R ENDICONTI
del
S EMINARIO M ATEMATICO
della
U NIVERSITÀ DI P ADOVA
T UDOR Z AMFIRESCU Area contractions in the plane
Rendiconti del Seminario Matematico della Università di Padova, tome 46 (1971), p. 49-52
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TUDOR ZAMFIRESCU
*)
The Picard-Banach contraction
principle
admitsgeneralizations
inwhich volume contractions appear instead of distance
contractions,
asin the
original
theorem. Our purpose in this note is to obtain such results in theplane
case; this will becompletely
done for boundeddomains of
definition,
the nonbounded casebeing only
touchedby
Theorems 2 and 3. To
generalize
these results for the n-dimensionalcase is routine.
We restrict ourselves to
considering
area contractions in aquite
usual Euclidean
plane
R 2.There,
we have a set M and a functionf :
M ~ M which will be called area contractionbecause,
for somenumber
1),
for each
triple
ofpoints a, b, ceM,
being
thetriangle-area
function on R~.The sequence of
points
is called the orbit ofueM,
and its set of limitpoints
is denotedby L(u). Put £ =
UL(u).
A set in R~ will be called linear if it is included in some
straight
line.
*) Indirizzo dell’A.: Mathematisches Institut der Universitat Dortmund - Rep.
Federale Tedesca.
50
THEOREM 1.
If
M isbounded, then £
is linear.PROOF.
Suppose £
is notlinear,
i.e. there exists a set{ x,
y,z } c ~
which is not linear. Let
X, Y,
Z beneighbourhoods
of x, y, z respec-tively,
such thatI > o,
whereLet
and
Suppose x E L(u),
Then there exist three natural numbers p, q, r > N such thatfP(u)eX,
It followswhich
provides
a contradiction.For M
nonbounded,
one may stillconjecture
that£,
ifnonempty,
is linear. We canonly
prove here that under certain additional assump-tions, C
is linear indeed.THEOREM 2. Il two
points
u and vfrom
M have bounded orbits andL(u)
isdisjoint from L(v), then C
is linear.PROOF.
Following
Theorem1, L(u)
uL(v)
is linear.Suppose
thereexists a
point w in £
which does notbelong
to the linecontaining L(u) u L(v).
ThenA(w,
x,y) > 0
for everyxeL(u), yeL(v).
The pro- duct{ w
} XL(u)
XL(v) being compact,
the infimumis
attained,
hence v>0.Take the open sets
W, X,
Yrespectively including f w), L(u), L(v),
such that
A(a, b, c) > v/2
for eachaeW, beX,
c E Y. For some natural numbers N, andN2, implies
andn > N2 implies fn(v)EY.
Let For some natural number
N~ , n > N3 yields
Taking
now n’ > max(Ni , N2, N3 1
such that(fn’(u)
and¡n’(v) respectively lying
in X andY),
which contradicts thepreceding inequality.
THEOREM 3.
If £
islinear,
x and y are two distinctpoints
in.2,
u, veM are such that
xeL(u), yeL(v),
and the orbitsof
u and v are bounded(u
and v mycoincide),
then the lineincluding £
is afixed
lineunder the
mapping f.
PROOF. Consider the line
Suppose f(M
n8) ct:
S. Then there exists apoint
zeM n 6 such thatTake the
neighbourhoods
X of x and Y of y, and the number v > 0 such thatfor each
couple
ofpoints (a, b)eX
X Y.Let A be an open set
including L(u)
uL(v)
uz }
such thatfor each three
points
a,b,
ceA. There is a number N such that n >_ Nimplies f(u), jn(v)eM
n A .Consider now two natural numbers p, such that
fP(u) e X,
fq(v) c Y.
Then52
and
which both
together
contradict the contractioninequality.
COROLLARY.
I f ~ is
not asingle point,
and M isbounded,
then theline
including ~
is afixed
line under themapping f .
Before
ending
the paper, we wish to thank here Prof. A. Cornea for hissuggestions
connected with thistopic,
and Prof. L. Lamberti for his most valuablehelp
inpointing
outsimplifications
inproofs
ofthe
original
version of this note.Manoscritto pervenuto in redazione il 16 marzo 1971.