Residue Class-Wise Affine Mappings

This chapter gives the basic definitions, describes how to enter residue class-wise affine mappings and how to compute with them. The functionality of this package which is dedicated to residue class-wise affine groups is described in detail in the next chapter. The reader is encouraged to look there already after having read the first few pages of this chapter, and to look up definitions as he needs to.

Basic definitions
rcwa mapping
rcwa group

Residue class-wise affine groups, or for short rcwa groups, are permutation groups whose elements are bijective residue class-wise affine mappings.

A mapping f: \Z \rightarrow \Z is called residue class-wise affine, or for short an rcwa mapping, if there is a positive integer m such that the restrictions of f to the residue classes r(m) \in \Z/m\Z are all affine, i.e. given by f|_{r(m)}: \ r(m) \rightarrow \Z, \ \ \ n \ \mapsto \ \frac{a_{r(m)} \cdot n + b_{r(m)}}{c_{r(m)}} a_r(m) * n + b_r(m) f|_r(m): r(m) -> Z, n |-> ------------------- c_r(m) for certain coefficients a_{r(m)}, b_{r(m)}, c_{r(m)} \in \Z.

modulus
rcwa mapping
multiplier
rcwa mapping
divisor
rcwa mapping

The smallest possible m is called the modulus of f. It is understood that all fractions are reduced, i.e. that \gcd( a_{r(m)}, b_{r(m)}, c_{r(m)} ) = 1, and that c_{r(m)} > 0. The lcm of the coefficients a_{r(m)} is called the multiplier of f, and the lcm of the coefficients c_{r(m)} is called the divisor of f.

It is easy to see that the residue class-wise affine mappings form a monoid under composition, and that the residue class-wise affine permutations form a countable subgroup of Sym(\Z). We denote the latter group by RCWA(\Z).

tame
tame
wild
wild
rcwa mapping
rcwa group
rcwa mapping
rcwa group

An rcwa mapping is called tame if the set of moduli of its powers is bounded, or equivalently if it permutes a partition of \Z into finitely many residue classes on all of which it is affine. An rcwa group is called tame if there is a common such partition for all of its elements, or equivalently if the set of moduli of its elements is bounded. Rcwa mappings and -groups which are not tame are called wild. Tame rcwa mappings and -groups are something which one could call the trivial cases or basic building blocks, while wild rcwa groups are the objects of primary interest.

The definitions of residue class-wise affine mappings and -groups can be generalized in an obvious way to suitable rings other than \Z. In fact, this package provides at least rudimentary support for residue class-wise affine groups over semilocalizations of \Z and over univariate polynomial rings over finite fields.

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Entering residue class-wise affine mappings

Entering an rcwa mapping of \Z into &RCWA; requires specifying the modulus m and the coefficients a_{r(m)}, b_{r(m)} and c_{r(m)} for r(m) running over the residue classes (mod m).

This can be done easiest by RcwaMapping( coeffs ), where coeffs is a list of m coefficient triples coeffs[r+1] = [a_{r(m)}, b_{r(m)}, c_{r(m)}], with r running from 0 to m-1.

If some coefficient c_{r(m)} is zero or if images of some integers under the mapping to be defined would not be integers, an error message is printed and a break loop is entered. For example, the coefficient triple [1,4,3] is not allowed at the first position. The reason for this is that not all integers congruent to 1 \cdot 0 + 4 = 4 mod m are divisible by 3.

For the general constructor for rcwa mappings, see .

gap> T := RcwaMapping([[1,0,2],[3,1,2]]); # The Collatz mapping. <rcwa mapping of Z with modulus 2> gap> [ IsSurjective(T), IsInjective(T) ]; [ true, false ] gap> SetName(T,"T"); Display(T);

Surjective rcwa mapping of Z with modulus 2

n mod 2 | n^T ---------------------------------------+-------------------------------------- 0 | n/2 1 | (3n + 1)/2

gap> a := RcwaMapping([[2,0,3],[4,-1,3],[4,1,3]]); SetName(a,"a"); <rcwa mapping of Z with modulus 3> gap> IsBijective(a); true gap> Display(a); # This is Collatz' permutation:

