1Zemαγ★3．10． It is easy to see that the property（1）does not depend on the coordinates used to set I in the z−axis． Similarly， the properties（2）and（3）
do not depend on the coordinates used to set Sμin the∬y−plane．
When det（H∫レ。）（μ。， o）＞0（resp．＜0）in De丘nition 3．9（3）， we say that the tangency r＝（μo，uo，0）is of e↓吻κc（resp．んyperbo∬c）type． The Taylor expansion ofノ㌦o around（μo， uo）is
F士om（3．3）together with the classi丘cation of quadratic surfaces in IR3， we know that｝τ。 has the fbrm near r＝（μo，uo，0）as illustrated in Fig．3．3．
z z z
X lU ・ X l X l」一ぺ
（1） （2） （3）
UO UO
Figure 3．3：（1）ris of elliptic type and∂2ん。（μo，uo）／∂ω2〈0． （2）ris of elliptic type and∂2戊ソo（rμo，ruo）／∂∬2＞0．（3）ris of hyperbolic type．
3．12 Generic conditions
Throughout the remainder of this paper， we suppose thatψis a 3−dimensional
O2 dif昆omorphism with saddle丘xed points p of index（p）＝1and 40f
index（9）＝2， and such that仰ノ8（ρ）and『日ソμ（σ）have a nondegenerate heterodi−
mensional tangency r， Wμ（ρ）and W8（σ）have a quasi−transverse intersection 3．Theψis locα11y O21仇eαr疹zαble in a neighborhoodσ（σ）ofσif there exists aO21inearizing coordinate（エ，31， z）onσ（9）， that is，
q＝（0，0，0）， （ρ（∬，ッ，づ＝（α∬，βy，7z） （3．4）
fbr any（エ， y， z）∈σ（σ）withψ（猛，y， z）∈σ（σ）， whereα，βand・γare eigen−
values of（（1（ρ）σ．
One can take a local unstable manifbld Wi：c（σ）so that it is an open disk
in the plane｛z＝0｝centered at㊤，y）＝（0，0）． We may assume that the
32
both points r，3are contained inσ（9）if necessary replacing r（resp．8）by ゾn（r）（resp．9π（3））with su伍ciently largeπ∈N． We set
r＝（μo，Uo，0）
with respect to the linearizing coordinate onσ（4）．
We suppose moreover that｛ψμ，〃｝is a two−parameter family in Di音2（M）
withψo，o・＝ψand satisfying the conditions of Theorem 3．1． In particular，
ψμμis locally O21inearizable in a small neighborhoodσ（σμ，〃）of g防． in M and henceψμhas the form as（3．4）inσ（gμμ）， whereα，β，7are O2 fUnctions onμ，〃， i・e・，α＝αμμ，β＝βμ，〃／γ＝7μμ・
We will put the fbllowing generic conditions（C1）一（C4）as the hypotheses in Theorem 3．1．
（C1）（Generic condition fbr g）Theψis locally O21inearizable atσgiven as in（3．4）． Fbr simplicity， we suppose that every eigenvalues of（吻）g is positive， that is，
0＜7＜1＜β＜α．
（C2）（Generic unfblding property fbrりThe nondegenerate heterodimen−
sional tangency r ofルγ％（・）and W5（p）unfblds generically with respect to theμ一parameter families｛肝（9μ，o）｝and｛W8（ρμρ）｝．
（C3）（Generic unfblding property fbr 8）The quasi−transverse intersection 3 0fルγ8（σ）and W％（ρ）unfblds generically with respect to the〃−parameter 塩milies｛w8（σ・，．）｝and｛W％（ρoμ）｝．
