f the equalit y ψ ( q ( f )) = ∂ D 2 should hold true. By the denition of the diagram P ( f ) an y tree Ψ i orresp onds to a onneted omp onen t of some ritial or semiregular lev el set of f . A n um b er of trees is the same as a n um b er of su h omp onen ts whi h on tain ritial or b oundary ritial p oin ts. Denote b y P c f = P f \ ∂ D 2 the union of su h omp onen ts. Let ψ : P ( f ) → D 2 b e an em b edding whi h is onsisten t with f . If the endp oin ts v ′ and v ′′ of some edge e = e ( v ′ , v ′′ ) of P ( f ) are non omparable, then ˚ e = e \ { v ′ , v ′′ } ∈ P ( f ) \ q ( f ) ⊆ S i Ψ i . Th us ψ ( ˚ e ) ⊆ P f ∩ In t D 2 ⊆ P c f . Then there exists c = c ( e ) ∈ R su h that ϕ ( e ) ⊂ f − 1 ( c ) . An y onneted set ψ (Ψ i ) b elongs to some onneted omp onen t of P c f . F rom the fats that a map ψ is an em b edding and ψ ( q ( f )) = ∂ D 2 follo w the equalities ψ [ i Ψ i = ψ P ( f ) \ q ( f ) = ψ ( P ( f )) \ ψ ( q ( f )) = P f \ ∂ D 2 = P c f . By the denition the n um b er of onneted omp onen ts of sets S i Ψ i and P c f oinides th us an y set ψ (Ψ i ) is a onneted omp onen t of P c f . Let us om bine together orollaries of Conditions C 1 C 3 whi h w e obtained ab o v e. Prop osition 2.1. L et P ( f ) b e a ombinatorial diagr am of pseudo- harmoni funtion f and ψ : P ( f ) → D 2 b e an emb e dding whih is onsistent with f . Then the fol lowing onditions hold true: • ψ ( q ( f )) = ∂ D 2 ; • for any tr e e Ψ i the set ψ (Ψ i ) is a omp onent of riti al or semir e gular level set of f . Let us pro v e Condition C 4 . Prop osition 2.2. L et P ( f ) b e a ombinatorial diagr am of pseudo- harmoni funtion f and ψ : P ( f ) → D 2 b e an emb e dding suh that ψ ( q ( f )) = ∂ D 2 . 10 The set F r Σ = F r Σ is an image of a simple yle Q of P ( f ) for any onne te d omp onent Σ of D 2 \ ψ ( P ( f )) . Pr o of. All v erties of Ψ j ⊂ P ( f ) whi h do not b elong to C r -yle q ( f ) orresp ond to ritial p oin ts of f for an y j , th us they ha v e ev en degree no smaller than 2. Therefore the set V j ter of all v erties of Ψ j of degree 1 is on tained in q ( f ) and w e an apply Lemma 1.1 to a map ψ Ψ j . By indution on the n um b er of trees Ψ i em b edded in to disk from Lemma 1.1 it follo ws that a b oundary F r Σ of Σ is simple Jordan urv e. Let us pro v e that its preimage Q = ψ − 1 (F r Σ) is a subgraph of P ( f ) . It sues to v erify the follo wing assertion. Let e = e ( v 1 , v 2 ) b e some edge of P ( f ) and x ∈ ˚ e = e \ { v 1 , v 2 } b e an inner p oin t of e . If x ∈ Q , then e ⊂ Q . It is ob vious that the set Q is a simple losed urv e. Therefore Q \ { x ′ } is onneted for an y x ′ ∈ Q . Th us Q \ e 6 = ∅ (an y p oin t of segmen t e exept its endp oin ts splits it, see [22℄). Supp ose that an edge e is supp ort of simple on tin uous urv e α : I → P ( f ) , α (0) = v 1 , α (1) = v 2 . Therefore x = α ( τ ) for some τ ∈ (0 , 1) . W e should remark that e is one-dimensional ell of CW-omplex P ( f ) , th us ˚ e = α ( ˚ I ) is an op en subset of P ( f ) (w e denoted ˚ I = (0 , 1) ). F or an y in terv al ˚ I ( t 1 , t 2 ) = ( t 1 , t 2 ) , t 1 , t 2 ∈ I , t 1 < t 2 , the set α ( ˚ I ( t 1 , t 2 )) is an op en subset of P ( f ) . It is also ob vious that P ( f ) \ α ([ t 1 , t 2 ]) , where t 1 , t 2 ∈ I , t 1 < t 2 is op en in P ( f ) . Let us sho w that at least one of sets α ([0 , τ ]) , α ([ τ , 1 ]) b elong to Q . Supp ose that it do es not hold true. So, there exist t 1 ∈ [0 , τ ] and t 2 ∈ [ τ , 1] su h that α ( t 1 ) , α ( t 2 ) / ∈ Q . Then the nonempt y sets Q ∩ α ( ˚ I ( t 1 , t 2 )) ∋ x and Q \ α ([ t 1 , t 2 ]) ⊇ Q \ e op en in subspae Q of P ( f ) generate a partition of Q , but it is imp ossible sine Q is onneted. Supp ose that α ( t ) / ∈ Q for some t ∈ I . Without loss of generalit y w e an assume that t < τ . Then α ([ τ , 1 ]) ⊂ Q . Let us x t ′ ∈ ( τ , 1) and set x ′ = α ( t ′ ) . The nonempt y op en in Q sets Q ∩ α ( ˚ I ( t, t ′ )) ∋ x and Q \ α ( [ t, t ′ ]) ⊃ Q \ e generate in Q the partition of subset Q \ { x ′ } , but it is imp ossible sine Q \ { x ′ } is onneted. Th us e ∈ Q and Q is a subgraph of P ( f ) . The set Q is homeomorphi to irle th us it is a simple yle. Lemma 2.1. L et P ( f ) b e a diagr am onstrute d at pseudoharmoni funtion f and ψ : P ( f ) → D 2 b e an emb e dding that is onsistent with f . Then for any omp onent Θ of the omplement D 2 \ P ( f ) = D 2 \ P f its losur e Θ is home omorphi to disk and f is r e gular on Θ . Pr o of. Let Θ b e a onneted omp onen t of D 2 \ P ( f ) . Let us pro v e, at rst, that f is w eekly regular in Θ . F rom Prop ositions 2.1 and 2.2 it follo ws that a b oundary of Θ is a simple losed urv e. Th us Θ is a 11 losed disk and F r Θ = Θ ∩ P f = (Θ ∩ ψ ( q ( f ))) ∪ Θ ∩ ψ [ i Ψ i = Γ V ∪ Γ E ∪ Γ T , where Γ T = Θ ∩ ψ ( S i Ψ i ) ; Γ V = Θ ∩ M ( f ) is a set of p oin ts of ∂ D 2 ∩ F r Θ whi h orresp ond to v erties of P ( f ) of q ( f ) \ S i Ψ i ; Γ E are the op en ars of ∂ D 2 ∩ F r Θ whi h orresp ond to the edges of the yle q ( f ) without endp oin ts. It is ob vious that the sets Γ V , Γ E and Γ T are pairwise disjoin t. The set Γ V onsists of the isolated p oin ts of lev el sets of f . Ea h of them is a lo al extrem um of f in D 2 . The funtion f is lo ally onstan t on Γ T therefore an y onneted omp onen t K of su h set b elongs to ψ S i Ψ i and there exists c K ∈ R su h that K ∈ f − 1 ( c K ) . Let Γ K b e a onneted omp onen t of f − 1 ( c K ) ∩ Θ on taining K . Then Γ K ⊆ Θ ∩ ψ S i Ψ i = Γ T . Consequen tly Γ K = K . F rom the denition it follo ws that all p oin ts of Γ E are regular b ound- ary p oin ts of f in D 2 . It is easy to see that suien tly small anonial neigh b orho o d of an y p oin t of Γ E b elongs to Θ therefore all p oin ts of Γ E are regular b oundary p oin ts of f in Θ . By the denition the set Γ E has a nite n um b er of onneted omp o- nen ts (their n um b er is no more than a n um b er of the edges of the yle q ( f ) ) therefore there exists a nite olletion of p oin ts z 1 , . . . , z 2 n ∈ F r Θ whi h divide the irle F r Θ in to ars γ 1 , . . . , γ 2 n su h that Γ E = S n k =1 ˚ γ 2 k − 1 (some ars with ev en indies an degenerate in to p oin ts). It is lear that F r Θ \ S n k =1 ˚ γ 2 k − 1 = F n k =1 γ 2 k = Γ V ∪ Γ T . The sets Γ V and Γ T are losed and disjoin t th us an y ar γ 2 k , k ∈ { 1 , . . . , n } , b elongs to either Γ V or Γ T . F rom the preeding it follo ws that an y set γ 2 k , k ∈ { 1 , . . . , n } , is a onneted omp onen t of some lev el set of f on Θ . Therefore the olletion of p oin ts z 1 , . . . , z 2 n satises to Denition 1.7 and f is w eakly regular on Θ . F rom Lemma 1.2 it follo ws that n = N ( f | Θ ) = 2 . If ˚ γ 2 k 6 = ∅ , k ∈ { 1 , 2 } , then γ 2 k ∈ Γ T (the set Γ V is disrete therefore γ 2 k ∩ Γ V = ∅ , see ab o v e) and an y p oin t z ∈ ˚ γ 2 k either b elongs to In t D 2 or is b oundary ritial p oin t of f . If z ∈ ˚ γ 2 k ∩ In t D 2 , then there exist an op en neigh b orho o d W z of z in D 2 and a homeomorphism Φ z : W z → In t D 2 su h that Φ z ( z ) = 0 and f ◦ Φ − 1 z ( w ) = Rew m + f ( z ) for some m ≥ 2 . The set Φ z ( f − 1 ( f ( z ))) divides In t D 2 on to 2 m op en setors su h that ea h of them (for suf- ien tly small neigh b orho o d W z ) b elongs to D 2 \ P ( f ) . Th us for at least one of them its image under the ation of Φ − 1 z b elongs to Θ . It is ob vious that for ev ery su h setor there exists U -tra jetory of f whi h passes through the p oin t z and is on tained in the losure of the im- age of setor under the ation of Φ − 1 z . T aking that in to aoun t some U -tra jetory in Θ passes through z . 12 The n um b er of the b oundary ritial p oin ts of f on D 2 is nite therefore Γ = ˚ γ 2 k ∩ Int D 2 is a dense subset of an ar ˚ γ 2 k , k ∈ { 1 , 2 } , and funtion f is regular on Θ . Lemma 2.2. L et P ( f ) b e a ombinatorial diagr am of pseudoharmoni funtion; ψ 1 , ψ 2 : P ( f ) → D 2 b e emb e ddings suh that ψ i ( q ( f )) = ∂ D 2 , i = 1 , 2 . If an image ψ 1 ( Q ) of a simple yle Q ⊂ P ( f ) is a b oundary of some omp onent of the omplement D 2 \ ψ 1 ( P ( f )) , then an image ψ 2 ( Q ) is a b oundary of some omp onent of the omplement D 2 \ ψ 2 ( P ( f )) . Pr o of. Let us x an em b edding ϕ : P ( f ) → D 2 onsisten t with f . Let ψ : P ( f ) → D 2 b e an em b edding su h that ψ ( q ( f )) = ∂ D 2 . It is ob vious that lemma follo ws from the follo wing statemen t: an image ψ ( Q ) of a simple yle Q ⊆ P ( f ) is a b oundary of some omp onen t of the omplemen t D 2 \ ψ ( P ( f )) i a urv e ϕ ( Q ) is a b oundary of one of omp onen ts of D 2 \ P f = D 2 \ ϕ ( P ( f )) . Let us pro v e this statemen t. Supp ose that an image ϕ ( Q ) of the yle Q b ounds one of the omp o- nen ts Θ of the set D 2 \ P f . F rom Lemma 2.1 it follo ws that f is regular on disk Θ , therefore there exist p oin ts z 1 , . . . , z 4 ∈ F r Θ = ϕ ( Q ) whi h divide a urv e ϕ ( Q ) in to ars γ 1 , . . . , γ 4 satisfying the follo wing ondi- tions: • ˚ γ 1 6 = ∅ , ˚ γ 3 6 = ∅ , and the set ˚ γ 1 ∪ ˚ γ 3 is the set of b oundary regular p oin ts of f on Θ ; • γ 2 and γ 4 are the omp onen ts of lev el sets of f on Θ . F rom these onditions it follo ws that (see a pro of of Lemma 2.1) γ 2 ∪ γ 4 = F r Θ ∩ ϕ ( V ( P ( f )) ) ∪ ϕ [ i Ψ i ⊇ ⊇ F r Θ ∩ ϕ [ i Ψ i = F r Θ ∩ [ i ϕ (Ψ i ) . (4) Supp ose that γ 2 ⊆ f − 1 ( c ′ ) , γ 4 ⊆ f − 1 ( c ′′ ) for c ′ , c ′′ ∈ R . As ˚ γ 1 6 = ∅ and all p oin ts of this set are regular b oundary p oin ts of f on Θ the follo wing statemen t holds true: z 1 6 = z 2 (sine ˚ γ 1 = γ 1 \ { z 1 , z 2 } ) and f is stritly monotone on γ 1 . Th us c ′′ = f ( z 1 ) 6 = f ( z 2 ) = c ′ and the sets γ 2 and γ 4 b elong to dieren t lev el sets of f . Let us set w i = ψ ◦ ϕ − 1 ( z i ) , ν i = ψ ◦ ϕ − 1 ( γ i ) , i ∈ { 1 , . . . , 4 } . The urv e ψ ( Q ) b ounds an op en domain Σ . F rom ( 4) it follo ws that ν 2 ∪ ν 4 ⊇ F r Σ ∩ ψ ( S i Ψ i ) . Let us assume that a urv e ψ ( Q ) ⊆ ψ ( P ( f )) is not a b oundary of onneted omp onen t of D 2 \ ψ ( P ( f )) . Therefore Σ ∩ ψ ( P ( f )) 6 = ∅ . W e x z ∈ Σ ∩ ψ ( P ( f )) . It is ob vious that Σ ⊆ In t D 2 , therefore x = ψ − 1 ( z ) ∈ P ( f ) \ q ( f ) ⊆ S i Ψ i and z ∈ ψ (Ψ j ) for some j . All v erties of the tree Ψ j whi h do not b elong to C r -yle q ( f ) orresp ond to the ritial p oin ts of f th us they ha v e ev en degree no less than 2. 