Tie-points and fixed-points in N^*

TIE-POINTS AND FIXED-POIN T S IN N ∗ ALAN DOW AND SAHAR ON SHELAH Abstract. A po int x is a (b ow) tie-point of a space X if X \ { x } can be partitioned into (relativ ely) clop en sets each with x in its closure. Tie- po in ts ha ve app eared in the construction o f non- trivial autohomeomor phisms of β N \ N (e.g . [10, 8]) and in the recent study of (precisely) 2- to-1 maps on β N \ N . In these cases the tie-po int s hav e been the unique fixed p o int of an in volution o n β N \ N . This pap er is motiv ated b y the sear ch for 2-to-1 maps and obtaining tie-p oints of s trikingly differing ch ar acteristics. 1. Introductio n A p oin t x is a tie-p oint of a space X if there ar e closed sets A, B of X suc h that { x } = A ∩ B and x is an adheren t p oint of eac h of A and B . W e picture (and denote) this as X = A ⊲ ⊳ x B where A, B are the closed sets whic h hav e a unique common accum ulation p oint x and sa y that x is a tie-p o int as witnessed b y A, B . Let A ≡ x B mean that there is a homeomorphism from A to B with x as a fixed p oin t. If X = A ⊲ ⊳ x B and A ≡ x B , then there is an inv olution F of X (i.e. F 2 = F ) suc h that { x } = fix( F ). In this case w e will say that x is a symmetric tie-p oint of X . An a utohomeomorphism F of β N \ N (or N ∗ ) is said to b e trivial if there is a bijection f b et we en cofinite subsets of N such that F = β f ↾ β N \ N . If F is a trivial a utohomeomorphism, then fix( F ) is clop en; so of course β N \ N will ha v e no symmetric tie-p o in ts in this case if all autohomeomorphisms ar e trivial. If A and B are arbit r a ry compact spaces, and if x ∈ A and y ∈ B are accum ulation p oints, then let A ⊲ ⊳ x = y B denote the quotien t space o f Date : O c tobe r 29, 2018 . 1991 Mathematics Subje ct Classific ation. 03A35 . Key wor ds and phr ases. automorphism, Stone- Cech, fixed p oints. Research of the first author was supp or ted b y NSF gr ant No. NSF-. The resear ch of the s econd author was supp orted by The Isr ael Science F oundation founded by the Is r ael Academy of Sciences and Humanities, and by NSF grant No. NSF- . This is pa per n umber 91 6 in the second author ’s p er sonal listing. 1 2 A. DO W AND S. SHELA H A ⊕ B obtained by iden tifying x and y and let xy denote the colla psed p oin t. Clearly the p o in t xy is a tie-p oint o f this space. W e came to the study of t ie-p oin ts via the follo wing observ ation. Prop osition 1.1. If x, y ar e symm etric tie- p oints of β N \ N as wit- nesse d by A, B and A ′ , B ′ r esp e ctively, then ther e is a 2-to-1 mapping fr om β N \ N onto the sp ac e A ⊲ ⊳ x = y B ′ . The prop osition holds more g enerally if x a nd y a re fixed p oints of in v olutions F , F ′ resp ectiv ely . That is, replace A by the quotien t space of β N \ N obtained b y collapsing all sets { z , F ( z ) } to single p oin ts and similary replace B ′ b y the quotien t space induced by F ′ . It is a n op en problem to determine if 2- to-1 con tin uous images of β N \ N are homeomorphic to β N \ N [5]. It is known to b e true if CH [3 ] or PF A [2] holds. There are many interesting questions tha t arise naturally when con- sidering the concept of tie-p oints in β N \ N . Given a closed set A ⊂ β N \ N , let I A = { a ⊂ N : a ∗ ⊂ A } . Giv en an ideal I of subsets of N , let I ⊥ = { b ⊂ N : ( ∀ a ∈ I ) a ∩ b = ∗ ∅} and I + = { d ⊂ N : ( ∀ a ∈ I ) d \ a / ∈ I ⊥ } . If J ⊂ [ N ] ω , let J ↓ = S J ∈J P ( J ). Say t hat J ⊂ I is un b ounded in I if for eac h a ∈ I , there is a b ∈ J suc h that b \ a is infinite. Definition 1.1. If I is an ideal of subsets of N , set cf ( I ) to b e the cofinalit y of I ; b ( I ) is the minimum cardinalit y of an un b o unded family in I ; δ ( I ) is the minim um cardinalit y of a subset J of I suc h that J ↓ is dense in I . If β N \ N = A ⊲ ⊳ x B , then I B = I ⊥ A and x is the unique ultr a filter on N extending I + A ∩ I + B . The c haracter of x in β N \ N is equal to the maxim um o f cf ( I A ) and cf ( I B ). Definition 1.2. Say that a tie-p oint x has (i) b -type; (ii) δ - t yp e; re- sp ectiv ely (iii) b δ -type, ( κ, λ ) if β N \ N = A ⊲ ⊳ x B and ( κ, λ ) equals: (i) ( b ( I A ) , b ( I B )) (ii) ( δ ( I A ) , δ ( I B )); and (iii) eac h of ( b ( I A ) , b ( I B )) and ( δ ( I A ) , δ ( I B )). W e will adopt the conv en tion to put the smaller of the pair ( κ, λ ) in the first co or dina t e. Again, it is in teresting to note that if x is a tie-p oin t o f b -t yp e ( κ, λ ), then it is uniquely determined (in β N \ N ) by λ man y subsets of N since x will b e the unique p oin t extending the family ( ( J A ) ↓ ) + ∩ (( J B ) ↓ ) + where J A and J B are un b ounded subfamilies of I A and I B . Question 1.1. Can there b e a tie-p oint in β N \ N with δ -type ( κ, λ ) with κ ≤ λ less than the ch ara cter of the p oin t? TIE-POINTS AND FIX ED-POINTS IN N ∗ 3 Question 1.2. Can β N \ N hav e tie-p oin ts of δ -t yp e ( ω 1 , ω 1 ) and ( ω 2 , ω 2 )? Prop osition 1.2. If β N \ N has symmetric tie-p oints of δ -typ e ( κ, κ ) and ( λ, λ ) , but no tie-p oints of δ -typ e ( κ, λ ) , then β N \ N ha s a 2-to-1 image which is n o t home om orphic to β N \ N . One could say that a tie-p oin t x was r adio active in X ( i.e. ▽ ⊲ ⊳ ) if X \ { x } can b e similarly split into 3 (or more) relative ly clop en sets accum ulating to x . This is equiv alen t to X = A ⊲ ⊳ x B suc h that x is a tie-p oint in either A or B . Eac h p oint of character ω 1 in β N \ N is a radioactiv e p oint (in pa r t ic- ular is a tie-p oint). P-p oints of character ω 1 are symme tric tie-p oints of b δ -t yp e ( ω 1 , ω 1 ), while p o ints of character ω 1 whic h are not P-p oints will hav e b -type ( ω , ω 1 ) and δ -ty p e ( ω 1 , ω 1 ). If there is a t ie-p oin t of b -t yp e ( κ, λ ), then of course there are ( κ, λ )-gaps. If there is a tie-p oin t of δ -t yp e ( κ, λ ), then p ≤ κ . Prop osition 1.3. If β N \ N = A ⊲ ⊳ x B , then p ≤ δ ( I A ) . Pr o o f . If J ⊂ I A has cardinalit y less than p , t here is, b y Solo v ay’s Lemma (and Bell’s Theorem) an infinite set C ⊂ N suc h that C and N \ C each meet ev ery infinite set of the form J \ ( S J ′ ) where { J } ∪J ′ ∈ [ J ] <ω . W e may assume t ha t C / ∈ x , hence there are a ∈ I A and b ∈ I B suc h that C ⊂ a ∪ b . How ev er no finite union from J cov ers a sho wing that J ↓ can not b e dense in I A .  Although it do es not seem to b e completely tr ivial, it can b e sho wn that PF A implies there a r e no tie-p oin ts (the har dest case to eliminate is those of b -t yp e ( ω 1 , ω 1 ))). Question 1.3. Do es p > ω 1 imply there are no tie-p oin ts of b -type ( ω 1 , ω 1 )? Analogous to tie-p oin ts, w e also define a tie-set: sa y that K ⊂ β N \ N is a tie-set if β N \ N = A ⊲ ⊳ K B and K = A ∩ B , A = A \ K , and B = B \ K . Say that K is a symmetric tie-set if there is an in v olution F suc h that K = fix( F ) and F [ A ] = B . Question 1.4. If F is a n inv olution on β N \ N suc h that K = fix ( F ) has empt y in terior, is K a (symmetric) tie-set? Question 1.5. Is there some nat ura l restriction on whic h compact spaces can (or can not) b e homeomorphic to the fixed p oin t set o f some in v olution o f β N \ N ? Again, we note a p ossible application t o 2- to-1 maps. 4 A. DO W AND S. SHELA H Prop osition 1.4. Assume that F is an invo lution of β N \ N with K = fix( F ) 6 = ∅ . F urther assume that K h a s a symmetric tie-p oint x (i.e. K = A ⊲ ⊳ x B ), then β N \ N has a 2 -to-1 c ontinuous ima g e which has a symmetric tie-p oint (a nd p ossi b ly β N \ N do es not have such a tie-p oint). Question 1.6. If F is an inv olution of N ∗ , is the quotien t space N ∗ /F (in whic h eac h { x, F ( x ) } is collapsed to a single p o in t) a homeomorphic cop y of β N \ N ? Prop osition 1.5 (CH) . I f F i s an invol ution of β N \ N , then the quotient sp ac e N ∗ /F is home omorphic to β N \ N . Pr o o f . If fix( F ) is empt y , then N ∗ /F is a 2-to-1 image of β N \ N , and so is a cop y of β N \ N . If fix( F ) is not empty , then consider t w o copies, ( N ∗ 1 , F 1 ) and ( N ∗ 2 , F 2 ), o f ( N ∗ , F ). The quotient space of N ∗ 1 /F 1 ⊕ N ∗ 2 /F 2 obtained b y identifying the tw o homeomorphic sets fix( F 1 ) and fix( F 2 ) will b e a 2-to- 1-image o f N ∗ , hence again a cop y of N ∗ . Since N ∗ 1 \ fix( F 1 ) and N ∗ 2 \ fix( F 2 ) are disjoin t and ho meomorphic, it f ollo ws easily that fix( F ) m ust b e a P-set in N ∗ . It is trivial to v erify tha t a regular closed set of N ∗ with a P-set b oundary will b e (in a mo del of CH) a cop y of N ∗ . Therefore t he cop y of N ∗ 1 /F 1 in this final quotien t space is a cop y of N ∗ .  2. a s pectrum of tie-s ets W e a dapt a metho d from [1] to pro duce a mo del in whic h there are tie-sets of specified b δ - types. W e further arrange that these tie-sets will themselv es hav e tie-p oints but unfortunately w e are not a ble to mak e the tie-sets symmetric. In the next section w e mak e some prog ress in in v olving in volutions. Theorem 2.1. Assume GCH and that Λ is a set of r e gular unc ountable c ar dinals such that for e ach λ ∈ Λ , T λ is a < λ - close d λ + -Souslin tr e e. Ther e is a for ci n g extension in w hich ther e is a tie-se t K (of b δ -typ e ( c , c ) ) and f o r e ach λ ∈ Λ , ther e is a tie-s e t K λ of b δ -typ e ( λ + , λ + ) such that K ∩ K λ is a single p oi n t wh i ch is a tie-p oint of K λ . F urthermor e, for µ ≤ λ < c , if µ 6 = λ or λ / ∈ Λ , then ther e is n o tie-set of b δ -typ e ( µ, λ ) . W e will assume that our So uslin trees are we ll-pruned and are ev er ω - ary branc hing. That is, if T λ is a λ + -Souslin tree, w e assume t ha t for each t ∈ T , t has exactly ω immediate successors denoted { t ⌢ ℓ : ℓ ∈ ω } and that { s ∈ T λ : t < s } has cardinality λ + (and so has success ors on ev ery lev el). A p oset is <κ -closed if ev ery directed subset TIE-POINTS AND FIX ED-POINTS IN N ∗ 5 of cardinalit y less t han κ has a lo w er b o und. A p o set is <κ -distributiv e if the inte rsection of a ny fa mily of f ew er than κ dense op en subsets is again dense. F or a car dinal µ , let µ − b e the minim um cardinal such that ( µ − ) + ≥ µ (i.e. the predecess or if µ is a successor). The main idea of the construction is nicely illustrated by the follow- ing. Prop osition 2.2. Assume that β N \ N ha s no tie-sets of b δ -typ e ( κ 1 , κ 2 ) for some κ 1 ≤ κ 2 < c . A lso assume that λ + < c is such that λ + is distinct fr om o ne of κ 1 , κ 2 and that T λ is a λ + -Souslin tr e e and { ( a t , x t , b t ) : t ∈ T λ } ⊂ ([ N ] ω ) 3 satisfy that, f o r t < s ∈ T λ : (1) { a t , x t , b t } is a p artition of N , (2) x t ⌢ j ∩ x t ⌢ ℓ = ∅ for j < ℓ , (3) x s ⊂ ∗ x t , a t ⊂ ∗ a s , and b t ⊂ ∗ b s , (4) for e ach ℓ ∈ ω , x t ⌢ ℓ +1 ⊂ ∗ a t ⌢ ℓ and x t ⌢ ℓ +2 ⊂ ∗ b t ⌢ ℓ , then if ρ ∈ [ T λ ] λ + is a generic br anch (i.e. ρ ( α ) is an elemen t of the α -th level of T λ for e ach α ∈ λ + ), then K ρ = T α ∈ λ + x ∗ ρ ( α ) is a tie-set of β N \ N of b δ -typ e ( λ + , λ + ) , and ther e is no tie-set of b δ -typ e ( κ 1 , κ 2 ) . (5) Assume further that { ( a ξ , x ξ , b ξ ) : ξ ∈ c } is a family of p a rtitions of N such that { x ξ : ξ ∈ c } is a mo d fin ite d e sc ending family of subsets of N such that for e ach Y ⊂ N , ther e is a ma ximal antichain A Y ⊂ T λ and some ξ ∈ c such that for e ach t ∈ A Y , x t ∩ x ξ is a pr op er subset of either Y or N \ Y , then K = T ξ ∈ c x ∗ ξ me ets K ρ in a sin gle p oint z λ . (6) If we assume further