（C4）（Additional generic conditions）The tangency r is not on the∬−axis
慨：：（4），that is，
Uo≠0． （3．5）
There exists a regular parametrization I（オ）＝＠（τ），y（τ），z（τ））（τ∈∫）of asmall curve in 1〃勉（ρ）∩σ（σ）with respect to the linearizing coordinate
（ω，y， z）onσ（σ）wi七h s＝1（0）and
∂z
㌃（o）≠ぴ （36）
where∫is an open interval centered at O．
There exists a O2 function∫：0→Rdefined on an open disk O in the
∬y＿plane centered atγ・such that∫（μo，uo）＝0，｛（ω， y，∫（エ， y））；（ちy）∈
0｝⊂「レγ8（ρ）∩σ（q）and
；裟M≠α （37）
Note that the condition（3．7）is automatically satis丘ed when r is of elliptic type．
3．2 Some lemmas about parametrization and curva−
tures
The goal of this section is to prove three lemmas needed fbr the proof of Theorem 3．1． These play important roles in Section 3．3．
●Lemma 3．11presents a new parameter（A Aμ， ）such that， fbr anyρnear O，
there exists a quasi−transverse intersection sρ1，00f「レγ8（9ρ、，o）and W％（ρρ，o）
which unfblds generically atρ＝Owith respect to theρ一parame七er．
After Lemma 3．11， we denote the new parameter（∧ Aμ， ）again by（μ，μ）
fbr simplicity．
●In Lemma 3．12， we show that， fbr anyμo near O， there exists a regu−
lar curve Im in「レレw（Pμo，o）containing the quasi−transverse intersection ψ㌶，o（8μo，o）and arbitrarily O2 close to明：：（4μo，o）・ In particular， this implies that the curvature of Im can be taken arbitrarily close to O with respect to the linearizing coordinate（3．4）onσ（σμo，o）．
●Lemma 3．13 gives a connection between the curvature and quadratic
tangencies． In fact， we show that a tangencyτof a regular curve I and a regular surface 3 in R3 is quadratic if the curvature of I atア is dif飴rent from the normal curvature of S at 7−along the direction tangent to I．For any（μ，〃）near（0，0）， we may assume thatσ（gμ， ）is equal to
D（δ）：＝（一δ，δ）3
with respect to the linearizing coordinate given in Subsection 3．1．2｛br some constantδ＞0． Since 8 is a quasi−transverse intersection which unfblds gener−
ically with respect to the〃−parameter families｛W8（40， ）｝and｛W％（po，．）｝by the condition（C3）， there exists a O2 continuation 8〃∈▽μ（ρo，〃）∩D（δ）with 80＝3and such that 9 satis丘es the conditions same as those fbr 5 in De丘ni−
tion 3・9（1）・By（3・6）， fbr any〃near O， the component Z． of Wμ（ρo，．）∩D（δ）
containing 9 meets transversely the yz−plane at a point 8． which de丘nes aO2 continuations｛3〃｝with 80＝8， see Fig．3．4． Note that d8〃／（∫μ（0）＝
∂3．／d〃（0）十ωfbr someω∈万（lo）＝万（W％（p））， where∂ん／∂〃（0）denotes