13 So, the set V j ter of all v erties of degree one of tree Ψ j is on tained in q ( f ) . It is easy to see that this set has at least t w o elemen ts. By easy he k w e an see that for an y p oin t u of a subspae Ψ j of the spae P ( f ) there exist v ′ u , v ′′ u ∈ V j ter and path P ( v ′ u , v ′′ u ) ⊂ Ψ j whi h onnets v ′ u with v ′′ u and passes through u . Let us x for a p oin t x = ψ − 1 ( z ) v erties v ′ x , v ′′ x ∈ V j ter and a path P ( v ′ x , v ′′ x ) ⊂ Ψ j whi h onnets them and passes through a p oin t x . W e also x a simple on tin uous urv e α : I → P ( f ) whose supp ort is a path P ( v ′ x , v ′′ x ) . Supp ose that α (0) = v ′ x , α (1) = v ′′ x , α ( τ ) = x . It is kno wn that ψ ( V j ter ) ⊂ ψ ( q ( f )) = ∂ D 2 , but z = ψ ( x ) ∈ Σ ⊆ In t D 2 . Therefore τ ∈ (0 , 1) . F urthermore ψ ( v ′ x ) , ψ ( v ′′ x ) / ∈ Σ , so that ea h of the sets ψ ◦ α ([0 , τ ]) and ψ ◦ α ([ τ , 1]) should in terset ψ ( Q ) = F r Σ . Supp ose that t ′ = inf { t ∈ [0 , τ ] | ψ ◦ α ([ t, τ ]) ∈ Σ } , t ′′ = sup { t ∈ [ τ , 1] | ψ ◦ α ([ τ , t ]) ∈ Σ } . Then τ ∈ ( t ′ , t ′′ ) ⊆ ( ψ ◦ α ) − 1 (Σ) but ψ ◦ α ( t ′ ) , ψ ◦ α ( t ′′ ) ∈ ψ ( Q ) = F r Σ . It is lear that for ev ery e = e ( w ′ , w ′′ ) ∈ P ( v ′ x , v ′′ x ) there exist t ′ , t ′′ ∈ I , t ′ < t ′′ su h that w ′ = α ( t ′ ) , w ′′ = α ( t ′′ ) and e = α ([ t ′ , t ′′ ]) . So, there exist n um b ers t 0 = 0 < t 1 < · · · < t k = 1 , v erties v 0 = v ′ x , v 1 , . . . , v k = v ′′ x , and the edges e 1 , . . . , e k of the tree Ψ j su h that v i = α ( t i ) , i ∈ { 0 , . . . , k } , and e i = α ([ t i − 1 , t i ]) , i ∈ { 1 , . . . , k } . F rom the hoie of the n um b ers t ′ and t ′′ it follo ws that only the p oin ts α ( t ′ ) and α ( t ′′ ) b elong to the in tersetion of the set α ([ t ′ , t ′′ ]) and the subgraph Q . Th us α ( t ′ ) = v r and α ( t ′′ ) = v s for some r , s ∈ { 0 , . . . , k } , r < s . Hene the path P ( v r , v s ) = α ([ t ′ , t ′′ ]) onnets the v erties v r 6 = v s of the yle Q and in tersets Q along the set { v r , v s } . W e kno w already that ψ ( v r ) , ψ ( v s ) ∈ ν 2 ∪ ν 4 . Observ e that the p oin ts ψ ( v r ) and ψ ( v s ) an not b elong to the dieren t ars ν 2 , ν 4 . Really from Prop osition 2.1 it follo ws that there is c ∈ R su h that ϕ (Ψ j ) ⊆ f − 1 ( c ) , therefore f ◦ ϕ ( v r ) = f ◦ ϕ ( v s ) = c . On the other hand, as w e he k ed ab o v e, the sets γ 2 = ϕ ◦ ψ − 1 ( ν 2 ) and γ 4 = ϕ ◦ ψ − 1 ( ν 4 ) b elong to the dieren t lev el sets of f . Without loss of generalit y , supp ose that ψ ( v r ) , ψ ( v s ) ∈ ν 2 . The set ν 2 is onneted, moreo v er ν 2 ⊆ ψ (Ψ j ) , ν 4 ∩ ψ (Ψ j ) = ∅ and ν 2 ∪ ν 4 ⊇ ψ ( Q ) ∩ ψ ( S i Ψ i ) . Therefore the onneted set ψ − 1 ( ν 2 ) = Q ∩ Ψ j is a subgraph of P ( f ) . Hene there exists the path ˆ P ( v r , v s ) onneting the v erties v r and v s in Q ∩ Ψ j . F rom the onstrution w e ha v e P ( v r , v s ) ∪ ˆ P ( v r , v s ) ⊆ Ψ j and P ( v r , v s ) 6 = ˆ P ( v r , v s ) . Sine Ψ j is a tree then the v erties v r and v s an b e on- neted b y the unique path in Ψ j . So, w e obtained the on tradi- tion whi h pro v es that ψ ( P ( f )) ∩ Σ = ∅ . Considering that F r Σ = ψ ( Q ) ⊆ ψ ( P ( f )) it follo ws that Σ is a onneted omp onen t of the set D 2 \ ψ ( P ( f )) . 14 Supp ose no w that for some simple yle Q ′ ⊆ P ( f ) the urv e ψ ( Q ′ ) b ounds a onneted omp onen t Σ ′ of the set D 2 \ ψ ( P ( f )) , but the urv e ϕ ( Q ′ ) is not a b oundary of the onneted omp onen t of the set D 2 \ ϕ ( P ( f )) = D 2 \ P f . Let us pro v e that in this ase Q ′ ⊆ S i Ψ i . If it do es not hold true, then there exists an edge e 0 ⊆ Q ′ ∩ q ( f ) . Eviden tly , there exists a onneted omp onen t Θ of the set D 2 \ ϕ ( P ( f )) whose b oundary on tains the set ϕ ( e 0 ) . Supp ose that Q = ϕ − 1 (F r Θ ) . F rom Prop ositions 2.1 and 2.2 it follo ws that Q is a simple yle. As w e pro v ed ab o v e the set ψ ( Q ) is a b oundary of some onneted omp onen t Σ of the set D 2 \ ψ ( P ( f )) . Ob viously , e 0 ⊆ Q ∩ Q ′ . Let x b e an inner p oin t of an edge e 0 , z = ψ ( x ) . By the onditions of prop osition w e ha v e z ∈ ψ ( q ( f )) = ∂ D 2 . It is easy to see that for su- ien tly small neigh b orho o d W of the p oin t z in D 2 whi h is homeomor- phi to half-disk the set W \ ψ ( P ( f )) = W \ ψ ( e 0 ) is onneted. There- fore W \ ψ ( P ( f )) ⊆ Σ ∩ Σ ′ 6 = ∅ . F rom Σ ∩ F r Σ ′ ⊆ Σ ∩ ψ ( P ( f )) = ∅ it follo ws that Σ ⊆ Σ ′ . By a parallel argumen t Σ ′ ⊆ D 2 \ ψ ( P ( f )) , th us Σ ′ ∩ F r Σ ⊆ Σ ′ ∩ ψ ( P ( f ) ) = ∅ and Σ ′ ⊆ Σ . Hene Σ ′ = Σ , Q ′ = Q and the urv e ϕ ( Q ′ ) b ounds a onneted omp onen t of the set D 2 \ P f , but it on tradits to the hoie of the yle Q ′ . Therefore Q ′ ⊆ S i Ψ i . The set Q ′ is onneted th us there is j su h that Q ′ ⊆ Ψ j . But Ψ j is tree and no one yle is on tained in it. This on tradition is a nal step of pro of. Corollary 2.1. In the onditions of L emma 2.2 ther e exists a home o- morphism Φ : D 2 → D 2 suh that Φ ◦ ψ 1 = ψ 2 . Pr o of. Let Σ 1 , . . . , Σ k b e the onneted omp onen ts of the set D 2 \ ψ 1 ( P ( f )) and Q 1 , . . . , Q k b e the yles of the graph P ( f ) su h that ψ 1 ( Q i ) = F r Σ i , i ∈ { 1 , . . . , k } , see Prop osition 2.2. It is lear that P ( f ) = S k i =1 Q i . By Lemma 2.2, ev ery set ψ 2 ( Q i ) is a b oundary of some onneted omp onen t Σ ′ i of D 2 \ ψ 2 ( P ( f )) . By using Lemma 2.2 one again it is easy to see that D 2 \ ψ 2 ( P ( f )) = S k i =1 Σ ′ i . By S ho enies's theorem, for ev ery i ∈ { 1 , . . . , k } , the homeomorphism ψ 2 ◦ ψ − 1 1 ψ 1 ( Q i ) : ψ 1 ( Q i ) → ψ 2 ( Q i ) an b e extended to a homeomorphism of disks Φ i : Σ i → Σ ′ i , see [22℄. It is easy to see that the map Φ : D 2 → D 2 , Φ( z ) = Φ i ( z ) , for z ∈ Σ i , is w ell dened and maps D 2 on to itself bijetiv ely . A nite family of the losed sets { Σ i } generates the fundamen tal o v er of D 2 and on ea h of them Φ is on tin uous. Hene Φ is on tin uous on D 2 , see [23℄. It is kno wn that a on tin uous bijetiv e map of ompatum to a Hausdor spae is a homeomorphism. 15 3. The onditions of topologial equiv alene Let f : D 2 → R b e a pseudoharmoni funtion and a 1 < · · · < a N b e all its ritial and semiregular v alues. Let us onsider a homeomor- phism h f : [ a 1 , a N ] → [1 , N ] su h that h f ( a j ) = j for all j ∈ { 1 , . . . , N } . It is easy to see that a on tin uous funtion ˆ f = h f ◦ f is pseudo- harmoni, set of its ritial and semiregular v alues is { 1 , . . . , N } , and P ( ˆ f ) = P ( f ) . F untion ˆ f is alled a standar dization of f . Theorem 3.1. Two pseudoharmoni funtions f and g ar e top olo gi- al ly e quivalent i ther e exists an isomorphism of ombinatorial dia- gr ams ϕ : P ( f ) → P ( g ) whih pr eserves a strit p artial or der dene d on them and the orientation. Pr o of. Ne essity. Supp ose that t w o pseudoharmoni funtions f : D 2 → R and g : D 2 → R are top ologially equiv alen t. Then there exist homeomorphisms H : D 2 → D 2 and h : R → R su h that f = h − 1 ◦ g ◦ H . Also to f and g orresp ond their om binatorial diagrams P ( f ) and P ( g ) with the strit partial order and the orien ta- tion whi h onform to f and g . Let ψ 1 and ψ 2 b e em b eddings of P ( f ) and P ( g ) in to D 2 whi h are onsisten t with f and g , resp etiv ely (re- all that the partial orien tations on C r -subgraphs of P ( f ) and P ( g ) are the same as the orien tation of ∂ D 2 ). Eviden tly , the homeomorphism H maps the regular p oin ts of f on to the regular p oin ts of g and the ritial p oin ts of f on to the ritial p oin ts of g , resp etiv ely . F rom h ◦ f = g ◦ H and bijetivit y of h it follo ws that the homeomorphism H maps the regular, ritial and semiregular lev els of f on to regular, ritial and semiregular lev els of g , resp etiv ely . Th us H ◦ ψ 1 ( P ( f )) = ψ 2 ( P ( g )) and the bijetiv e map ϕ = ψ − 1 2 ◦ H ◦ ψ 1 : P ( f ) → P ( g ) is dened. So, w e ha v e the follo wing omm utativ e diagram: P ( f ) ψ 1 − − − → D 2 f − − − → R ϕ y H y y h P ( g ) − − − → ψ 2 D 2 − − − → g R It is easy to see that ϕ denes an isomorphism of graphs. Let us pro v e that the maps ϕ and ϕ − 1 are monotone. W e should remind that only the preserving orien tation homeomorphisms R are onsidered th us the map h : R → R preserv es an order of p oin ts of R . Let v 1 and v 2 b e t w o v erties of the diagram P ( f ) . By the denition of the diagram P ( f ) an inequalit y v 1 < v 2 is equiv alen t to f ◦ ψ 1 ( v 1 ) < f ◦ ψ 1 ( v 2 ) , so, that is also equiv alen t to h ◦ f ◦ ψ 1 ( v 1 ) < h ◦ f ◦ ψ 1 ( v 2 ) . This inequalit y is equiv alen t to g ◦ ψ 2 ◦ ϕ ( v 1 ) < g ◦ ψ 2 ◦ ϕ ( v 2 ) sine h ◦ f ◦ ψ 1 = g ◦ H ◦ ψ 1 = g ◦ ψ 2 ◦ ϕ . By the denition of the relation of order on P ( g ) , the last inequalit y is equiv alen t to ϕ ( v 1 ) < ϕ ( v 2 ) . So, the inequalities v 1 < v 2 and ϕ ( v 1 ) < ϕ ( v 2 ) are equiv alen t. Finally , w e remind that ϕ is bijetiv e 16 b y the onstrution th us ϕ and ϕ − 1 are monotone. F rom what w e said it follo ws that ϕ : P ( f ) → P ( g ) is an isomorphism of diagrams. Suieny. Supp ose that f , g : D 2 → R are pseudoharmoni fun- tions and P ( f ) , P ( g ) are their diagrams su h that there exists an iso- morphism φ : P ( f ) → P ( g ) preserving a strit partial order on the set of v erties and the partial orien tations on