EXPONENTIALS FORM A BASIS OF DISCRETE HOLOMORPHIC FUNCTIONS

Christian Mercat

Description

EXPONENTIALS FORM A BASIS OF DISCRETE HOLOMORPHIC FUNCTIONS by Christian Mercat Abstract. — We show that discrete exponentials form a basis of discrete holomorphic functions on a critical map. On a combinatorially convex set, the discrete polynomials form a basis as well. Resume (Les exponentielles forment une base des fonctions holomorphes discretes) Nous montrons que les exponentielles forment une base des fonctions holomorphes discretes sur une carte critique. Sur un convexe, les polynomes discrets forment egalement une base. 1. Introduction The notion of discrete Riemann surfaces has been defined in [7]. A good basis for the associated space of holomorphic functions was still missing. This article discuss an interesting one in the simply connected critical case. We are interested in a cellular decomposition ? of the complex plane or a simply connected portion U of it, by rhombi (equilateral quadrilaterals, or lozenges). In other words, we have a map from the set of vertices ?0 to the complex plane Z : ?0 ? C such that for each oriented face (x, y, x?, y?) ? ?2, its image is a positively oriented rhombus (Z(x), Z(y), Z(x?), Z(y?)) of side length ? > 0. It defines a straightforward Cauchy-Riemann equation for a function f ? C0(?) of the vertices, and similarly for 2000 Mathematics Subject Classification.