Bijective rcwa mapping of Z with modulus 3

n mod 3 | n^a ---------------------------------------+-------------------------------------- 0 | 2n/3 1 | (4n - 1)/3 2 | (4n + 1)/3

gap> Support(a); Z \ [ -1, 0, 1 ] gap> Cycle(a,44); [ 44, 59, 79, 105, 70, 93, 62, 83, 111, 74, 99, 66 ]
\pagebreak[4]
\noindent
There is computational evidence for the conjecture that any residue class-wise affine permutation of \Z can be factored into members of the following three series of rcwa mappings of particularly simple structure (cp. ):

The class shift \nu_{r(m)}.
The class shift \nu_{r(m)} is the rcwa mapping of \Z which maps n \in r(m) to n + m and which fixes \Z \setminus r(m) pointwise. Enclosing the argument list in list brackets is permitted. gap> Display(ClassShift(5,12)); Tame bijective rcwa mapping of Z with modulus 12, of order infinity n mod 12 | n^ClassShift(5,12) ---------------------------------------+-------------------------------------- 0 1 2 3 4 6 7 8 9 10 11 | n 5 | n + 12
The class reflection \varsigma_{r(m)}.
The class reflection \varsigma_{r(m)} is the rcwa mapping of \Z which maps n \in r(m) to -n + 2r and which fixes \Z \setminus r(m) pointwise. Enclosing the argument list in list brackets is permitted. gap> Display(ClassReflection(5,9)); Bijective rcwa mapping of Z with modulus 9, of order 2 n mod 9 | n^ClassReflection(5,9) ---------------------------------------+-------------------------------------- 0 1 2 3 4 6 7 8 | n 5 | -n + 10
\pagebreak[4]
The class transposition \tau_{r_1(m_1),r_2(m_2)}.
Given two disjoint residue classes r_1(m_1) and r_2(m_2) of the integers, the class transposition \tau_{r_1(m_1),r_2(m_2)} \in RCWA(\Z) is defined by \tau_{r_1(m_1),r_2(m_2)}: \ \ \ n \ \longmapsto \ \begin{cases} (m_2 n + m_1 r_2 - m_2 r_1)/m_1 & \text{if} \ n \in r_1(m_1), \\ (m_1 n + m_2 r_1 - m_1 r_2)/m_2 & \text{if} \ n \in r_2(m_2), \\ n & \text{otherwise}, \end{cases} / | (m_2 n + m_1 r_2 - m_2 r_1)/m_1 if n in r_1(m_1), n |-> < (m_1 n + m_2 r_1 - m_1 r_2)/m_2 if n in r_2(m_2), | otherwise, \ where it is understood that 0 \leq r_1 < m_1 and that 0 \leq r_2 < m_2. The class transposition \tau_{r_1(m_1),r_2(m_2)} is an involution which interchanges the residue classes r_1(m_1) and r_2(m_2) and which fixes the complement of their union pointwise.

Enclosing the argument list in list brackets is permitted. The residue classes r_1(m_1) and r_2(m_2) are stored as an attribute TransposedClasses. gap> Display(ClassTransposition(1,2,8,10)); Bijective rcwa mapping of Z with modulus 10, of order 2 n mod 10 | n^ClassTransposition(1,2,8,10) ---------------------------------------+-------------------------------------- 0 2 4 6 | n 1 3 5 7 9 | 5n + 3 8 | (n - 3)/5
\noindent
The set of all class transpositions generates the simple group mentioned in the preface.

The permutations of the following kind play an important role in factoring bijective rcwa mappings into class shifts, class reflections and class transpositions (cp. ):

In the one-argument form the prime switch \sigma_p := \tau_{0(8),1(2p)} \cdot \tau_{4(8),-1(2p)} \cdot \tau_{0(4),1(2p)} \cdot \tau_{2(4),-1(2p)} \cdot \tau_{2(2p),1(4p)} \cdot \tau_{4(2p),2p+1(4p)}, and in the two-argument form the restriction of \sigma_p by n \mapsto kn.
For an odd prime p, the prime switch \sigma_p is a bijective rcwa mapping of \Z with modulus 4p, multiplier p and divisor 2. gap> Factorization(PrimeSwitch(3)); [ ClassTransposition(1,6,0,8), ClassTransposition(5,6,4,8), ClassTransposition(0,4,1,6), ClassTransposition(2,4,5,6), ClassTransposition(2,6,1,12), ClassTransposition(4,6,7,12) ]

There are properties IsClassShift, IsClassReflection, IsClassTransposition and IsPrimeSwitch which indicate whether a given rcwa mapping belongs to the corresponding series.