that fo r e ach ξ < η < c , a ξ ⊂ ∗ a η and b ξ ⊂ ∗ b η , and for e ach t ∈ T λ , η may b e chosen so that x t me ets e ach of ( a η \ a ξ ) and ( b η \ b ξ ) , then z λ is a tie-p oint of K ρ . Pr o o f . T o show t hat K ρ is a tie-set it is sufficien t to sho w tha t K ρ ⊂ S α ∈ λ + a ∗ α ∩ S α ∈ λ + b ∗ α . Since T λ is a λ + -Souslin tree, no new subset of λ is added when forcing with T λ . Of course w e use that ρ is T λ is generic, so assume that Y ⊂ N and that some t ∈ T λ forces that Y ∗ ∩ K ρ is not empt y . W e mus t show that there is some t < s suc h that s forces that a s ∩ Y and b s ∩ Y are b oth infinite. How ev er, we know tha t x t ⌢ ℓ ∩ Y is infinite for eac h ℓ ∈ ω since t ⌢ ℓ  T λ “ K ρ ⊂ x ∗ t ⌢ ℓ ”. Therefore, b y condition 4, f o r eac h ℓ ∈ ω , Y ∩ a t ⌢ ℓ and Y ∩ b t ⌢ ℓ are b oth infinite. No w let κ 1 , κ 2 b e regular cardinals at least one of whic h is distinct from λ + . Recall that forcing with T λ preserv es car dina ls. Assum e that in V [ ρ ], K ⊂ N ∗ and N ∗ = C ⊲ ⊳ K D with b ( I C ) = δ ( I C ) = κ 1 and b ( I D ) = δ ( I D ) = κ 2 . In V , let { c γ : γ ∈ κ 1 } b e T λ -names fo r the increasing cofinal sequence in I C and let { d ξ : ξ ∈ κ 2 } b e T λ -names for the increasing cofinal sequence in I D . Again using t he fact that T λ adds 6 A. DO W AND S. SHELA H no new subsets of N and the fact that ev ery dense op en subset of T λ will contain an entire lev el of T λ , w e ma y c ho ose ordinals { α γ : γ ∈ κ 1 } and { β ξ : ξ ∈ κ 2 } suc h that each t ∈ T λ , if t is on lev el α γ it will force a v alue o n c γ and if t is on lev el β ξ it will force a v alue on d ξ . If κ 1 < λ + , then sup { α γ : γ ∈ κ 1 } < λ + , hence there a re t ∈ T λ whic h force a v alue on each c γ . If λ + < κ 2 , then t here is some β < λ + , suc h t ha t { ξ ∈ κ 2 : β ξ ≤ β } has cardinality κ 2 . Therefore there is some t ∈ T λ suc h that t forces a v alue on d ξ for a cofinal set of ξ ∈ κ 2 . Of course, if neither κ 1 nor κ 2 is equal to λ + , then we ha v e a condition t hat decided cofinal families of eac h of I C and I D . This implies tha t N ∗ already has tie-sets of b δ -t yp e ( κ 1 , κ 2 ). If κ 1 < κ 2 = λ + , then fix t ∈ T λ deciding C = { c γ : γ ∈ κ 1 } , and let D = { d ⊂ N : ( ∃ s > t ) s  T λ “ d ∗ ⊂ D ” } . It follows easily that D = C ⊥ . But also, since f o rcing with T λ can not raise b ( D ) and can not low er δ ( D ), w e again hav e that there a re tie-sets of b δ -t yp e in V . The case κ 1 = λ + < κ 2 is similar. No w assume w e hav e the family { ( a ξ , x ξ , b ξ ) : ξ ∈ c } as in (5) and (6) a nd set K = T ξ x ∗ ξ , A = { K } ∪ S { a ∗ ξ : ξ ∈ c } , and B = { K } ∪ S { b ∗ ξ : ξ ∈ c } . It is routine to see that (5) ensures that the family { x ξ ∩ x ρ ( α ) : ξ ∈ c and α ∈ λ + } generates an ultrafilter when ρ meets eac h maximal an tic hain A Y ( Y ⊂ N ). Condition (6) clearly ensures that A \ K and B \ K eac h meet ( x ξ ∩ x ρ ( α ) ) ∗ for eac h ξ ∈ c and α ∈ λ + . T hus A ∩ K ρ and B ∩ K ρ witness that z λ is a tie-p o int of K ρ .  Let θ b e a regular cardinal greater than λ + for all λ ∈ Λ. W e will need the following w ell-kno wn Easton lemma (see [4, p234]). Lemma 2.3. L et µ b e a r e gular c ar di n al and assume that P 1 is a p oset satisfying the µ -c c. Then any <µ - c lose d p os e t P 2 r emains <µ - distributive after fo r cing with P 1 . F urthermor e an y < µ -distributive p oset r emai n s <µ -d istributive after for cing with a p o set of c ar dina lity less than µ . Pr o o f . Recall t ha t a p oset P is <µ -distributiv e if fo r cing with it do es not add, for a ny γ < µ , an y new γ -sequences of o r dina ls. Since P 2 is <µ -closed, forcing with P 2 do es not add any new a n tic hains to P 1 . Therefore it follow s that forcing with P 2 preserv es that P 1 has the µ -cc and that for ev ery γ < µ , each γ -sequence of o r dinals in the fo r cing extension by P 2 × P 1 is really just a P 1 -name. Since forcing with P 1 × P 2 is the same as P 2 × P 1 , this sho ws that in the extension by P 1 , there are no new P 2 -names of γ - sequence s o f ordinals. TIE-POINTS AND FIX ED-POINTS IN N ∗ 7 No w supp ose that P 2 is µ - distributiv e and t hat P 1 has cardina lity less than µ . Let ˙ D b e a P 1 -name of a dense op en subset o f P 2 . F or eac h p ∈ P 1 , let D p ⊂ P 2 b e the set o f all q suc h that some extension of p forces t ha t q ∈ ˙ D . Since p forces that ˙ D is dense a nd tha t ˙ D ⊂ D p , it fo llo ws that D p is dense (and op en). Since P 2 is µ -distributiv e, T p ∈ P 1 D p is dense and is clearly g oing to b e a subset of ˙ D . Rep eating this ar g umen t f or at most µ man y P 1 -names o f dense op en subsets o f P 2 completes the pro of .  W e recall the definition of Easton supp ort ed pro duct of p o sets (see [4, p233]). Definition 2.1. If Λ is a set of cardinals a nd { P λ : λ ∈ Λ } is a set of p osets, t hen we will use Π λ ∈ Λ P λ to denote the collection o f partial functions p suc h that (1) dom( p ) ⊂ Λ , (2) | dom( p ) ∩ µ | < µ for all regular cardinals µ , (3) p ( λ ) ∈ P λ for all λ ∈ dom( p ). This collection is a p oset when ordered by q < p if dom ( q ) ⊃ dom ( p ) and q ( λ ) ≤ p ( λ ) for all λ ∈ do m( p ) . Lemma 2.4. F or e ach c ar dinal µ , Π λ ∈ Λ \ µ + T λ is <µ + -close d and, if µ is r e gular, Π λ ∈ Λ ∩ µ T λ has c ar dina lity at most 2 <µ ≤ min(Λ \ µ ) . Lemma 2.5. If P is c c c and G ⊂ P × Π λ ∈ Λ T λ is generic, then in V [ G ] , for any µ and any family A ⊂ [ N ] ω with |A | = µ : (1) if µ ≤ ω , then A is a memb er of V [ G ∩ P ] ; (2) if µ = λ + , λ ∈ Λ , then ther e is an A ′ ⊂ A of c ar dinali ty λ + such that A ′ is a me m b er of V [ G ∩ ( P × T λ )] ; (3) if µ − / ∈ Λ , then ther e is an A ′ ⊂ A of c ar dina l i ty µ whic h is