• agree point-wise


• connected critical


• discrete exponentials


• converging discrete


• wise multiplication


• cauchy-riemann equation

Sujets

• Kenyon
• Cauchy?Riemann equations

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 15
• Langue: English

Extrait

EXPONENTIALS FORM A BASIS OF DISCRETE
HOLOMORPHIC FUNCTIONS
by
Christian Mercat
Abstract. — Weshowthatdiscreteexponentialsformabasisofdiscreteholomorphic
functionsonacriticalmap. Onacombinatoriallyconvexset,thepolynomials
form a basis as well.
R´esum´e (Les exponentielles forment une base des fonctions holomorphes
discr`etes)
Nousmontronsquelesexponentiellesformentunebasedesfonctions
discr`etes sur une carte critique. Sur un convexe, les polynˆomes discrets forment
´egalement une base.
1. Introduction
The notion of discrete Riemann surfaces has been defined in [7]. A good basis for
the associated space of holomorphic functions was still missing. This article discuss
an interesting one in the simply connected critical case.
We are interested in a cellular decomposition } of the complex plane or a simply
connectedportionU ofit,byrhombi (equilateralquadrilaterals,orlozenges). Inother
words, we have a map from the set of vertices } to the complex plane Z :} !C0 0
0 0such that for each oriented face (x;y;x;y ) 2 } , its image is a positively oriented2
0 0rhombus (Z(x);Z(y);Z(x);Z(y )) of side length – > 0. It defines a straightforward
0Cauchy-Riemann equation for a function f 2C (}) of the vertices, and similarly for
2000 Mathematics Subject Classification. — 30G25,52C26,31C20,39A12.
Keywordsandphrases. — discreteholomorphicfunctions,discreteanalyticfunctions,monodriffic
functions, exponentials.
This research is supported by the Deutsche Forschungsgemeinschaft in the framework of Sonder-
forschungsbereich 288, “Differential Geometry and Quantum Physics”. I thank Richard Kenyon for
his comments on the draft of this article.2 CH. MERCAT
1-forms:
0 0f(y )¡f(y) f(x)¡f(x)
(1.1) = :
0 0Z(y )¡Z(y) Z(x)¡Z(x)
0y
0x x
–
y
Figure 1. The discrete Cauchy-Riemann equation takes place on each rhombus.
We call such a data a critical map of U. The relevance of this kind of maps in
the context of discrete holomorphy was first pointed out and put to use by Duffin [2].
0 0 0The rhombi can be split in four, yielding a finer critical map }; Z with – = –=2.
In [7, 6] we proved that a converging sequence of discrete holomorphic functions on
a refining sequence of critical maps converges to a continuous function
and any holomorphic function on U can be approximated by a converging sequence
of discrete functions. The proof was based on discrete polynomials and
series. In the present article we are going to show that the vector space spanned by
discrete polynomials is the same as the one spanned by discrete exponentials and the
main result is the following
Theorem 1. — On a combinatorially convex critical map, the discrete exponentials
form a basis of discrete holomorphic functions.
On a non combinatorially convex map we define some special exponentials which
supplement this basis.
The article is organized as follows. After recalling some basic features of discrete
Riemann surfaces at criticality in Sec. 2, we define discrete exponentials in Sec. 3 and
showsomeoftheirbasicproperties,relatedtopolynomialsandseries. Wegiveinpar-
ticularaformulafortheexpressionofagenericexponentialinabasisofexponentials.
In Sec. 4, we introduce the notion of convexity, related to a geometrical construction
called train-tracks, and we prove the main result. Finally we study the general caseDISCRETE EXPONENTIALS 3
in Sec. 5 where we define special exponentials and show they form a basis. The ap-
pendix lists some other interesting properties of the discrete exponentials which are
not needed in the proof.
We note that it is possible to use the wonderful machinery defined in [3, 4] to
prove that discrete exponentials form a basis of discrete holomorphic functions on a
critical compact. Indeed, Richard Kenyon gives an expression of the discrete Green’s
function (the discrete logarithm) as an integral over a loop in the space of discrete
exponentials:
I –log ‚1 2(1.2) G(O;x)=¡ Exp(:‚:x) d‚
28… i ‚C
where the integration contour C contains all the points in P (the possible poles}
of Exp(:‚:x)) but avoids the negative real line. It is real (negative) on half of the
vertices and imaginary on the others. Because of the logarithm, this imaginary part
is multivalued, it has a (discrete) logarithmic singularity at the origin: the Laplacian
is 1, and null elsewhere. On a compact, considering points on the boundary as
origins, thesefunctionscanbesinglevaluedandcanbeformedintoabasisofdiscrete
holomorphic functions. They clearly belong in the space of discrete exponentials.
The approach we will present here is much more pedestrian and simplistic.
2. Integration and Derivation at criticality
2.1. Integration. — GivenanisometriclocalmapZ :U\}!C,wheretheimage
of the quadrilaterals are lozenges inC, any holomorphic function f 2Ω(}) gives rise
to an holomorphic 1-form fdZ defined by the formula,
Z
f(x)+f(y)
(2.1) fdZ := (Z(y)¡Z(x));
2(x;y)
where (x;y)2} is an edge of a lozenge. It fulfills the Cauchy-Riemann equation for1
forms which is, in the same conditions as Eq. (1.1):
0 1
Z Z Z Z
1 B C
(2.2) @ + + + A fdZ
0Z(y )¡Z(y)
0 0 0 0(y;x) (x;y ) (y;x ) (x ;y )
0 1
Z Z Z Z
1 B C
= + + + fdZ:@ A0Z(x)¡Z(x)
0 0 0 0(x;y) (y;x ) (x;y ) (y ;x )
Once an origin O is chosen, it provides a way to integrate a function Int(f)(z):=
Rz
fdZ. We proved in [6] that the integrals of converging discrete holomorphic func-
O
k ktions (f ) on a refining sequence (} ) of critical maps of a compact converge tok k
k 2the integral of the limit. If the original limit was of order f(z) = f (z)+O(– ), itk4 CH. MERCAT
R Rz z k 2stays this way for the integrals, f(u)du = f dZ +O(– ), where the left handkO O
side is the usual continuous integral and the right hand side the discrete ones.
Following Duffin [1, 2], we define by inductive integration the analogues
k :k:of the integer power monomials z , that we denote Z :
:0:(2.3) Z := 1;
Z
:k: :k¡1:(2.4) Z := k Z dZ:
O
The discrete polynomials of degree less than three agree point-wise with their
:2: 2continuous counterpart, Z (x)=Z(x) so that by repeated integration, the discrete
polynomials in a refining sequence of a compact converge to the continuous ones and
2the limit is of order O(– ). We will see (Eq. (3.6)) that a closed formula can be
obtained for these monomials.
2.2. Derivation. — The combinatorial surface being simply connected and the
⁄graph } having only quadrilateral faces, it is bi-colorable. Let Γ and Γ the two
⁄sets of vertices and " be the biconstant "(Γ) = +1, "(Γ ) =¡1. For a holomorphic
function f, the equality fdZ·0 is equivalent to f =‚" for some ‚2C.
Following Duffin [1, 2], we introduce the
Definition 2.1. — For a holomorphic function f, define on a flat simply connected
y 0map U the holomorphic functions f , the dual of f, and f , the derivative of f, by
the following formulae:
y ¯(2.5) f (z):="(z)f(z);
¯where f denotes the complex conjugate, "=§1 is the biconstant, and
? ¶Z yz40 y(2.6) f (z):= f dZ +‚";
2– O
defined up to ".
We proved in [7] the following
0Proposition 2.2. — The derivative f fulfills
0(2.7) df =f dZ:
3. Exponential
3.1. Definition. —
Definition 3.1. — The discrete exponential Exp(:‚:Z) is the solution of
Exp(:‚:O) = 1
(3.1) d Exp(:‚:Z) = ‚ Exp(:‚:Z)dZ:DISCRETE EXPONENTIALS 5
We define its derivatives with respect to the continuous parameter ‚:
k@:k:(3.2) Z Exp(:‚:Z) := Exp(:‚:Z):
k@‚
The discrete exponential was first defined in [5] and put to a very interesting use
in [3, 4]. Forj‚j = 2=–, an immediate check shows that it is a rational fraction in ‚
P i?kat every point: For the vertex x= –e ,
‚– i?kY 1+ e2(3.3) Exp(:‚:x)=
‚– i?k1¡ e2k
i?kwhere(? )aretheanglesdefining(–e ),thesetof(Z-imagesof)}-edgesbetweenxk
and the origin. Because the map is critical, Eq. (3.3) only depends on the end points
(O;x). It is a generalization of a well known formula, in a slightly better version,
ˆ !n? ¶n ‚x2 2 3 3‚x ‚ x 1+ ‚ x2n(3.4) exp(‚x)= 1+ +O( )= +O( )
‚x 2n n n1¡ 2n
P Pn x i?kto the case when the path from the origin to the point x = = –e is not1 n
restricted to straight equal segments but to a general path of O(jxj=–) segments of
any directions.
:¡k:Theintegrationwithrespectto‚givesaninterestinganalogueofZ Exp(:‚:Z).
Itisdefineduptoagloballydefineddiscreteholomorphicmap. Onewaytofixitisto
integrate from a given ‚ of modulus 2=–, which is not a pole of the rational fraction,0
along a path that doesn’t cross the circle of radius 2=– again.
Proposition 3.2. — For point-wise multiplication, at ever