In the sequel, a description of the general-purpose constructor for rcwa mappings is given. This might look a bit technical on a first glance, but knowing all possible ways of entering an rcwa mapping is by no means necessary for understanding this manual or for using this package.

An rcwa mapping.
In all cases the argument R is the underlying ring, m is the modulus and coeffs is the coefficient list. A coefficient list for an rcwa mapping with modulus m consists of |R/mR| coefficient triples [a_{r(m)}, b_{r(m)}, c_{r(m)}]. Their ordering is determined by the ordering of the representatives of the residue classes (mod m) in the sorted list returned by AllResidues(R, m). In case R = \mathbb{Z} this means that the coefficient triple for the residue class 0(m) comes first and is followed by the one for 1(m), the one for 2(m) and so on. In case one or several of the arguments R, m and coeffs are omitted or replaced by other arguments, the former are either derived from the latter or default values are taken. The meaning of the other arguments is defined in the detailed description of the particular methods given in the sequel.

The above methods return the rcwa mapping (a) of R with modulus modulus and coefficients coeffs, resp. (b) of R = \mathbb{Z} or R = \mathbb{Z}_{(\pi)} with modulus Length(coeffs) and coefficients coeffs, resp. (c) of R = \mathbb{Z} with modulus Length(coeffs) and coefficients coeffs, resp. (d) of R = \mathbb{Z}, acting on any set range+k*Length(range) like the permutation perm on range, resp. (e) of R = \mathbb{Z} with modulus modulus and values prescribed by the list val, which consists of 2\cdot modulus pairs giving preimage and image for 2 points per residue class (mod modulus), resp. (f) of R = \Z_{(\pi)} with modulus Length(coeffs) and coefficients coeffs (the set of primes \pi denoting the underlying ring is passed as argument pi), resp. (g) of R = GF(q)[x] with modulus modulus and coefficients coeffs, resp. (h) a bijective rcwa mapping which induces a bijection between the partitions P1 and P2 of R into residue classes and which is affine on the elements of P1, resp. (i) a bijective rcwa mapping with residue class cycles as given by cycles. The latter is a list of lists of pairwise disjoint residue classes which the mapping should permute cyclically, each. The methods for the operation RcwaMapping perform a number of argument checks, which can be skipped by using RcwaMappingNC instead.

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gap> x := Indeterminate(GF(2),1);; SetName(x,"x"); gap> R := PolynomialRing(GF(2),1); GF(2)[x] gap> RcwaMapping(R,x+1,[[1,0,x+One(R)],[x+One(R),0,1]]*One(R)); # (a) <rcwa mapping of GF(2)[x] with modulus x+Z(2)^0> gap> RcwaMapping(Z_pi(2),[[1/3,0,1]]); # (b) Rcwa mapping of Z_( 2 ): n -> 1/3 n gap> a := RcwaMapping([[2,0,3],[4,-1,3],[4,1,3]]); # (c) <rcwa mapping of Z with modulus 3> gap> RcwaMapping((1,2,3),[1..4]); # (d) <bijective rcwa mapping of Z with modulus 4, of order 3> gap> T = RcwaMapping(2,[[1,2],[2,1],[3,5],[4,2]]); # (e) true gap> RcwaMapping([2],[[1/3,0,1]]); # (f) Rcwa mapping of Z_( 2 ): n -> 1/3 n gap> RcwaMapping(2,x+1,[[1,0,x+One(R)],[x+One(R),0,1]]*One(R)); # (g) <rcwa mapping of GF(2)[x] with modulus x+Z(2)^0> gap> a = RcwaMapping(List([[0,3],[1,3],[2,3]],ResidueClass), > List([[0,2],[1,4],[3,4]],ResidueClass)); # (h) true gap> RcwaMapping([List([[0,2],[1,4],[3,8],[7,16]],ResidueClass)]); # (i) <bijective rcwa mapping of Z with modulus 16, of order 4> gap> Cycle(last,ResidueClass(0,2)); [ 0(2), 1(4), 3(8), 7(16) ]
View
Display
Print
String
\noindent
In most cases the output of the View method does not describe an rcwa mapping completely. In these cases the output is enclosed in brackets. There are methods installed for Display, Print and String. The Printed representation of an rcwa mapping is &GAP; - readable if and only if the Printed representation of the elements of the underlying ring is so. There is also a method for LaTeX:

A &LaTeX; representation of the rcwa mapping f.
The output makes use of the &LaTeX; macro package `amsmath`. If the option Factorization is set, a factorization of f into class shifts, class reflections, class transpositions and prime switches is printed (cp. ). For rcwa mappings with modulus greater than 1, an indentation by Indentation characters can be specified by setting this option value accordingly. gap> Print(LaTeXObj(T)); n \ \longmapsto \ \begin{cases} \frac{n}{2} & \text{if} \ n \in 0(2), \\ \frac{3n + 1}{2} & \text{if} \ n \in 1(2). \end{cases} gap> Print(LaTeXObj(Comm(a,ClassShift(0,4)):Factorization)); &\nu_{4(8)} \cdot \nu_{0(8)}^{-1} \cdot \tau_{0(8),4(8)} \cdot \tau_{5(16),11(16)} \cdot \tau_{4(8),11(16)}
LaTeXAndXDVI
There is an operation LaTeXAndXDVI which displays an rcwa mapping in an `xdvi` window. This works as follows: The string returned by the LaTeXObj - method described above is inserted into a &LaTeX; template file. This file is &LaTeX;'ed, and the result is shown with `xdvi`. Calling Display with option xdvi has the same effect. The operation LaTeXAndXDVI is only available on UNIX systems, and requires suitable installations of &LaTeX; and `xdvi`.

Basic functionality for rcwa mappings
rcwa mapping
Order
IsTame

Checking whether two rcwa mappings are equal is cheap. Rcwa mappings can be multiplied, thus there is a method for *. Bijective rcwa mappings can also be inverted, thus there is a method for Inverse. The latter method is usually accessed by raising a mapping to some power with negative exponent. Multiplying, inverting and computing powers of tame rcwa mappings is cheap. Computing powers of wild mappings is usually expensive -- runtime and memory requirements normally grow approximately exponentially with the exponent. How expensive multiplying a couple of wild mappings is, varies very much. In any case, the amount of memory required for storing an rcwa mapping is proportional to its modulus. Whether a given mapping is tame or wild can be determined by the operation IsTame. There are methods for Order, which can not only compute a finite order, but can also detect infinite order.

gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; # The Collatz mapping. gap> a := RcwaMapping([[2,0,3],[4,-1,3],[4,1,3]]);; # Collatz' permutation. gap> List([-4..4],k->Modulus(a^k)); [ 256, 64, 16, 4, 1, 3, 9, 27, 81 ] gap> IsTame(T) or IsTame(a); false gap> IsTame(ClassShift(0,1)) and IsTame(ClassTransposition(0,2,1,2)); true gap> T^2*a*T*a^-3; <rcwa mapping of Z with modulus 768> gap> (ClassShift(1,3)*ClassReflection(2,7))^1000000; <bijective rcwa mapping of Z with modulus 21>
IsInjective
IsSurjective
IsBijective
Image

There are methods installed for IsInjective, IsSurjective, IsBijective and Image.

gap> [ IsInjective(T), IsSurjective(T), IsBijective(a) ]; [ false, true, true ] gap> Image(RcwaMapping([[2,0,1]])); 0(2)
rcwa mapping

Images of elements, of finite sets of elements and of unions of finitely many residue classes of the source of an rcwa mapping can be computed with &circum;, the same symbol as used for exponentiation and conjugation. The same works for partitions of the source into a finite number of residue classes.

gap> 15^T; 23 gap> ResidueClass(1,2)^T; 2(3) gap> List([[0,3],[1,3],[2,3]],ResidueClass)^a; [ 0(2), 1(4), 3(4) ]
PreImageElm
PreImagesElm
PreImage

For computing preimages of elements under rcwa mappings, there are methods for PreImageElm and PreImagesElm. The preimage of a finite set of ring elements or of a union of finitely many residue classes under an rcwa mapping can be computed using PreImage.