a memb er of V [ G ∩ P ] . Corollary 2.6. If P is c c c and G ⊂ P × Π λ ∈ Λ T λ is generic, then for any κ ≤ µ < c such that either κ 6 = µ or κ / ∈ { λ + : λ ∈ Λ } , if ther e is a tie-set of b δ -typ e ( κ, µ ) in V [ G ] , then ther e i s such a tie-set in V [ G ∩ P ] . Pr o o f . Assume that β N \ N = A ⊲ ⊳ K B in V [ G ] with µ = b ( A ) and λ = b ( B ). Let J A ⊂ I A b e an increasing mo d finite ch ain, of or der type µ , whic h is dense in I A . Similarly let J B ⊂ I B b e such a c hain of order t yp e λ . By Lemma 2.5, J A and J B are subsets of [ N ] ω ∩ V [ G ∩ P ] = [ N ] ω . Cho ose, if p ossible µ 1 ∈ Λ suc h that µ + 1 = µ and λ 1 ∈ Λ suc h t ha t λ + 1 = λ . Also by Lemma 2.5 , w e can, b y pa ssing to a sub collection, assume that J A ∈ V [ G ∩ ( P × T µ 1 )] (if there is no µ 1 , then let T µ 1 denote 8 A. DO W AND S. SHELA H the trivial order). Similarly , w e may assume that J B ∈ V [ G ∩ ( P × T λ 1 )]. Fix a condition q ∈ G ⊂ ( P × Π λ ∈ Λ T λ ) whic h forces that ( J A ) ↓ is a ⊂ -dense subset of I A , that ( J B ) ↓ is a ⊂ -dense subset of I B , and that ( I A ) ⊥ = I B . W orking in the mo del V [ G ∩ P ] then, there is a fa mily { ˙ a α : α ∈ µ } of T µ 1 -names for the mem b ers of J A ; and a family { ˙ b β : β ∈ λ 1 } of T λ 1 -names for the mem b ers of J B . Of course if µ = λ and T µ 1 is the trivial o r der, then J A and J B are alr eady in V [ G ∩ P ] and w e hav e our tie-set in V [ G ∩ P ]. Otherwise, w e assume tha t µ 1 < λ 1 . Set A to b e the set of all a ⊂ N suc h that there is some q ( µ 1 ) ≤ t ∈ T µ 1 and α ∈ µ suc h that t  T µ 1 “ a = ˙ a α ”. Similarly let B b e the set of all b ⊂ N such that there is some q ( λ 1 ) ≤ s ∈ T λ 1 and β ∈ λ suc h that s  T λ 1 “ b = ˙ b β ”. It follo ws from the construction that, in V [ G ], for an y ( a ′ , b ′ ) ∈ J A × J B , there is an ( a, b ) ∈ A × B suc h that a ′ ⊂ ∗ a and b ′ ⊂ ∗ b . Therefore the ideal generated by A ∪ B is certainly dense. It r emains only to sho w that B ⊂ ( A ) ⊥ . Consider an y ( a, b ) ∈ A × B , and c ho ose ( q ( µ 1 ) , q ( λ 1 )) ≤ ( t, s ) ∈ T µ 1 × T λ 1 suc h that t  T µ 1 “ a ∈ J A ” and s  T λ 1 “ b ∈ J B ”. It follo ws tha t for an y condition ¯ q ≤ q with ¯ q ∈ ( P × Π λ ∈ Λ T λ ), ¯ q ( µ 1 ) = t , ¯ q ( λ 1 ) = s , w e hav e that ¯ q  ( P × Π λ ∈ Λ T λ ) “ a ∈ J A and b ∈ J B ” . It is ro utine no w to c hec k tha t , in V [ G ∩ P ], A and B generate ideals that witness that T { ( N \ ( a ∪ b )) ∗ : ( a, b ) ∈ A × B} is a tie-set of b δ - t yp e ( µ, λ ).  Let T be the ro oted tree {∅} ∪ S λ ∈ Λ T λ and w e will force an em- b edding of T in to P ( N ) mo d finite. In fact, w e force a structure { ( a t , x t , b t ) : t ∈ T } satisfying the conditions (1)-(4) of Prop osition 2.2. Definition 2.2. The p oset Q 0 is defined a s the set of elemen ts q = ( n q , T q , f q ) where n q ∈ N , T q ∈ [ T ] <ω , and f q : n q × T q → { 0 , 1 , 2 } . The idea is that x t will b e S q ∈ G { j ∈ n q : f q ( j, t ) = 0 } , a t will b e S q ∈ G { j ∈ n q : f q ( j, t ) = 1 } and b t = N \ ( a t ∪ x t ). W e set q < p if n q ≥ n p , T q ⊃ T p , f q ⊃ f p and for t, s ∈ T p and i ∈ [ n p , n q ) (1) if t < s and f q ( i, t ) ∈ { 1 , 2 } , then f q ( i, s ) = f q ( i, t ); (2) if t < s and f q ( i, s ) = 0, then f q ( i, t ) = 0; (3) if t ⊥ s , then f q ( i, t ) + f q ( i, s ) > 0. (4) if j ∈ { 1 , 2 } and { t ⌢ ℓ, t ⌢ ( ℓ + j ) } ⊂ T p and f q ( i, t ⌢ ( ℓ + j )) = 0, then f q ( i, t ⌢ ℓ ) = j . The next lemma is v ery routine but w e record it fo r reference. TIE-POINTS AND FIX ED-POINTS IN N ∗ 9 Lemma 2.7. Th e p oset Q 0 is c c c and if G ⊂ Q 0 is generic, the family X T = { ( a t , x t , b t ) : t ∈ T } satisfies the c onditions of Pr op osition 2.2. W e will need some other com binatoria l prop erties of t he family X T . Definition 2.3. F o r an y ˜ T ∈ [ T ] <ω , w e define the follow ing ( Q 0 - names). (1) for i ∈ N , [ i ] ˜ T = { j ∈ N : ( ∀ t ∈ ˜ T ) i ∈ x t iff j ∈ x t } , (2) the collection fin( ˜ T ) is the set of [ i ] ˜ T whic h are finite. W e abuse notat ion and let fin( ˜ T ) ⊂ n abbreviate fin( ˜ T ) ⊂ P ( n ). Lemma 2.8. F or e ach q ∈ Q 0 and e ach ˜ T ⊂ T q , fin( ˜ T ) ⊂ n q and for i ≥ n q , [ i ] ˜ T is infin i te. Definition 2.4. A sequence S W = { ( a ξ , x ξ , b ξ ) : ξ ∈ W } is a t o w er of T -splitters if for ξ < η ∈ W and t ∈ T : (1) { a ξ , x ξ , b ξ } is a partition of N , (2) a ξ ⊂ ∗ a η , b ξ ⊂ ∗ b η , (3) x t ∩ x ξ is infinite. Definition 2.5. If S W is a tow er of T -splitters and Y is a subset N , then the p oset Q ( S W , Y ) is defined as follows. Let E Y b e the (p ossibly empt y) set of minimal elemen ts of T such that there is some finite H ⊂ W such that x t ∩ Y ∩ T ξ ∈ H x ξ is finite. Let D Y = E ⊥ Y = { t ∈ T : ( ∀ s ∈ E Y ) t ⊥ s } . A condition q ∈ Q ( S W , Y ) is a tuple ( n q , a q , x q , b q , T q , H q ) where (1) n q ∈ N and { a q , x q , b q } is a partition of n q , (2) T q ∈ [ T ] <ω and H q ∈ [ W ] <ω , (3) ( a ξ \ a η ), ( b ξ \ b η ), and ( x η \ x ξ ) are all con tained in n q for ξ < η ∈ H q . W e define q < p to mean n p ≤ n q , T p ⊂ T q , H p ⊂ H q , and (4) for t ∈ T p ∩ D Y , x t ∩ ( x q \ x p ) ⊂ Y , (5) x q \ x p ⊂ T ξ ∈ H p x ξ , (6) a q \ a p is disjoin t from b max( H p ) , (7) b q \ b p is disjoint from a max( H p ) . Lemma 2.9. If W ⊂ γ , S W is a tower of T -splitters, and if G is Q ( S W , Y ) -generic, then S W ∪ { ( a γ , x γ , b γ ) } is also a tower of T -splitters wher e a γ = S { a q : q ∈ G } , x γ = S { x q : q ∈ G } , and b γ = S { b q : q ∈ G } . In addition, for e ach t ∈ D Y , x t ∩ x ξ ⊂ ∗ Y (and x t ∩ x ξ ⊂ ∗ N \ Y for t ∈ E Y ). Lemma 2.10. I f W do es not have c ofinal i ty ω 1 , then Q ( S W , Y ) is σ -c enter e d. 10 A. DO W AND S. SHELA H As usual