their C r subgraphs. A t rst, w e w an t to replae the funtions f and g on the normal- ized pseudoharmoni funtions ˆ f and ˆ g with the same om binatorial diagrams as f and g . Let a 1 < · · · < a N b e the ritial and the semiregular v alues of f and b 1 < · · · < b M b e ritial and semiregular v alues of g . W e x homeomorphisms h f : [ a 1 , a N ] → [1 , N ] and h g : [ b 1 , b M ] → [1 , M ] su h that h f ( a i ) = i and h g ( b j ) = j for all i ∈ { 1 , . . . , N } and j ∈ { 1 , . . . , M } . Ob viously , the maps h f and h g are orien tation preserv- ing. Required normalized pseudoharmoni funtions ha v e the follo wing forms ˆ f = h f ◦ f and ˆ g = h g ◦ g . Let us x an em b edding ψ f : P ( f ) → D 2 whi h is onsisten t with f and an em b edding ψ g : P ( g ) → D 2 whi h is onsisten t with g (w e should remark that the em b eddings ψ f and ψ g are also onsisten t with ˆ f and ˆ g , resp etiv ely). Let us pro v e that for an y v ertex v of P ( f ) the follo wing ondition holds true (5) ˆ f ◦ ψ f ( v ) = ˆ g ◦ ψ g ◦ φ ( v ) . W e remark that f ◦ ψ f ( V ( P ( f ))) = { a 1 , . . . , a N } , g ◦ ψ g ( V ( P ( g ))) = { b 1 , . . . , b M } . Hene ˆ f ◦ ψ f ( V ( P ( f ))) = { 1 , . . . , N } , ˆ g ◦ ψ g ( V ( P ( g ))) = { 1 , . . . , M } . Fix the sequene of v erties u 1 , . . . , u s ∈ V ( P ( f )) su h that ˆ f ◦ ψ f ( u 1 ) = 1 , . . . , ˆ f ◦ ψ f ( u s ) = s = ˆ f ◦ ψ f ( v ) . Then u 1 < · · · < u s in P ( f ) hene φ ( u 1 ) < · · · < φ ( u s ) in P ( g ) and ˆ g ◦ ψ g ( u 1 ) ◦ φ < · · · < ˆ g ◦ ψ g ◦ φ ( u s ) . Th us j ≤ ˆ g ◦ ψ g ◦ φ ( u j ) , j ∈ { 1 , . . . , s } . Hene ˆ f ◦ ψ f ( v ) = ˆ f ◦ ψ f ( u s ) = s ≤ ˆ g ◦ ψ g ◦ φ ( u s ) = ˆ g ◦ ψ g ◦ φ ( v ) . By replaing f at g , w e ha v e ˆ f ◦ ψ f ( v ) ≥ ˆ g ◦ ψ g ◦ φ ( v ) . So, (5) holds true. F rom ( 5 ) it follo ws that M = N and ˆ f ( D 2 ) = ˆ g ( D 2 ) = [1 , N ] . Let us onstrut a homeomorphism ϕ : P ( f ) → P ( g ) su h that it realizes an isomorphism φ and satises the follo wing relation on the spae P ( f ) (6) ˆ f ◦ ψ f = ˆ g ◦ ψ g ◦ ϕ . By the denition w e ha v e ϕ ( v ) = φ ( v ) , v ∈ V ( P ( f ) ) , on the set of v erties. 17 Supp ose that an edge e = e ( v ′ , v ′′ ) ∈ E ( P ( f )) b elongs to a subgraph S i Ψ i ( f ) = P ( f ) \ q ( f ) . The set e is onneted and ˆ f ◦ ψ f is lo ally onstan t on S i Ψ i ( f ) , therefore ψ f ( e ) ⊆ ˆ f − 1 ( c ) for some c ∈ R . In partiular, ˆ f ◦ ψ f ( v ′ ) = ˆ f ◦ ψ f ( v ′′ ) = c and v erties v ′ and v ′′ are non omparable in P ( f ) . So, the v erties φ ( v ′ ) and φ ( v ′′ ) are non omparable in P ( g ) and φ ( e ) ⊂ S j Ψ j ( g ) = P ( g ) \ q ( g ) . Hene ψ g ◦ φ ( e ) ⊂ ˆ g − 1 ( c ′ ) for some c ′ ∈ R , in partiular, ˆ g ◦ ψ f ◦ φ ( v ′ ) = ˆ g ◦ ψ f ◦ φ ( v ′′ ) = c ′ . But from (5) it follo ws that ˆ g ◦ ψ f ◦ φ ( v ′ ) = ˆ f ◦ ψ f ( v ′ ) = c , therefore e ⊂ ( ˆ f ◦ ψ f ) − 1 ( c ) and φ ( e ) ⊂ ( ˆ g ◦ ψ g ) − 1 ( c ) . Fix a homeomorphism ϕ e : e → φ ( e ) su h that ϕ ( v ′ ) = φ ( v ′ ) and ϕ ( v ′′ ) = φ ( v ′′ ) . Ob viously , ˆ f ◦ ψ f ( x ) = ˆ g ◦ ψ g ◦ ϕ e ( x ) = c , x ∈ e . Supp ose that an edge e = e ( v ′ , v ′′ ) ∈ E ( P ( f )) b elongs to C r -subgraph q ( f ) . Then ev ery p oin t of a set ψ f ( e ) \ { ψ f ( v ′ ) , ψ f ( v ′′ ) } is a regular b oundary p oin t of ˆ f , hene ˆ f is stritly monotone on the ar ψ f ( e ) and maps it homeomorphially on [ c ′ , c ′′ ] , where c ′ = min( ˆ f ◦ ψ f ( v ′ ) , ˆ f ◦ ψ f ( v ′′ )) , c ′′ = max( ˆ f ◦ ψ f ( v ′ ) , ˆ f ◦ ψ f ( v ′′ )) . Sine φ is an isomorphism of the om binatorial diagrams then from C 1 it follo ws that φ ( e ) ∈ q ( g ) . Th us ˆ g maps the set ψ g ( φ ( e )) in R homeomorphially . F rom (5 ) it follo ws that ˆ g ◦ ψ g ( φ ( e )) = [ c ′ , c ′′ ] . Supp ose that ϕ e = ˆ g ◦ ψ g φ ( e ) − 1 ◦ ˆ f ◦ ψ f : e → φ ( e ) . It is easy to see that this map is a homeomorphism and satises the follo wing relation ˆ f ◦ ψ f ( x ) = ˆ g ◦ ψ g ◦ ϕ e ( x ) , x ∈ e . Let us dene a map ϕ : P ( f ) → P ( g ) as ϕ ( x ) = ϕ e ( x ) , for x ∈ e . By the onstrution ϕ e ( v ) = φ ( v ) for v ∈ e ∩ V ( P ( f )) therefore ϕ e ′ ( x ) = ϕ e ′′ ( x ) for ev ery pair of edges e ′ , e ′′ ∈ E ( P ( f )) and x ∈ e ′ ∩ e ′′ ⊆ V ( P ( f )) . So, the map ϕ is dened orretly . It is easy to see that ϕ satises (6). The olletion of edges { e ∈ E ( P ( f )) } generate a nite losed o v er of a spae P ( f ) th us it is fundamen tal. Hene ϕ is on tin uous sine ea h of maps ϕ e is on tin uous b y denition, where e ∈ E ( P ( f )) , see [23℄. It is easy to see that ϕ is a bijetiv e map and the spaes P ( f ) and P ( g ) are ompat. Therefore ϕ maps P ( f ) on P ( g ) homeomorphially . Moreo v er, sine φ preserv es orien tation of q ( f ) , then an orien tation on q ( g ) = ϕ ( q ( f )) indued b y ϕ oinides with the orien tation of q ( g ) in P ( g ) . W e set H 0 = ψ g ◦ ϕ ◦ ψ − 1 f P f : P f → P g . 