∂9〃／（1〃レ＝o． Let 31（〃） be the y−coordinate of 8〃． If（ら／〔1〃（0） ＝ 0， then
34
Z ／ cひ●●・・「
ぷ＼』ぷ靭〆
Figure 3．4：
d3μ／d！〃（0）would be tangent to the z−axis慨さc（σ）at 3 and hence（膳〃／∂〃（0）∈
Z』（W8（σ））㊥宏（ルγ秘（p））． This contradicts（3．1）． Thus， we have
砦（・）≠α （38）
For any（μ，〃）close to（0，0）， let 8μ，〃be a transverse intersection point of レγμ（ρ〃，〃）∩D（δ）with the解一plane such that｛8μ，〃｝is a O2 continuation with
30，〃：＝8〃・
Lemma 3．11．τみere e斑￡3αco耐励ρ＞0α閲α02ルηcτion〃：（一ρ，ρ）→
R3μch君んαちプ6rαηyμ∈（一ρ，ρ），8μ，ρ（μ）i3α4鋤α3i一診7 αγあ3uer8e仇τer3εC渉勿η0『
W8（4μ，ρ（μ））αηd肝（ρμ，ρ（μ））晒cゐ吻bld3 geηer乞Cα吻ω軌re3畑彦・彦乃eμ一 ραrαmeτeゆ嚇e3｛W5（・μ（丘。，d），り｝αη4｛肝（ρ，（丘。，d）， ）｝・
Proq声Let y（μ，〃）be the y−coordinate of 3μ，〃． By（3．8），∂y／∂μ（0，0）≠0．
Hence， by the Implicit Function Theorem， there exists a O2 function〃：
（一ρ，ρ）→Rfbr someρ＞Osuch that
ρ（・）一咀（仏〃（μ））一曙（呼（μ））≠・
fbr anyμ∈（一ρ，ρ）． This implies that 3μ，ρ（μ）is a quasi−transverse intersection
unfblding generically at〃＝〃（μ）with respect to the〃−parameter families
｛W8（（1μ，レ）｝and｛「レγμ（Pμ，〃）｝・ □
Anew parametrization
Consider the coordinate（A Aμ， ）on the parameter space de丘ned byρ＝μ，ρ＝
〃一〃（μ）．Fbr simplicityl we denote the new coordinate again by（μ，り． Thus，
there exists a continuation｛8μ，o｝μ∈（＿ρ，ρ）of quasi−transverse intersections of
W8（qμ，o）and Wμ（ρμ，o）such that each 8μ，o unfblds generically at〃＝Owith respect to theμ一parameter families｛W5（gμ，．）｝and｛Wμ（ρμ， ）｝．
Fixμo with suf五ciently small lμo l arbitrarily By the properties（3．4）and
（3．6），there exists mo∈Nsuch that， fbr any m≧mo， one can parameterize the component Im of Wu（ρμo，0）∩D（δ）containingψ㌶，o（8μ。，o）so that Im（0）＝
ψ㌶，o（8μo，o）and
Zm（τ）＝＝（τ，9／m（τ），2∫m（τ）） （τ∈ （一δ，δ））．
Lemma 3．12． TWe 3e卿eηce｛lm｝02 co噺卿e3肌ぴorm1垣o Wi器（gμo，o）α8 m→oc．∫ηρ励 cμ励∫brαnyε＞0，τんere e蹴τ3命o≧mo 3刎cれんατ 輪醐りα伽eα‡卿ρ0 ηオ0『lm 31e83 Z九αηεω軌アe8ρec杜0τんe 8ταηdαrd
lDμc∬∂eαηη1eτric oγzσ（σμo，o）＝・D（δ） 勾Fm≧仇o．
Proo∫By（3．4）， fbr any m≧mo，
Im（τ）＝（舌，βπym。（α一ητ），7ηZm。（α一ητ）），
whereη＝m−mo，α＝αμ。，o，β＝βμ。，o，7＝7μ。，o・Thus we have
藁1：：ll：簿∴）翼曇ぎ∵∴）ω）
as m→○◇． Since｛lm（0）｝篇＝m。 converges to qμo，o＝（0，0，0）， it fbllows from
（3．9）that｛Zm｝02 converges unifbrmly to theコc−axis in D（δ）． □
Lemma 3．13． Leτ8beαre鋼αr 8μ加ce仇診んe」仇c掘eαη3−3ραce R3αη∂↓
αre鋼αr c脚e諺αηgeη診τ・Sα診ア． S卿・3eオんα崩e c脚α加reκ1（τ）・∫1ατ 丁づ81e33抗α励ゐeαb5・施彦euα九e（ゾτんeη・rmαIC惚uα加reκS（ア，ω）・∫Sατア α10η9αη0η一zero uec彦or祖ταηgeηオτo l．7We川αηgeηcy oゾSαη∂1ατアi3 9μαd耐ic．
Proq￡By changing the coordinate（∬， y， z）on R3 by an isometry， we may assume thatア＝（0，0，0）， the tangent space of 8 atτis the巧一plane and ω／llωll＝（1，0，0）． Then one can suppose that S（resp．のis parameterized as（∬， y，ψ（エ， y））（resp．㊤，∫1（エ），克＠））in a small neighborhood of（0，0，0）