with ( ω 1 , ω 1 )-gaps, Q ( S W , Y ) may not (in general) b e ccc if W has a cofinal ω 1 sequence . Let 0 / ∈ C ⊂ θ b e cofinal and assume that if C ∩ γ is cofinal in γ and cf ( γ ) = ω 1 , then γ ∈ C . Definition 2.6. Fix any w ell- ordering ≺ of H ( θ ). W e define a fi- nite supp ort iteration sequence { P γ , ˙ Q γ : γ ∈ θ } ⊂ H ( θ ). W e abuse notation and use Q 0 rather than ˙ Q 0 from definition 2.2 . If γ / ∈ C , then let ˙ Q γ b e the ≺ -least among the list of P γ -names of ccc p osets in H ( θ ) \ { ˙ Q ξ : ξ ∈ γ } . If γ ∈ C , then let ˙ Y γ b e the ≺ -least P γ -name of a subset N whic h is in H ( θ ) \ { ˙ Y ξ : ξ ∈ C ∩ γ } . Set ˙ Q γ to b e the P γ name of Q ( S C ∩ γ , ˙ Y γ ) adding the partition { ˙ a γ , ˙ x γ , ˙ b γ } and, where S C ∩ γ is the P γ -name of the T -splitting tow er { ( a ξ , x ξ , b ξ ) : ξ ∈ C ∩ γ } . W e view the mem b ers o f P θ as f unctions p with finite domain ( o r supp ort) denoted dom( p ). The main difficult y to the pro of of Theorem 2.1 is to prov e tha t the iteration P θ is ccc. Of course, since it is a finite supp ort it era t io n, this can b e prov en by induction at success or o rdinals. Lemma 2.11. F or e ach γ ∈ C such that C ∩ γ has c ofinality ω 1 , P γ +1 is c c c. Pr o o f . W e pro ceed by induction. F or each α , define p ∈ P ∗ α if p ∈ P α and there is an n ∈ N suc h that (1) for eac h β ∈ dom( p ) ∩ C , with H β = dom( p ) ∩ C ∩ β , there are subsets a β , x β , b β of n and T β ∈ [ T ] <ω suc h that p ↾ β  P β “ p ( β ) = ( n, a β , x β , b β , T β , H β )” Assume that P ∗ β is dense in P β and let p ∈ P β +1 . T o sho w that P ∗ β +1 is dense in P β +1 w e mu st find some p ∗ ≤ p in P ∗ β +1 . If β / ∈ C a nd p ∗ ∈ P ∗ β is b elo w p ↾ β , then p ∗ ∪ { ( β , p ( β ) } is the desired elemen t of P ∗ β +1 . Now assume that β ∈ C and a ssume that p ↾ β ∈ P ∗ β and that p ↾ β forces that p ( β ) is the tuple ( n 0 , a, x, b, ˜ T , ˜ H ) . By an easy densit y argumen t, w e may assume tha t ˜ H ⊂ dom ( p ). Let n ∗ b e the intege r witnessing that p ↾ β ∈ P ∗ β . Let ζ b e the maxim um elemen t of dom( p ) ∩ C ∩ β and let p ↾ ζ  P ζ “ p ( ζ ) = ( n ∗ , a ζ , x ζ , b ζ , T ζ , H ζ )” as p er the definition of P ∗ ζ +1 . Notice that since ˜ H ⊂ H ζ w e hav e that p ↾ β  P β “( n ∗ , a ∗ , x, b ∗ , T ζ ∪ ˜ T , H ζ ∪ { ζ } ) ≤ p ( β )” where a ∗ = a ∪ ([ n 0 , n ∗ ) \ b ζ ) and b ∗ = b ∪ ([ n 0 , n ∗ ) ∩ b ζ ). Defining p ∗ ∈ P β +1 b y p ∗ ↾ β = p ↾ β and p ∗ ( β ) = ( n ∗ , a ∗ , x, b ∗ , T ζ ∪ ˜ T , H ζ ∪ { ζ } ) TIE-POINTS AND FIX ED-POINTS IN N ∗ 11 completes the pro o f tha t P ∗ β +1 is dense in P β +1 , and b y induction, that this holds fo r β = γ . No w assume that { p α : α ∈ ω 1 } ⊂ P ∗ γ +1 . By passing to a sub collec- tion, w e ma y a ssume that (1) the collection { T p α ( γ ) : α ∈ ω 1 } f o rms a ∆- system with ro ot T ∗ ; (2) the collection { dom( p α ) : α ∈ ω 1 } also forms a ∆-system with ro ot R ; (3) there is a tuple ( n ∗ , a ∗ , x ∗ , b ∗ ) so that for all α ∈ ω 1 , a p α ( γ ) = a ∗ , x p α ( γ ) = x ∗ , and b p α ( γ ) = b ∗ . Since C ∩ γ ha s a cofinal sequence of o rder t yp e ω 1 , there is a δ ∈ γ suc h that R ⊂ δ and, w e ma y assume, (dom( p α ) \ δ ) ⊂ min (dom( p β ) \ δ ) for α < β < ω 1 . Since P δ is ccc, there is a pair α < β < ω 1 suc h that p α ↾ δ is compatible with p β ↾ δ . Define q ∈ P γ +1 b y (1) q ↾ δ is any elemen t of P δ whic h is b elo w eac h of p α ↾ δ and p β ↾ δ , (2) if δ ≤ ξ ∈ γ ∩ dom ( p α ), then q ( ξ ) = p α ( ξ ), (3) if δ ≤ ξ ∈ dom( p β ) \ C , then q ( ξ ) = p β ( ξ ), (4) if δ ≤ ξ ∈ dom( p β ) ∩ C , then q ( ξ ) = ( n ∗ , a p β ( ξ ) , x p β ( ξ ) , b p β ( ξ ) , T p β ( ξ ) , H p β ( ξ ) ∪ H p α ( γ ) ) . The main non-trivial fa ct ab out q is that it is in P γ +1 whic h dep ends on the fact t ha t, by induction on η ∈ C ∩ γ , q ↾ η forces that ( a η \ a ξ ) ∪ ( b η \ b ξ ) ∪ ( x ξ \ x η ) ⊂ n ∗ for ξ ∈ C ∩ η . It no w fo llo ws trivially that q is b elow eac h of p α and p β .  Pr o o f of The or em 2 . 1. This completes the construction o f the ccc p oset P ( P θ as ab ov e). Let G ⊂ ( P × Π λ ∈ Λ T λ ) b e generic. It follows that V [ G ∩ P ] is a mo del o f Martin’s Axiom a nd c = θ . F urt hermore b y applying Lemma 2.4 with µ = ω a nd Lemma 2.3, w e hav e t ha t P 2 = Π λ ∈ Λ T λ is ω 1 -distributiv e in the mo del V [ G ∩ P ]. Therefore all subsets of N in the mo del V [ G ] are also in the mo del V [ G ∩ P ]. Fix an y λ ∈ Λ and let ρ λ denote the g eneric bra nch in T λ giv en b y G . Let G λ denote the generic filter on P × Π { T µ : λ 6 = µ ∈ Λ } and w ork in the mo del V [ G λ ]. It follows easily b y Lemma 2.4 and Lemma 2 .3, that T λ is a λ + -Souslin tree in this mo del. Therefore b y Prop osition 2.2, K λ = T α<λ + x ∗ ρ λ ( α ) is a tie-set of b δ - type ( λ + , λ + ) in V [ G ]. By the definition o f the iteration in P , it fo llo ws that condition (4) o f Lemma 2.2 is also satisfied, hence the tie-set K = T ξ ∈ C x ∗ ξ meets K λ in a single p oin t z λ . A simple genericit y argumen t confirms that conditions (5) and (6) of Prop osition 2.2 also holds, hence z λ is a tie-p oin t of K λ . 12 A. DO W AND S. SHELA H It follow s from Corollary 2 .6 that there are no unwante d tie-sets in β N \ N in V [ G ], a t least if there are none in V [ G ∩ P ]. Since p = c in V [ G ∩ P ], it f o llo ws fro m Prop o sition 1.3 that indeed there are no suc h tie-sets in V [ G ∩ P ].  