gap> PreImagesElm(T,8); [ 5, 16 ] gap> PreImage(T,ResidueClass(Integers,3,2)); Z \ 0(6) U 2(6) gap> M := [1];; l := [1];; gap> while Length(M) < 10000 do M := PreImage(T,M); Add(l,Length(M)); od; l; [ 1, 1, 2, 2, 4, 5, 8, 10, 14, 18, 26, 36, 50, 67, 89, 117, 157, 208, 277, 367, 488, 649, 869, 1154, 1534, 2039, 2721, 3629, 4843, 6458, 8608, 11472 ]
Support
MovedPoints
RestrictedPerm

There is a method for the operation Support for computing the support of a bijective rcwa mapping. A synonym for Support is MovedPoints. There is also a method for RestrictedPerm for computing the restriction of a bijective rcwa mapping to a union of residue classes which it fixes setwisely.

gap> List([a,a^2],Support); [ Z \ [ -1, 0, 1 ], Z \ [ -3, -2, -1, 0, 1, 2, 3 ] ] gap> RestrictedPerm(ClassShift(0,2)*ClassReflection(1,2),ResidueClass(0,2)); <rcwa mapping of Z with modulus 2> gap> last = ClassShift(0,2); true

Rcwa mappings can be added and subtracted pointwise. However, please note that the set of rcwa mappings of a ring does not form a ring under + and *.

gap> b := ClassShift(0,3) * a;; gap> [ Image((a + b)), Image((a - b)) ]; [ 2(4), [ -2, 0 ] ]
Modulus
Mod
Coefficients

There are operations Modulus (abbreviated Mod) and Coefficients for extracting the modulus resp. the coefficient list of a given rcwa mapping. The meaning of the return values is as described in the previous section. General documentation for most operations mentioned in this section can be found in the &GAP; reference manual. For rcwa mappings of rings other than \mathbb{Z}, not for all operations applicable methods are available.

Factoring rcwa mappings

Factoring group elements into elements of some nice set of generators is often helpful. The following can be seen as an attempt towards getting a satisfactory solution of this problem for the group RCWA(\Z):

A factorization of the bijective rcwa mapping g into class shifts, class reflections and class transpositions, provided that such a factorization exists and the method finds it.
The method may return fail, stop with an error message or run into an infinite loop. If it returns a result, this result is always correct.

By default, prime switches are taken as one factor. If the option ExpandPrimeSwitches is set, they are each decomposed into the 6 class transpositions given in the definition (see ). By default, the factoring process begins with splitting off factors from the right. This can be changed by setting the option Direction to "from the left". By default, a reasonably coarse respected partition of the integral mapping occuring in the final stage of the algorithm is computed. This can be suppressed by setting the option ShortenPartition equal to false. By default, at the end it is checked whether the product of the determined factors indeed equals g. This check can be suppressed by setting the option NC.

The problem of obtaining a factorization as described is algorithmically difficult, and this factorization routine is currently perhaps the most sophisticated part of the &RCWA; package. Information about the progress of the factorization process can be obtained by setting the info level of the Info class to 2. gap> Factorization(Comm(ClassShift(0,3)*ClassReflection(1,2),ClassShift(0,2))); [ ClassReflection(2,3), ClassShift(2,6)^-1, ClassTransposition(0,6,2,6), ClassTransposition(0,6,5,6) ]

For purposes of demonstrating the capabilities of the factorization routine, in Section Collatz' permutation is factored. Lothar Collatz has investigated this permutation in 1932. Its cycle structure is unknown so far.

Class transpositions can be written as products of any given number k of class transpositions, as long as the underlying ring has a residue class ring of cardinality k. Such a decomposition can be obtained by SplittedClassTransposition(ct,k).

SplittedClassTransposition

Obtaining a factorization of a bijective rcwa mapping into class shifts, class reflections and class transpositions is particularly difficult if multiplier and divisor are coprime. A prototype of permutations which have this property has been introduced in a different context in :

The permutation g_m as introduced in .
The argument m must be an odd integer \geq 3. gap> Display(mKnot(5)); Wild bijective rcwa mapping of Z with modulus 5 n mod 5 | n^mKnot(5) ---------------------------------------+-------------------------------------- 0 | 6n/5 1 | (4n + 1)/5 2 | (6n - 2)/5 3 | (4n + 3)/5 4 | (6n - 4)/5
\noindent
In his article, Timothy P. Keller shows that a permutation of this type cannot have infinitely many cycles of any given finite length.