in R3． Since the graph of z・＝ψ（エ，0）is the cross section of S along the
∬z−plane，
1κ8（ 19 （oア，ω）ト （9 （o）2＋）｝）3／2−1〆（・）1，
36
where g（り＝ψ（猛，0）． Since the graph of z＝∫2＠）coincides with the
orthogonal projection I of I into the zz−plane，κ1（ア）≧κτ（ア）一 i∬（耀｛）3／2−1κ（・）L
It fbllows from our assumption lκs（ア，⇒1＞κ1（丁）that lg （0）1＞げ； （0）1． This shows that the tangency atτis quadratic． □
3．3 Proof of Theorem 3．1
In this section， we give the proof of Theorem 3．1．
●In Subsection 3．3．1， we show that， for anyμo in either（一ε，0）or（0，ε）
and any su伍ciently large m∈N， there exists〃m with limm→。。 z！m＝O such that 仰ノ勉（ρμo，〃m） and W5（pμo，〃m）have a quadratic tangency 7玩 （Assertion 3．15 and Assertion 3．16）． Here the sigll ofμo is chosen so thatμo・bμ。，o＜0（resp・μo・bμo，o＞0）if the tangency r is of elliptic （resp． hyperbolic）type， where 6μo，o is the coefHcient of（g−〃μo，o）2．
term of the Taylor expansion（3．10）． See Fig．3．5 and Fig．3．8． As is suggested in Fig．3．6， the existence of the homoclinic tangency偏is proved by using the Intermediate Value Theorem． By Lemma 3．12， the curvature of「レγμ（Pμoμm）atτm converges to zero as m→∞・ On the other hand， we will show that the normal curvature of W8（pμo， m）atτm along the direction tangent to Im is bounded away f士om zero． Hence，
by Lemma 3．13， the tangency石is quadratic．
●In Subsection 3．3．2， we show that the quadratic homoclinic tangency偏 皿fblds generically at〃＝〃m with respect to the〃−parameter families ｛W8（Pμo，〃）｝and｛「レγu（Pμo，〃）｝by representing a neighborhood ofτ肌in W8（Pμo，〃）as the graph of a fUnction of（エ， z）・
3．3．1 Existence of quadratic homoclinic tangencies
Let｛￠）μ，〃｝be the family given in Subsection 3．1．2． In particular， r＝
（μo，uo，0）is a nondegenerate heterodimensional tangency of刊7％（句and 1〃3（p）
which unfblds generically with respect to theμ一parameter families｛Wμ（σμ，o）｝
and｛w5（Pμ，o）｝． By our settings in Sections 3．1 and 3．2， there exist O2 func−
tionsん，μ：0⊂IR2→RO2 depending on（μ，〃）with／b，o＝∫and
Σ（μ，〃）：＝｛（∬，y，ん，〃（∬，y））；（∬，y）∈0｝⊂仰ノ8（ρμ，〃）∩D（δ）
fbr any（μ，〃）near（0，0）． Since det（H∫）（μoψo）≠0， there exists a uniquely determined O2 continuation（μμ，μ，uμ，〃）with（獅o，uqo）＝＠o，uo）and
警』の一寄』D−・
Proposition 3．14．」Fbrα3μガic eη吻5mα〃ε＞0α閲αηyμ η， e 仇θr（0，ε）
・r（一ε，0），仇ere e斑8〃αrb伽α晦cZ・3eτ・03μc励鋤μんα8α4μα伽彦乞c
ん・m・C甑Cταη9ε卿α38・C ατeれ・ρ拓 ．By the condition（3．5），ris not in theひaxis． One can take the linearizing coordinate on D（δ）so that 3（resp． r）is in the upPer half space｛z＞0｝（resp，