18 By the onstrution H 0 maps the set P f = ψ f ( P ( f )) on P g = ψ g ( P ( g )) homeomorphially . Moreo v er, from (6 ) it follo ws that (7) ˆ g ◦ H 0 = ˆ f . As orien tations indued on ∂ D 2 = ψ f ( q ( f )) = ψ g ( q ( g )) b y ψ f and ψ g from q ( f ) and q ( g ) resp etiv ely oinide with the p ositiv e orien tation of ∂ D 2 b y denition, then H 0 preserv es the orien tation of ∂ D 2 . Our aim is to extend H 0 to a homeomorphism H : D 2 → D 2 su h that ˆ g ◦ H = ˆ f . Let Θ b e one of onneted omp onen ts of D 2 \ P ( f ) . F rom Prop o- sitions 2.1 and 2.2 it follo ws that there exists a simple yle Q ⊆ P ( f ) su h that ψ f ( Q ) = F r Θ . Eviden tly , φ ( q ( f )) = ϕ ( q ( f )) = q ( g ) . Th us ψ g ◦ ϕ ( q ( f )) = ψ g ( q ( g )) = ∂ D 2 and from Lemma 2.2 it follo ws that a set ψ g ◦ ϕ ( Q ) is a b oundary of some onneted omp onen t Σ of D 2 \ P g . Denote b y R F r ( f ) a set of all regular b oundary p oin ts of ˆ f . It is easy to see that, on one hand, R F r ( f ) = ψ f ( q ( f ) \ V ( P ( f ))) , on the other hand the set W f of all regular b oundary p oin ts of ˆ f | Θ oinides with R F r ( f ) ∩ Θ = R F r ( f ) ∩ F r Θ = R F r ( f ) ∩ ψ f ( Q ) . Therefore W f is an image of a set Q ∩ ( q ( f ) \ V ( P ( f ))) . Let R F r ( g ) b e a set of regular b oundary p oin ts of ˆ g . By analogy , w e an onlude that the set W g of regular b oundary p oin ts of ˆ g | Σ is an image of a set ϕ ( Q ) ∩ ( q ( g ) \ V ( P ( g ))) . But a map ϕ is bijetiv e and, also, it is kno wn that q ( g ) = ϕ ( q ( f )) and V ( P ( g )) = ϕ ( V ( P ( f ))) . Therefore ϕ ( Q ) ∩ ( q ( g ) \ V ( P ( g ))) = ϕ ( Q ∩ ( q ( f ) \ V ( P ( f )))) and W g = ψ g ◦ ϕ ◦ ψ − 1 f ( W f ) = H 0 ( W f ) . F rom Lemma 2.1 it follo ws that the funtion ˆ f is regular on the set Θ . Let z 1 , . . . , z 4 and γ 1 , . . . , γ 4 b e the p oin ts and the ars, resp etiv ely , from Denition 1.7. Prop osition 1.1 guaran tees that W f = ˚ γ 1 ∪ ˚ γ 3 holds true. W e set K f = γ 2 ∪ γ 4 = F r Θ \ W f . Similarly , the funtion ˆ g is regular on the set Σ . Let w 1 , . . . , w 4 and ν 1 , . . . , ν 4 b e the p oin ts and the ars, resp etiv ely , from Denition 1.7. Then W g = ˚ ν 1 ∪ ˚ ν 3 . W e set K g = ν 2 ∪ ν 4 = F r Σ \ W g . W e already v eried that W g = H 0 ( W f ) . The map H 0 is bijetiv e, th us K g = H 0 ( K f ) . Hene for the funtions ˆ f | Θ and ˆ g | Σ Theorem 1.1 is satised with the same set D ′ ∈ { I 2 , D 2 + , D 2 } and its subset K ′ = ( x, y ) ∈ D ′ | y ∈ { y 1 , y 2 } ; y 1 = min { y | ( x, y ) ∈ D ′ } , y 2 = max { y | ( x, y ) ∈ D ′ } . Fix a homeomorphism χ f : F r Θ → F r D ′ su h that χ f ( K f ) = K ′ . W e set χ g = χ f ◦ ψ f ◦ ϕ − 1 ◦ ψ − 1 g = χ f ◦ H − 1 0 : F r Σ → F r D ′ . 19 The map χ g is a omp osition of homeomorphisms therefore χ g is a homeomorphism. Moreo v er, χ g ( K g ) = χ f ◦ H − 1 0 ( K g ) = χ f ( K f ) = K ′ . F rom Theorem 1.1 it follo ws that there exist n um b ers a f , b f , a g , b g ∈ R and homeomorphisms F Q : Θ → D ′ and G Q : Σ → D ′ su h that F Q K f = χ f , G Q K g = χ g , and ˆ f ◦ F − 1 Q ( x, y ) = a f y + b f , ˆ g ◦ G − 1 Q ( x, y ) = a g y + b g , ( x, y ) ∈ D ′ . W e set K 1 = { ( x, y ) ∈ D ′ | y = y 1 } , K 2 = { ( x, y ) ∈ D ′ | y = y 2 } . Due to the hoie of D ′ the sets K 1 and K 2 are onneted. Hene F − 1 Q ( K i ) , i = 1 , 2 , are also onneted. F rom F − 1 Q ( K 1 ) = χ − 1 f ( K 1 ) ⊆ K f ⊆ ψ f ( V ( P ( f )) ∪ S i Ψ i ( f )) it follo ws that there exists c 1 ∈ R su h that F − 1 Q ( K 1 ) ⊆ ˆ f − 1 ( c 1 ) (w e remind that ˆ f is lo ally onstan t on the set ψ f ( V ( P ( f )) ∪ S i Ψ i ( f )) ). On the other hand, G − 1 Q ( K 1 ) = χ − 1 g ( K 1 ) = ( χ f ◦ H − 1 0 ) − 1 ( K 1 ) , therefore ˆ g ◦ G − 1 Q ( K 1 ) = ˆ g ◦ H 0 ◦ χ − 1 f ( K 1 ) = ˆ f ◦ χ − 1 f ( K 1 ) = ˆ f ◦ F − 1 Q ( K 1 ) = c 1 sine (7 ) and G − 1 Q ( K 1 ) ⊆ ˆ g − 1 ( c 1 ) . Sim- ilarly , there exists c 2 ∈ R su h that F − 1 Q ( K 2 ) ⊆ ˆ f − 1 ( c 2 ) and G − 1 Q ( K 2 ) ⊆ ˆ g − 1 ( c 2 ) . Hene for an y ( x 1 , y 1 ) ∈ K 1 and ( x 2 , y 2 ) ∈ K 2 the follo wing onditions hold true (8) ˆ f ◦ F − 1 Q ( x 1 , y 1 ) = a f y 1 + b f = c 1 , ˆ g ◦ G − 1 Q ( x 1 , y 1 ) = a g y 1 + b g = c 1 , ˆ f ◦ F − 1 Q ( x 2 , y 2 ) = a f y 2 + b f = c 2 , ˆ g ◦ G − 1 Q ( x 2 , y 2 ) = a g y 2 + b g = c 2 . It is easy to see that a determinan t of this system of linear equations with v ariables a f , b f , a g and b g equals to ( y 2 − y 1 ) 2 . By the onstrution y 1 6 = y 2 th us ( y 2 − y 1 ) 2 6 = 0 and the system (8) has the unique solution whi h an b e easily alulated: a f = a g = c 2 − c 1 y 2 − y 1 , b f = b g = c 1 y 2 − c 2 y 1 y 2 − y 1 . So, on the set D ′ the follo wing equalit y holds true (9) ˆ f ◦ F − 1 Q = ˆ g ◦ G − 1 Q . It is lear that G − 1 Q ◦ F Q (F r Θ ) = F r Σ . W e remind that G − 1 Q ◦ F Q K f = χ − 1 g ◦ χ f K f = H 0 K f . Sine G − 1 Q ◦ F Q ( K f ) = H 0 ( K f ) = K g , then a homeomorphism G − 1 Q ◦ F Q satises to relations G − 1 Q ◦ F Q ( W f ) = G − 1 Q ◦ F Q (F