Determinant and sign
The determinant of the bijective rcwa mapping g.
The determinant of an affine mapping n \mapsto (an+b)/c whose source is a residue class r(m) is defined by b/|a|m. This definition is extended additively to determinants of rcwa mappings and their restrictions to unions of residue classes.

Using the notation from the definition of an rcwa mapping, the determinant det(\sigma) of an rcwa mapping \sigma is given by (see LaTeX version of the manual). \frac{1}{m} \left( \sum_{r(m) \in R/mR} \frac{b_{r(m)}}{|a_{r(m)}|} \right). The determinant mapping is an epimorphism from the group of all class-wise order-preserving bijective rcwa mappings of \Z to (\Z,+), see , Theorem 2.11.9.

If a residue class union S is given as an additional argument, the method returns the determinant of the restriction of g to S. gap> List([ClassTransposition(0,4,5,12),ClassShift(3,7)],Determinant); [ 0, 1 ] gap> Determinant(ClassTransposition(0,4,5,12)*ClassShift(3,7)^100); 100
The sign of the bijective rcwa mapping g.
Using the notation from the definition of an rcwa mapping, the sign of a bijective rcwa mapping \sigma of \Z is defined by (see LaTeX version of the manual). (-1)^{\displaystyle{{\rm det}(\sigma) + \frac{1}{m} \left( \sum_{r(m): \ a_{r(m)} < 0} (m - 2r) \right)}}. The sign mapping is an epimorphism from RCWA(\Z) to the group \Z^\times of units of \Z, see , Theorem 2.12.8. Therefore the kernel of the sign mapping is a normal subgroup of RCWA(\Z) of index 2. gap> List([ClassTransposition(3,4,2,6),ClassShift(0,3),ClassReflection(2,5)],Sign); [ 1, -1, -1 ] gap> Sign (ClassTransposition(3,4,2,6)*ClassShift(0,3)*ClassReflection(2,5)); 1
Attributes and properties derived from the coefficients
Multiplier
Mult
Divisor
Div
PrimeSet
integral
rcwa mapping
IsIntegral
balanced
rcwa mapping
IsBalanced
rcwa mapping
IsClassWiseOrderPreserving

A number of basic attributes and properties of an rcwa mapping are derived immediately from the coefficients of its affine partial mappings. This holds for example for the multiplier and the divisor. These two values are stored as attributes Multiplier and Divisor, or in short Mult resp. Div. The prime set of an rcwa mapping is the set of prime divisors of the product of its modulus and its multiplier. It is stored as an attribute PrimeSet. An rcwa mapping is called integral if its divisor equals 1. An rcwa mapping is called balanced if its multiplier and divisor have the same prime divisors. An integral mapping has the property IsIntegral and a balanced mapping has the property IsBalanced. An rcwa mapping of the ring of integers or of one of its semilocalizations is called class-wise order-preserving if and only if all coefficients a_{r(m)} in the numerators of the affine partial mappings are positive. The corresponding property is IsClassWiseOrderPreserving.

\pagebreak[4]
gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);; gap> IsBijective(u);; Display(u);

Bijective rcwa mapping of Z with modulus 5

n mod 5 | n^f ---------------------------------------+-------------------------------------- 0 | 3n/5 1 | (9n + 1)/5 2 | (3n - 1)/5 3 | (9n - 2)/5 4 | (9n + 4)/5