｛エ＞0｝）．The Taylor expansion of∫防レaround（輪．，uぽ）has the form： ．
whereαqo＝＝Oand
Since the tangency r unfblds generically with respect toψ＝ψoρby（C1），
恥一 ̲＿）≠・ （3．11）
If necessary replacingμby一μ， we may assume thatηo＞0． By the condition
（3．7），bo式）≠Oand hence b防〃≠0丘）r any（μ，のnear（0，0）．
Proposition 3．14 is divided to the fbllowing two assertions．
Assertion 3．15（Elliptic case）．、ぴr 8〔ぴe1吻オ c勅）θ，抗eη」Proρ03臨oπ314
九〇ld8．
、Proq￡First we consider the case of bμμ〈Ofor any（μ，〃）near（0，0）． By
（3．11），負）rany su伍ciently smallμo＞0， the intersection（九。＝Σ（μo，0）∩
｛z＝0｝is a circle di司oint ffom the伽axis． For a suf丘ciently sma11力o＞0，
A＝Σ（μo，0）∩｛0≦z≦んo｝is an annulus in D（δ），see Fig．3．5（1）． Replacing mo by an integer greater than mo if necessary， we may assume that zm（0）＜
んo／2fbr any m≧7ηo． By Lemma 3．12， the curve Im⊂Wμ（Pμo，0）∩D（δ）
given in Section 3．2 is su」田ciently O2 close to the∬−axis． Thus one can
suppose thatπy（ん）∩πy（A）＝の， whereπy：D（δ）一→Ris the orthogonal
projection de丘ned byπy（∬， y， z）＝y． For any su伍ciently small〃， let Imμ38
Z z
x ↑1 x ↑）
（1） （2）
Figure 3．5：（1）The case ofμo＞0，〃＝0， bμ。，o＜0．（2）The case ofμo一〈
0，〃＝0，bμo，o＞0． Each shaded region represents、4．
be the component of Wμ（pμo， ）∩D（δ）such that｛lm，．｝is an〃−continuation with Im，o＝Im， and set、4．＝Σ（μo，〃）∩｛0≦z≦んo｝． Moreover， one can supPose that Im， is parameterized as
Im，〃（τ）＝（ちym（Zノ，τ），Zη♪（Zノ，τ）） （τ∈（一δ，δ））．
By the condition（C3）， one can takeρ≠Owith arbitrarily small lρl such that
O＜πy（Zmo，ρ（0））≦sup｛πy（lmo，の｝＜min｛πy（．4り｝．
We may assume thatρ＞Oif necessary replacing〃by−〃． For any integer
msuf丘ciently greater than mo， there exists O＜ρm、＜ρsuch the continuation｛lm，〃｝o≦〃≦ρm is well de丘ned and
max｛πy（・4ρm）｝＜inf｛πy（lm，ラm）｝
holds， see Fig．3．6（1）． By the Intermediate Value Theorem， there exists O＜〃η｝＜ρmsuch that Iw♪，〃m and／1〃m have a tangency石， see Fig．3．6（2）．
Since脇 m⊂ルγμ（ρμo，〃m）andんm⊂ルγ5（ρμ。，μm），ηηis a homoclinic tangency associated to Pμo，〃m・
When bμ， ＞Ofbr any（μ，〃）near（0，0）， one can prove the existence of
ahomoclinic tangencyτm near r associated toρμo，μm by arguments quite
similar to those as above fbr anyμo withμo＜0．It remains to show that the tangency玩is quadratic． SinceΣ（μo，〃m）
is of elliptic type and limm→。。〃m＝0， any normal curvature ofΣ（μo，〃m）
atηn is greater than some positive constantκo independent of m． On the other hand， by an argument quite similar to that in Lemma 3．12， for any
2 z