Unfortunately the next result sho ws that the construction do es not pro vide us with o ur desired v ariet y of tie-p oints (ev en with v ar ia tions in the definition of the iteration). W e do no t know if b δ -t yp e can b e impro v ed to δ -t yp e (or simply exclude tie-p oin ts altog ether). Prop osition 2.12. In the mo del c onstructe d in Th e or e m 2.1, ther e ar e no tie-p oin ts wi th b δ -typ e ( κ 1 , κ 2 ) for any κ 1 ≤ κ 2 < c , Pr o o f . Assume that β N \ N = A ⊲ ⊳ x B and that δ ( I A ) = κ 1 and δ ( I B ) = κ 2 . It f ollo ws f rom Corollary 2.6 that w e can assume that κ 1 = κ 2 = λ + for some λ ∈ Λ. Also, following the pro of of Corolla ry 2 .6, there are P × T λ -names J A = { ˜ a α : α ∈ λ + } and P × T λ + -names J B = { ˜ b β : β ∈ λ + } suc h tha t the v aluation of these names b y G result in increasing (mo d finite) c hains in I A and I B resp ectiv ely whose do wn w ard closures are dense. P assing to V [ G ∩ P ], since T λ has the θ - cc, there is a Bo olean subalgebra B ∈ [ P ( N )] <θ suc h that eac h ˜ a α and ˜ b β is a name of a mem b er of B . F urthermore, there is an infinite C ⊂ N suc h that C / ∈ x and eac h of b ∩ C and b \ C are infinite for a ll b ∈ B . Since C / ∈ x , there is a Y ⊂ N (in V [ G ]) suc h that C ∩ Y ∈ I A and C \ Y ∈ I B . Now c ho ose t 0 ∈ T λ whic h forces this ab out C and Y . Bac k in V [ G ∩ P ], set A = { b ∈ B : ( ∃ t 1 ≤ t 0 ) t 1  T λ “ b ∈ J A ∪ J B ” } . Since V [ G ∩ P ] satisfies p = θ and A ↓ is f o rced b y t 0 to b e dense in [ N ] ω , there m ust b e a finite subset A ′ of A whic h co v ers C . It a lso follo ws easily then tha t there must b e some a, b ∈ A ′ and t 1 , t 2 eac h b elo w t 0 suc h that t 1  T λ + “ a ∈ J A ”, t 2  T λ + “ b ∈ J B ”, and a ∩ b is infinite. The final contradiction is tha t we will now hav e that t 0 fails to force tha t C ∩ a ⊂ ∗ Y and C ∩ b ⊂ ∗ ( N \ Y ).  3. T -involu tions In this section we strengthen the result in Theorem 2.1 b y making eac h K ∩ K λ a symmetric tie-p oint in K λ (at the exp ense of w eak ening Martin’s Axiom in V [ G ∩ P ]). This is progress in pro ducing inv olutions with some con trol ov er the fixed p oint set but we are still not able to mak e K the fixed p oint set o f an in v olution. A p oset is said to b e σ - link ed if there is a countable collection o f link ed (elemen ts are pairwise compatible) whic h union to the p oset. The statemen t MA( σ − link ed) TIE-POINTS AND FIX ED-POINTS IN N ∗ 13 is, of course, the assertion that Martin’s Axiom holds when restricted to σ - link ed p osets. Our approac h is to replace T - splitting tow ers b y the follow ing notio n. If f is a (partia l) in v olution on N , let min( f ) = { n ∈ N : n < f ( n ) } and max( f ) = { n ∈ N : f ( n ) < n } (hence dom( f ) is partitioned into min( f ) ∪ fix( f ) ∪ max( f )). Definition 3.1. A sequenc e T = { ( A ξ , f ξ ) : ξ ∈ W } is a to we r of T -in v olutions if W is a set of ordinals and f o r ξ < ν ∈ W and t ∈ T (1) A ν ⊂ ∗ A ξ ; (2) f 2 ξ = f ξ and f ξ ↾ ( N \ fix( f ξ )) ⊂ ∗ f η ; (3) f ξ [ x t ] = ∗ x t and fix( f ξ ) ∩ x t is infinite; (4) f ξ ([ n, m )) = [ n, m ) for n < m b oth in A ξ . Sa y that T , a tow er o f T -inv olutions, is full if K = K T = T { fix( f ξ ) ∗ : ξ ∈ W } is a t ie-set with β N \ N = A ⊲ ⊳ K B where A = K ∪ S { min( f ξ ) ∗ : ξ ∈ W } and B = K ∪ S { max( f ξ ) ∗ : ξ ∈ W } . If T is a tow er of T -in v olutions, then there is a natural inv olution F T on S ξ ∈ W ( N \ fix( f ξ )) ∗ , but t his F T need not extend to a n in v olution o n the closure of the union - ev en if the tow er is full. In this section w e prov e the fo llo wing theorem. Theorem 3.1. Assume GCH and that Λ is a set of r e gular unc ountable c ar dinals such that for e ach λ ∈ Λ , T λ is a < λ - close d λ + -Souslin tr e e. L et T denote the tr e e sum of { T λ : λ ∈ Λ } . Ther e is fo r cing extension in which ther e is T , a ful l tower of T -invo l utions, such that the a sso ciate d tie-set K has b δ -typ e ( c , c ) and such that for e ach λ ∈ Λ , ther e is a tie-set K λ of b δ -typ e ( λ + , λ + ) such that F T do es induc e an involution on K λ with a sin gleton fixe d p oint set { z λ } = K ∩ K λ . F urthermor e , for µ ≤ λ < c , if µ 6 = λ or λ / ∈ Λ , then ther e is n o tie-set of b δ -typ e ( µ, λ ) . Question 3.1. Can the tow er T in Theorem 3.1 b e constructed so that F T extends to an in v olution o f β N \ N with fix( F ) = K T ? W e in tro duce T -tow er extending forcing. Definition 3.2. If T = { ( A ξ , f ξ ) : ξ ∈ W } is a tow er of T -inv olutions and Y is a subset of N , w e define the p oset Q = Q ( T , Y ) as fo llo ws. Let E Y b e the (p ossibly empt y) set of minimal elemen ts of T suc h that there is some finite H ⊂ W suc h t ha t x t ∩ Y ∩ T ξ ∈ H fix( f ξ ) is finite. Let D Y = E ⊥ Y = { t ∈ T : ( ∀ s ∈ E Y ) t ⊥ s } . A tuple q ∈ Q if q = ( a q , f q , T q , H q ) where: 14 A. DO W AND S. SHELA H (1) H q ∈ [ W ] <ω , T q ∈ [ T ] <ω , and n q = max( a q ) ∈ A α q where α q = max( H q ), (2) f q is an in v olution on n q , (3) ( A α q \ n q ) ⊂ A ξ for eac h ξ ∈ H q , (4) fin( T q ) ⊂ n q , (5) f ξ ↾ ( N \ (fix( f ξ ) ∪ n q )) ⊂ f α q for ξ ∈ H q , (6) f α q [ x t \ n q ] = x t \ n q for t ∈ T q , W e define p < q if n p ≤ n q , and for t ∈ T p and i ∈ [ n p , n q ): (7) a p = a q ∩ n p , T p ⊂ T q , and H p ⊂ H q , (8) a q \ a p ⊂ A α p , (9) f α p ( i ) 6 = i implies f q ( i ) = f α p ( i ), (10) f q ([ n, m )) = [ n, m ) for n < m b oth in a q \ a p , (11) f q ( x t ∩ [ n p , n q )) = x t ∩ [ n p , n q ), (12) if t ∈ D p and i ∈ x t ∩ fix( f q ), then i ∈ Y It should b e clear that the inv olution f in tro duced b y Q ( T , Y ) sat- isfies that for eac h t ∈ D Y , fix( f ) ∩ x t ⊂ ∗ Y , and, with the help o f the following densit y argumen t, that T ∪ { ( γ , A, f ) } is again a to w er of T -in v olutions where A is the infinite set introduced b y the first co or- dinates of the conditions in the generic filter. Lemma 3.2. If W ⊂ γ , Y ⊂ N , and T = { ( A ξ , f ξ ) : ξ ∈ W } is a tower of T -invo lutions and p ∈ Q ( T , Y ) , then for any ˜ T ∈ [ T ] <ω , ζ ∈ W , and any m ∈ N , ther e is a q < p such that n q ≥ m , ζ ∈ H q , T q ⊃ ˜ T , and fix( f q ) ∩ ( x t \ n p ) is not empty for e ach t ∈ T p . Pr o o f . Let β denote the maxim um α p and ζ and let η denote the min- im um. Cho ose a n y n q ∈ A α q \ m large enough so t hat (1) f α p [ x t \ n q ] = x t \ n q for t ∈ ˜ T , (2) f η ↾ ( N \ ( n q ∪ fix( f η ))) ⊂ f β , (3) A β \ A η is con tained in n q , (4) n q ∩ [ i ] T p ∩ fix( f α p ) is non-empt y for eac h i ∈ N suc h t hat [ i ] T p is in t he finite set { [ i ] T p : i ∈ N } \ fin( T p ), (5) if i ∈ x t ∩ n q \ n p for some t ∈ D Y ∩ T p , then Y meets [ i ] T p ∩ n q \ n p in at least t w o p oin ts. Naturally w e also set H q = H p ∪ { ζ } and T q = T p ∪ ˜ T . The c hoice of n q is large enough t o satisfy (3), (4), ( 5 ) and (6) of Definition 3.2. W e will set a q = a p ∪ { n q } ensuring (1) of Definition 3.2. Therefore for a n y f q ⊃ f p whic h is a n in v olution on n q , w e will hav e that q = ( a q , f q , T q , H q ) is in the p oset. W e hav e to choose f q more carefully to ensure that q ≤ p . Let S = [ n p , n q ) ∩ fix ( f α p ), and S ′ = [ n p , n q ) \ S . W e c ho ose ¯ f an in v olution o n S and set f q = f p ∪ ( f α p ↾ S ′ ) ∪ ¯ f . W e lea ve TIE-POINTS AND FIX ED-POINTS IN N ∗ 15 it to the reader to c hec k that it suffices to ensure that ¯ f sends [ i ] T p ∩ S to itself for each t ∈ T p and that fix( ¯ f ) ∩ x t ⊂ Y f o r each t ∈ T p ∩ D Y . Since the mem b ers of { [ i ] T p ∩ S : i ∈ N } are pairwise disjoint w e can define ¯ f on eac h separately . F or each [ i ] T p ∩ S whic h has ev en cardinalit y , choose tw o p oints y i , z i from it so tha t if there is a p ∈ D Y ∩ T p suc h that [ i ] T p ⊂ x t , then { y i , z i } ⊂ Y . Let ¯ f b e any inv olution on [ i ] T p ∩ S so that y i , z i are the only fixed p oin ts. If [ i ] T p ∩ S has o dd cardinality then c ho ose a p oin t y i from it so t hat if [ i ] T p is con tained in x t for some t ∈ D y ∩ T p , then y i ∈ Y ∩ [ i ] T p ∩ S . Set ¯ f ( y i ) = y i and c ho ose ¯ f to b e any fixed-p o in t free in volution on [ i ] T p ∩ S \ { y i } .  Let P θ no w b e the finite supp ort iteration defined a s in Definition 2.6 except for tw o imp ort a n t c hanges. F or γ ∈ C , w e replace T -splitting to w ers by the ob vious inductiv e definition of tow ers o f T -in v olutions when we replace the p osets ˙ Q ( S C ∩ γ , ˙ Y γ ) by ˙ Q ( T C ∩ γ , ˙ Y γ ). F o r γ / ∈ C we require that  P γ “ ˙ Q γ is σ - link ed.” Sp ecial (parity ) prop erties o f the family { x t : t ∈ T } are needed to ensure that  P γ “ ˙ Q ( S C ∩ γ , ˙ Y γ ) is ccc ” ev en for cases when cf ( γ ) is not ω 1 . The pro of of Theorem 3 .1 is virtually the same as the pro of of The- orem 2.1 (so we skip) once w e ha v e established tha t the iteration is ccc. Lemma 3.3. F or e ach γ ∈ C , P γ +1 is c c c. Pr o o f . W e a g ain define P ∗ α to b e those p ∈ P α for which there is an n ∈ N suc h that for eac h β ∈ do m( p ) ∩ C , t here are n ∈ a β ⊂ n +1, f β ∈ n n , T β ∈ [ T ] <ω , and H β = dom( p ) ∩ C ∩ β suc h that p ↾ β  P β “ p ( β ) = ( a β , f β , T β , H β )”. Ho w ev er, in this pro o f we m ust also mak e some sp ecial assumptions in co ordinat es other than t ho se in C . F or eac h ξ ∈ γ \ C , w e fix a collection { ˙ Q ( ξ , n ) : n ∈ ω } of P ξ -names so that 1  P ξ “ ˙ Q ξ = [ n ˙ Q ( ξ , n ) and ( ∀ n ) ˙ Q ( ξ , n ) is link ed.” The final restriction on p ∈ P ∗ α is t ha t for eac h ξ ∈ α \ C , there is a k ξ ∈ ω suc h t hat p ↾ ξ  P ξ “ p ( ξ ) ∈ ˙ Q ( ξ , k ξ )”. Just as in Lemma 2.11, Lemma 3.2 can b e used to sho w b y induction that P ∗ α is a dense subset of P α . This time though, we a lso demand that dom( f p (0) ) = n × T p (0) is suc h that T β ⊂ T p (0) for a ll β ∈ dom ( p ) ∩ C and some extra argumen t is needed b ecause of needing to decide v alues in the name ˙ Y γ as in the pro of o f Lemma 3.2. Let p ∈ P β +1 and assume that P ∗ β is dense in P β . By densit y , w e may assume that p ↾ 16 A. DO W AND S. SHELA H β ∈ P ∗ β , H p ( β ) ⊂ dom( p ), T p ( β ) ⊂ T p (0) , and that p ↾ β has decided the mem b ers o f the set D ˙ Y β ∩ T p ( β ) . W e can assume further tha t for eac h t ∈ D ˙ Y β ∩ T p ( β ) , p ↾ β has fo rced a v alue y t ∈ ˙ Y β ∩ x t \ S { x s : s ∈ T p and s 6≤ t } suc h that y t > n p ( β ) . W e are using that T is not finitely branc hing to deduce that if t ∈ D ˙ Y β , then p ↾ β  P β “ ˙ Y β ∩ x t \ S { x s : s ∈ T p and s 6≤ t } is non- empt y” ( which follows since ˙ Y β m ust meet x s for each immediate succes sor s of t ). Cho o se an y m larger t ha n y t for eac h t ∈ T p ( β ) . Witho ut loss of generality , w e may assume that the integer n ∗ witnessing that p ↾ β ∈ P ∗ β is at least as large as m and that n ∗ ∈ T ξ ∈ H p ( β ) A ξ . Construct ¯ f just as in Lemma 3.2, except that this time there is no requiremen t to actually ha v e fixed p o in ts so o ne member of ˙ Y β in each appro priate [ i ] T p ( β ) is all that is required. Let ζ = max(dom( p ) ∩ β ). No new forcing decisions are required of p ↾ β in order to construct a suitable ¯ f , hence this shows that p ↾ β ∪ { ( β , q ) } (where q is constructed b elo w p ( β ) as in Corollary 3.2 in whic h H p ( ζ ) ∪ { ζ } is add to H q ) is the desired extension o f p whic h is a mem b er of P ∗ β +1 . No w to sho w that P γ +1 is ccc, let { p α : α ∈ ω 1 } ⊂ P ∗ γ +1 . Clearly w e may assume t ha t the family { p α (0) : α ∈ ω 1 } are pairwise compat- ible and tha t there is a single integer n suc h that, for each α ∈ ω 1 , dom( p α (0)) = n × T α for some T α ∈ [ T ] <ω . Also, we ma y a ssume that there is some ( a, h ) suc h that, for each α , p α ↾ γ  P γ “ p ( γ ) = ( a, h, T α , H α )” where H α = dom( p α ) ∩ C ∩ γ . The family { dom( p α ) ∩ γ : α ∈ ω 1 } ma y b e assumed to form a ∆- system with ro ot R . F or each ξ ∈ R , w e ma y a ssume t ha t , if ξ / ∈ C , there is a single k ξ ∈ ω suc h that, fo r all α , p α ↾ ξ  P ξ “ p α ( ξ ) ∈ ˙ Q ( ξ , k ξ )”, and if ξ ∈ C , then there is a single ( a ξ , h ξ ) suc h that p α ↾ ξ  P ξ “ p α ( ξ ) = ( a ξ , h ξ , T α , H α ∩ ξ ) ”. F or con v enience, for eac h ξ / ∈ C let ˙ r ξ b e a P ξ -name of a function from ω × ˙ Q 2 ξ suc h that, for eac h k ∈ ω , 1  P ξ “ ˙ r ξ ( k , q , q ′ ) ≤ q , q ′ ( ∀ q , q ′ ∈ ˙ Q ( ξ , k ))” . Fix an y α < β < ω 1 and let H = H α ∪ H β . Recall tha t p α (0) and p β (0) a re compatible. Recursiv ely define a P ξ -name q ( ξ ) for ξ ∈ TIE-POINTS AND FIX ED-POINTS IN N ∗ 17 dom( p α ) ∪ dom( p β ) so that q ↾ ξ  P ξ “ q ( ξ ) =              ( n, T α ∪ T β , f p α (0) ∪ f p β (0) ) ξ = 0 ˙ r ξ ( k ξ , p α ( ξ ) , p β ( ξ )) ξ ∈ R \ C p α ( ξ ) ξ ∈ dom( p α ) \ ( R ∪ C ) p β ( ξ ) ξ ∈ dom( p β ) \ ( R ∪ C ) ( a ξ , h ξ , T α ∪ T β , H ∩ ξ ) ξ ∈ C . ”. No w we c hec k t ha t q ∈ P ξ b y induction on ξ ∈ γ + 1 . The first thing to note is that no t only is this true for ξ = 1, but also that q (0)  Q 0 “ fin ( T α ∪ T β ) ⊂ n ”. Since p α and p β are eac h in P ∗ γ +1 , this sho w that condition (4) of Definition 3.2 will hold in all co ordinates in C . W e also prov e, b y induction on ξ , that q ↾ ξ forces that f o r η < δ b oth in H ∩ ξ and t ∈ T α ∪ T β , f δ [ x t \ n ] = x t \ n , f η ↾ ( N \ ( fix( f η ) ∪ n )) ⊂ f δ and A δ \ n ⊂ A η . Giv en ξ ∈ H and the assumption that q ↾ ξ ∈ P ξ , and α = α q ( ξ ) = max( H ∩ ξ ), condition (3), (5), and (6) of Definition 3.2 hold by the inductiv e h yp othesis a b o v e. It follows then that q ↾ ξ  P ξ “ q ( ξ ) ∈ ˙ Q ξ ”. By the definition of the o rdering on ˙ Q ξ , give n that H ∩ ξ = H q ( ξ ) and T α ∪ T β = T q ( ξ ) , it follows that the inductiv e hy p o thesis then holds for ξ + 1. It is trivial fo r ξ ∈ dom( q ) \ C , that q ↾ ξ ∈ P ξ implies that q ↾ ξ  P ξ “ q ( ξ ) ∈ ˙ Q ξ ”. This completes the pro of that q ∈ P γ +1 , and it is trivial that q is b elow each o f p α and p β .  R emark 1 . If w e a dd a trivial tree T 1 to the collection { T λ : λ ∈ Λ } (i.e. T 1 has only a ro ot), then the r o ot of T has a single extension which is a maximal no de t , and with no change to the pro of of Theorem 3.1, one obtains that F induces an automorphism on x ∗ t with a single fixed p oin t. Therefore, it is consisten t (and lik ely as constructed) that β N \ N will ha v e symmetric tie-p oin ts of t yp e ( c , c ) in the mo del V [ G ∩ P ] and V [ G ]. R emark 2 . In the pro of of Theorem 2 .1, it is easy to arrange t ha t eac h K λ ( λ ∈ Λ) is also K T λ for a ( T λ -generic) full tow er, T λ , of N - in v olutions. Ho w ev er the generic sets added by the forcing P will pre- v en t this tow er of in v olutions from extending to a full in v olution. 4. Questions In this section w e list a ll the questions with their original n um b ering. 18 A. DO W AND S. SHELA H Question 1.1. Can there b e a tie-p oint in β N \ N with δ -type ( κ, λ ) with κ ≤ λ less than the ch ara cter of the p oin t? Question 1.2. Can β N \ N hav e tie-p oin ts of δ -t yp e ( ω 1 , ω 1 ) and ( ω 2 , ω 2 )? Question 1.3. Do es p > ω 1 imply there are no tie-p oin ts of b -type ( ω 1 , ω 1 )? Question 1.4. If F is a n inv olution on β N \ N suc h that K = fix ( F ) has empt y in terior, is K a (symmetric) tie-set? Question 1.5. Is there some nat ura l restriction on whic h compact spaces can (or can not) b e homeomorphic to the fixed p oin t set o f some in v olution o f β N \ N ? Question 1.6. If F is an inv olution of N ∗ , is the quotien t space N ∗ /F (in whic h eac h { x, F ( x ) } is collapsed to a single p o in t) a homeomorphic cop y of β N \ N ? Question 3.1. Can the tow er T in Theorem 3.1 b e constructed so that F T extends to an in v olution o f β N \ N with fix( F ) = K T ? Reference s [1] J¨ org Brendle and Saharo n Shelah, Ultr afilters on ω —their ide als and their c ar- dinal char acteristics , T r ans. Amer. Math. So c. 351 (199 9), no. 7, 264 3 –2674 . MR 1686 797 (20 00m:031 1 1) [2] Alan Dow, Two to one images and PF A , Isr ael J. Math. 156 (2 006), 221– 241. MR 2282 377 [3] Alan Dow and Geta T echanie, Two-to-one c ont inu ous images of N ∗ , F und. Math. 186 (2005), no. 2 , 1 77–19 2. MR 216 2384 (2 006f:540 03) [4] Thomas Jech, Set the ory , Springer Monogr aphs in Mathematics, Springer- V erla g , Ber lin, 2 003, The third millennium edition, rev ised and expanded. MR 1940 513 (20 04g:03 071) [5] Ronnie Levy , The weight of c ertain images of ω , T o po logy Appl. 153 (200 6), no. 13, 227 2–227 7. MR MR223 8730 (2007e:54 034) [6] S. Shelah and J. Stepr¯ ans . Non-trivia l ho meomorphisms of β N \ N witho ut the Contin uum Hypothesis. F und. Math. , 132 :135–1 41, 1989. [7] S. Shelah and J. Stepr¯ ans. Somewher e trivial a utohomeomorphisms . J. L ondon Math. So c. (2) , 49:569 –580, 199 4. TIE-POINTS AND FIX ED-POINTS IN N ∗ 19 [8] Saharo n Shelah and Juris Stepr¯ ans, Martin ’s axiom is c onsistent with the ex - istenc e of n owher e trivial automorphisms , Pro c. Amer. Math. So c. 1 30 (2002), no. 7, 2097 –2106 (electronic). MR 1896 0 46 (200 3k:0306 3) [9] B. V eliˇ cko vi´ c. Definable a utomorphisms of P ( ω ) /f in . Pr o c. Amer. Math. So c. , 96:130 –135, 198 6. [10] Boban V eliˇ ckovi ´ c. OCA a nd automorphisms of P ( ω ) / fin. T op olo gy Appl. , 49(1):1–1 3, 19 93. Dep ar tment o f Ma thema tics, Rutgers University, Hil l Center, Pis- ca t a w a y, N ew Jersey, U. S.A. 08854-801 9 Curr ent addr ess : Institu te of Mathematics, Hebrew Univ ersity , Giv at Ram, Jerusalem 9190 4, Isr ael E-mail addr ess : she lah@ma th.rut gers.edu