gap> Multiplier(u); 9 gap> Divisor(u); 5 gap> PrimeSet(u); [ 3, 5 ] gap> IsIntegral(u) or IsBalanced(u); false gap> IsClassWiseOrderPreserving(u); true
Functionality related to the affine partial mappings
The coarsest partition of Source(f) on whose elements the rcwa mapping f is affine.
gap> LargestSourcesOfAffineMappings(ClassShift(3,7)); [ Z \ 3(7), 3(7) ] gap> LargestSourcesOfAffineMappings(ClassReflection(0,1)); [ Integers ] gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);; gap> List( [ u, u^-1 ], LargestSourcesOfAffineMappings ); [ [ 0(5), 1(5), 2(5), 3(5), 4(5) ], [ 0(3), 1(3), 2(9), 5(9), 8(9) ] ] gap> kappa := ClassTransposition(2,4,3,4) * ClassTransposition(4,6,8,12) > * ClassTransposition(3,4,4,6);; SetName(kappa,"kappa"); gap> LargestSourcesOfAffineMappings(kappa); [ 2(4), 1(4) U 0(12), 3(12) U 7(12), 4(12), 8(12), 11(12) ]
The union of the residue classes r(m) such that p^k||a_{r(m)} if k \geq 0, and the union of the residue classes r(m) such that p^k||c_{r(m)} if k \leq 0. In this context, m denotes the modulus and a_{r(m)} and c_{r(m)} denote the coefficients of f as introduced in the definition of an rcwa mapping.
gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; # The Collatz mapping. gap> [ Multpk(T,2,-1), Multpk(T,3,1) ]; [ Integers, 1(2) ] gap> u := RcwaMapping([[3,0,5],[9,1,5],[3,-1,5],[9,-2,5],[9,4,5]]);; gap> [ Multpk(u,3,0), Multpk(u,3,1), Multpk(u,3,2), Multpk(u,5,-1) ]; [ [ ], 0(5) U 2(5), Z \ 0(5) U 2(5), Integers ]
A list of the sets of fixed points of the affine partial mappings of the rcwa mapping f in the quotient field of its source.
The returned list contains entries for the restrictions of f to all residue classes modulo Mod(f). A list entry can either be an empty set, the source of f or a set of cardinality 1. The ordering of the entries corresponds to the ordering of the residues in AllResidues(Source(f),m). gap> FixedPointsOfAffinePartialMappings(ClassShift(0,2)); [ [ ], Rationals ] gap> List([1..3],k->FixedPointsOfAffinePartialMappings(T^k)); [ [ [ 0 ], [ -1 ] ], [ [ 0 ], [ 1 ], [ 2 ], [ -1 ] ], [ [ 0 ], [ -7 ], [ 2/5 ], [ -5 ], [ 4/5 ], [ 1/5 ], [ -10 ], [ -1 ] ] ]
SetOnWhichMappingIsClassWise
SetOnWhichMappingIsClassWise
SetOnWhichMappingIsClassWise

There are attributes SetOnWhichMappingIsClassWise (-OrderPreserving, -Constant, -OrderReversing) which store the union of the residue classes (mod Mod(f)) on which an rcwa mapping f of \Z or of a semilocalization thereof is class-wise order-preserving, class-wise constant resp. class-wise order-reversing.

gap> List([ClassTransposition(1,2,0,4),ClassShift(2,3),ClassReflection(2,5)], > SetOnWhichMappingIsClassWiseOrderPreserving ); [ Integers, Integers, Z \ 2(5) ] gap> SetOnWhichMappingIsClassWiseConstant(RcwaMapping([[2,0,1],[0,4,1]])); 1(2) gap> SetOnWhichMappingIsClassWiseOrderReversing(ClassReflection(13,17)); 13(17)
Transition graphs and transition matrices
The transition graph of the rcwa mapping f for modulus m.
rcwa mapping
\noindent
The transition graph \Gamma_{f,m} of f for modulus m is defined as follows: The vertices are the residue classes (mod m). There is an edge from r_1(m) to r_2(m) if and only if there is some n \in r_1(m) such that n^f \in r_2(m). The assignment of the residue classes (mod m) to the vertices of the graph corresponds to the ordering of the residues in AllResidues(Source(f),m). The result is returned in the format used by the package `GRAPE`. gap> TransitionGraph(ClassShift(0,1),6); rec( isGraph := true, order := 6, group := Group(()), schreierVector := [ -1, -2, -3, -4, -5, -6 ], adjacencies := [ [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], [ 1 ] ], representatives := [ 1, 2, 3, 4, 5, 6 ], names := [ 1, 2, 3, 4, 5, 6 ] )
The partition of AllResidues(Source(f),m) corresponding to the weakly connected components of the transition graph of the rcwa mapping f for modulus m.
gap> OrbitsModulo(ClassTransposition(0,2,1,4),8); [ [ 0, 1, 4 ], [ 2, 5, 6 ], [ 3 ], [ 7 ] ]
The set of restrictions of the rcwa mapping f to the weakly connected components of its transition graph \Gamma_{f,m}.
The product of the returned mappings is f. They have pairwise disjoint supports, hence any two of them commute. gap> sigma := ClassTransposition(1,4,2,4) * ClassTransposition(1,4,3,4) > * ClassTransposition(3,9,6,18) * ClassTransposition(1,6,3,9);; gap> List(FactorizationOnConnectedComponents(sigma,36),Support); [ 33(36) U 34(36) U 35(36), 9(36) U 10(36) U 11(36), <union of 23 residue classes (mod 36)> \ [ -6, 3 ] ]
The transition matrix of the rcwa mapping f for modulus m.
Let M be this matrix. Then for any two residue classes r_1(m), r_2(m) \in R/mR, the entry M_{r_1(m),r_2(m)} is defined by M_{r_1(m),r_2(m)} \ := \ \displaystyle{\frac{|R/mR|}{|R/\hat{m}R|}} \cdot \left|\left\{r(\hat{m}) \in R/\hat{m}R | \ r \in r_1(m) \wedge r^f \in r_2(m)\right\}\right|, where \hat{m} is the product of m and the square of the modulus of f. (See LaTeX version of the manual.) The assignment of the residue classes (mod m) to the rows and columns of the matrix corresponds to the ordering of the residues in AllResidues(Source(f),m).