r Θ \ K f ) = F r Σ \ K g = W g . As w e kno w, the set W f has t w o onneted omp onen ts ˚ γ 1 and ˚ γ 3 . Under the ation of the homeomorphism H 0 they ha v e to map on the 20 onneted omp onen ts ˚ ν 1 and ˚ ν 3 of the set W g . W e ylially hange a n umeration of the p oin ts w 1 , . . . , w 4 and the ars ν 1 , . . . , ν 4 so that H 0 ( ˚ γ 2 k − 1 ) = ˚ ν 2 k − 1 , k = 1 , 2 . The homeomorphism G − 1 Q ◦ F Q also has to map the sets ˚ γ 1 and ˚ γ 3 on the onneted omp onen ts of the set W g . Let us pro v e that under the ondition ˚ γ 2 ∪ ˚ γ 4 6 = ∅ the relations hold true G − 1 Q ◦ F Q ( ˚ γ 2 k − 1 ) = ˚ ν 2 k − 1 , k = 1 , 2 . W e remark that either G − 1 Q ◦ F Q ( ˚ γ 1 ) = ˚ ν 1 or G − 1 Q ◦ F Q ( ˚ γ 1 ) = ˚ ν 3 holds true. It is lear that an ar γ 2 k − 1 is a losure of a ar ˚ γ 2 k − 1 in D 2 (b y denition ˚ γ 2 k − 1 6 = ∅ ), k = 1 , 2 ; similarly , ν 2 k − 1 = ˚ ν 2 k − 1 . Th us if G − 1 Q ◦ F Q ( γ 2 k − 1 ) 6 = ν 2 j − 1 for some k , j ∈ { 1 , 2 } , then G − 1 Q ◦ F Q ( ˚ γ 2 k − 1 ) 6 = ˚ ν 2 j − 1 . Without loss of generalit y , supp ose that ˚ γ 2 6 = ∅ . Then z 2 ∈ γ 1 \ γ 3 and H 0 ( z 2 ) ∈ ν 1 \ ν 3 . But z 2 ∈ γ 2 ⊂ K f and H 0 ( z 2 ) = G − 1 Q ◦ F Q ( z 2 ) . Th us G − 1 Q ◦ F Q ( γ 1 ) 6 = ν 3 , hene G − 1 Q ◦ F Q ( ˚ γ 1 ) = ˚ ν 1 and G − 1 Q ◦ F Q ( ˚ γ 3 ) = ˚ ν 3 . Supp ose no w that ˚ γ 2 ∪ ˚ γ 4 = ∅ . Then b oth sets γ 2 and γ 4 are one-p oin t and D ′ = D 2 . Consider an in v olution I nv : D 2 → D 2 , I nv ( x, y ) = ( − x , y ) , ( x, y ) ∈ D 2 . Ob viously , it hanges the onneted omp onen ts of F r D ′ \ K ′ = F Q ( W f ) . Moreo v er, I nv | K ′ = I d sine K ′ = { (0 , − 1) , (0 , 1) } . If G − 1 Q ◦ F Q ( ˚ γ 1 ) = ν 3 , then the map G Q an b e replaed b y I nv ◦ G Q . It is easy to see that the follo wing onditions hold true • I nv ◦ G Q K g = χ g K g ; • ˆ g ◦ ( I nv ◦ G Q ) − 1 ( x, y ) = ˆ g ◦ G − 1 Q ( − x, y ) = ˆ g ◦ G − 1 Q ( x, y ) = a g y + b g ; • ( I nv ◦ G Q ) − 1 ◦ F Q ( ˚ γ 1 ) = W g \ ˚ ν 3 = ˚ ν 1 . So, w e pro v ed that the homeomorphisms F Q : Θ → D ′ and G Q : Σ → D ′ satisfy onditions • ˆ f ◦ F − 1 Q = ˆ g ◦ G − 1 Q ; • G − 1 Q ◦ F Q K f = H 0 K f ; • G − 1 Q ◦ F Q ( ˚ γ 2 k − 1 ) = ˚ ν 2 k − 1 , k = 1 , 2 . W e set H Q = G − 1 Q ◦ F Q : Θ → Σ . Let us v erify that H Q F r Θ = H 0 F r Θ . It is suien t to pro v e that H Q ( z ) = H 0 ( z ) for all z ∈ W f = F r Θ \ K f ⊂ γ 1 ∪ γ 3 . As w e kno w, the set ˚ γ 1 onsists of the regular b oundary p oin ts of the funtion ˆ f , therefore ˆ f is stritly monotone on the ar γ 1 and maps it on ˆ f ( γ 1 ) ⊂ R homeomorphially ( sine γ 1 is the ompatum and the spae R is Hausdor ). Similarly , a map ˆ g ν 1 : ν 1 → ˆ g ( ν 1 ) ⊂ R is a homeomorphism on to its image. 21 As a onsequene of γ 1 ⊂ P f , ν 1 ⊂ P g and from (7) it follo ws that ˆ f ( γ 1 ) = ˆ g ◦ H 0 ( γ 1 ) = ˆ g ( ν 1 ) . Th us the follo wing map is w ell dened ˆ g − 1 ◦ ˆ f γ 1 : γ 1 → ν 1 . By using (7 ) again w e ha v e H 0 | γ 1 = ˆ g − 1 ◦ ˆ f γ 1 . On the other hand, from (9 ) it follo ws that ˆ f = ˆ g ◦ G − 1 Q ◦ F Q = ˆ g ◦ H Q , therefore ˆ g − 1 ◦ ˆ f γ 1 = H Q | γ 1 . Hene H 0 | γ 1 = H Q | γ 1 . By analogy w e pro v e that H 0 | γ 3 = H Q | γ 3 . So, w e onstruted the homeomorphism H Q : Θ → Σ su h that ˆ f = ˆ g ◦ H Q and H Q | F r Θ = H 0 | F r Θ . Let us onstrut a homeomorphism H : D 2 → D 2 su h that ˆ f = ˆ g ◦ H . F or ev ery onneted omp onen t Θ of D 2 \ P f its b oundary F r Θ is an image of a simple yle Q (Θ) , see Prop ositions 2.1 and 2.2 . F or ev ery Θ w e x the homeomorphism H Q (Θ) su h that ˆ f = ˆ g ◦ H Q (Θ) and H Q (Θ) | F r Θ = H 0 | F r Θ . W e dene H b y the follo wing relations H ( z ) = H Q (Θ) , for z ∈ Θ . If z ∈ Θ ′ ∩ Θ ′′ , then z ∈ P f and H Q (Θ ′ ) ( z ) = H 0 ( z ) = H Q (Θ ′′ ) ( z ) . So, the map H is orretly dened. By Lemma 2.2 there exists a bijetiv e orresp ondene b et w een the onneted omp onen ts of the sets D 2 \ P f and D 2 \ P g . Th us H (Θ ′ ) ∩ H (Θ ′′ ) = ∅ for Θ ′ 6 = Θ ′′ and S Θ H ( Θ) = D 2 . Hene the map H is bijetiv e. Eviden tly , b y the onstrution w e ha v e ˆ f = ˆ g ◦ H . The losures of the onneted omp onen ts of D 2 \ P f generate a nite losed o v er of disk D 2 . It is kno wn [23℄ that this o v er is fundamen tal. Therefore the map H is on tin uous on D 2 sine b y onstrution it is on tin uous on ea h elemen t of its o v er. It is kno wn that a on tin uous bijetiv e map of ompatum in Haus- dor 's spae is a homeomorphism. So, H : D 2 → D 2 is a homeomor- phism. No w reall that map H 0 | ∂ D 2 = H | ∂ D 2 preserv es the orien tation. Consequen tly , H preserv es the orien tation on D 2 . W e remind that ˆ f = h f ◦ f and ˆ g = h g ◦ g for some homeomorphisms h f : f ( D 2 ) = [ a 1 , a N ] → [1 , N ] and h g : g ( D 2 ) = [ b 1 , b N ] → [1 , N ] whi h preserv e orien tation. It is ob vious that the map h 0 = h − 1 g ◦ h f : f ( D 2 ) → g ( D 2 ) is a homeomorphism of the segmen t f ( D 2 ) on the segmen t g ( D 2 ) whi h preserv es the orien tation. Let us x a homeomorphism h : R → 22 R , whi h preserv es the orien tation and satises h f ( D 2 ) = h 0 . It is easy to see that h ◦ f = h − 1 g ◦ h f ◦ f = h − 1 g ◦ ˆ f = h − 1 g ◦ ˆ g ◦ H = g ◦ H , so, the funtions f and g are top ologially equiv alen t. On Fig. 3 the diagrams of t w o pseudoharmoni funtions whi h ha v e t w o lo al minima, t w o lo al maxima on ∂ D 2 and one b oundary ritial p oin t are represen ted. But, these t w o funtions are not top ologially equiv alen t. 2 0 1 0 2 0 1 0 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 Figure 3. The diagrams of top ologially non equiv alen t pseudoharmoni funtions. Referenes [1℄ Morse M. T op ologial Metho ds in the Theory of F untions of a Complex V ariable. Annals of Mathematis Studies, no. 15. Prineton Univ ersit y Press, Prineton, N. J., 1947. 145 pp. [2℄ Sto low S. Le ons sur les prinip es top ologiques de la th eorie des fontions an- alytiques. Deuxi eme edition, augmen t ee de notes sur les fontions analytiques et leurs surfaes de Riemann. (F ren h) Gauthier-Villars, P aris, 1956. 194 pp. [3℄ Bo othby W.M. The top ology of regular urv e families with m ultiple saddle p oin ts. Amer. J. Math. 1951. v ol. 73 . pp. 405438. [4℄ Jenkins J.A, Morse M. Con tour equiv alen t pseudoharmoni funtions and pseudo onjugates. Amer. J. Math. 1952. v ol. 74 . pp. 23-51. [5℄ Kaplan T op ology of lev el urv es of harmoni funtions. T ransations of Amer. Math. So iet y v ol. 63 no. 3 (1948) pp. 514-522. [6℄ Morse M. The top ology of pseudo-harmoni funtions. Duk e Math. J. 1946. v ol. 13 . pp. 21-42. [7℄ Morse M., Jenkins J. The existene of pseudo onjugates on Riemann surfaes. F und. Math. 1952. v ol. 39 pp. 269287. [8℄ A rnol'd V. I. Snak e alulus and the om binatoris of the Bernoulli, Euler and Springer n um b ers of Co xeter groups. (Russian) Usp ekhi Mat. Nauk v ol. 47 (1992), no. 1(283), pp. 345, 240; translation in Russian Math. Surv eys v ol. 47 (1992), no. 1, pp. 151 [9℄ Maksimenko S. I. Classiation of m -funtions on surfaes. (Russian. English, Ukrainian summary) Ukra n. Mat. Zh. v ol. 51 (1999), no. 8, pp. 11291135; translation in Ukrainian Math. J. v ol. 51 (1999), no. 8, pp. 11751281 (2000) 23 [10℄ Oshemkov A. A. Morse funtions on t w o-dimensional surfaes. Co ding of sin- gularities. (Russian) T rudy Mat. Inst. Steklo v. v ol. 205 (1994), No vy e Rezult. v T eor. T op ol. Klassif. In tegr. Sistem, pp. 131140; translation in Pro . Steklo v Inst. Math. 1995, no. 4 (205), pp. 119127 [11℄ Prishlyak A. O. Classiation of three-dimensional gradien t-lik e Morse-Smale dynamial systems. (Russian) Some problems in on temp orary mathematis (Russian), pp. 305313, Pr. Inst. Mat. Nats. Ak ad. Nauk Ukr. Mat. Zastos., v ol. 25, Natsional. Ak ad. Nauk Ukra ni, Inst. Mat., Kiev, 1998. [12℄ Sharko V. V. Smo oth and top ologial equiv alene of funtions on surfaes. (Russian) Ukra n. Mat. Zh. v ol. 55 (2003), no. 5, pp. 687700; translation in Ukrainian Math. J. v ol. 55 (2003), no. 5, pp. 832846 [13℄ Sharko V. V. Morse funtions on nonompat surfaes. (Russian) Ukrainian Mathematis Congress2001 (Ukrainian), pp. 91102, Natsional. Ak ad. Nauk Ukra ni, Inst. Mat., Kiev, 2003. [14℄ A rnold V. I. Bernoulli Euler up do wn n um b ers, asso iated with funtion singularities, their om binatoris and a mathematis. Duk e Math. Journ. 1991. v ol. 63 . no. 2. pp. 537555. [15℄ Bolsinov A. V., Oshemkov A. A., Sharko V. V. On lassiation of o ws on manifolds I. Metho ds of F untional Analysis and T op ology , 1996. v ol. 2, no. 2 pp. 5160 [16℄ Bolsinov A. V., F omenko A. T. Exat top ologial lassiation of Hamiltonian o ws on smo oth t w o-dimensional surfaes. (Russian summary) Zap. Nau hn. Sem. S.-P eterburg. Otdel. Mat. Inst. Steklo v. 235 pp. (1996) [17℄ Prishlyak A. O. Regular funtions on losed three-dimensional manifolds. Dop o v. Nats. Ak ad. Nauk Ukr. Mat. Priro dozn. T ekh. Nauki 2003, no. 8 pp. 2124 [18℄ E.Polulyakh, I.Y ur huk On the riteria of D-planarit y of a tree. [19℄ Zieshang H., Ý. V o gt E., Coldewey H.- D. Surfaes and planar dison tin uous groups. Springer-v erlag, 1981. [20℄ Polulyakh Y e. On on tin uous funtions on t w o-dimensional disk whi h are reg- ular in its in terior p oin ts. [21℄ Mel'nikov O. V., R emeslennikov V. N., R oman 'kov V. A., Skornyakov L. A., Shestakov, I. P. General algebra. V ol. 1. (Russian) Mathematial Referene Library , Nauk a, Moso w, 1990. 592 pp. [22℄ Newman M. H. A. Elemen ts of the top ology of plane sets of p oin ts. Cam bridge: Cam bridge Univ. Press, 1964 214 pp. [23℄ F uks D. B., R okhlin V. A. Beginner's ourse in top ology . Geometri hapters. T ranslated from the Russian b y A. Iaob. Univ ersitext. Springer Series in So viet Mathematis. Springer-V erlag, Berlin, 1984. 519 pp.
Original Paper
Loading high-quality paper...
Comments & Academic Discussion
Loading comments...
Leave a Comment