The transition matrix is a weighted adjacency matrix of the corresponding transition graph TransitionGraph(f,m). The sums of the rows of a transition matrix are always equal to 1. gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; # The Collatz mapping. gap> Display(TransitionMatrix(T^3,3)); [ [ 1/8, 1/4, 5/8 ], [ 0, 1/4, 3/4 ], [ 0, 3/8, 5/8 ] ]
A list of unions of residue classes modulo the modulus m of the rcwa mapping f, as described below.
The returned list contains an entry for any strongly connected component of the transition graph of f for modulus Mod(f) which has only outgoing edges. The list entry corresponding to such a component is the union of the vertices belonging to it. gap> Sources(ClassTransposition(0,2,1,2)*ClassTransposition(0,2,1,4)); [ 0(4) ]
(See , with outgoing replaced by ingoing.)
gap> Sinks(ClassTransposition(0,2,1,2)*ClassTransposition(0,2,1,4)); [ 1(4) ]
The list of non-isolated vertices of the transition graph of the rcwa mapping f for modulus Mod(f) which carry a loop.
gap> Loops(ClassTransposition(0,2,1,2)*ClassTransposition(0,2,1,4)); [ 0(4), 1(4) ]
Trajectories
The first length iterates in the trajectory of the rcwa mapping f starting at n, resp. the initial part of the trajectory of the rcwa mapping f starting at n which ends at the first occurence of an iterate in the set terminal. If the argument m is given, the iterates are reduced (mod m).
To save memory when computing long trajectories containing huge iterates, the reduction (mod m) is done each time before storing an iterate. In place of the ring element n, the methods also accept a finite set of ring elements or a union of residue classes. gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; # The Collatz mapping. gap> Trajectory(T,27,16); Trajectory(T,27,25,5); [ 27, 41, 62, 31, 47, 71, 107, 161, 242, 121, 182, 91, 137, 206, 103, 155 ] [ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 0, 3, 0, 0, 3, 0, 3, 0, 0, 3 ] gap> Trajectory(T,15,[1]); Trajectory(T,15,[1],2); [ 15, 23, 35, 53, 80, 40, 20, 10, 5, 8, 4, 2, 1 ] [ 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1 ] gap> Trajectory(T,ResidueClass(Integers,3,0),Integers); [ 0(3), 0(3) U 5(9), 0(3) U 5(9) U 7(9) U 8(27), <union of 20 residue classes (mod 27)>, <union of 73 residue classes (mod 81)>, Z \ 10(81) U 37(81), Integers ]
Either the list c of triples of coprime coefficients such that for any k it holds that n&circum;(f&circum;(k-1)) = (c[k][1]*n + c[k][2])/c[k][3] or the last entry of that list, depending on whether whichcoeffs is "AllCoeffs" or "LastCoeffs".
The meanings of the arguments length and terminal are the same as in the methods for the operation Trajectory described above. In general, computing only the last coefficient triple (whichcoeffs = "LastCoeffs") needs considerably less memory than computing the entire list. gap> Trajectory(T,27,[1],"LastCoeffs"); [ 36472996377170786403, 195820718533800070543, 1180591620717411303424 ] gap> (last[1]*27+last[2])/last[3]; 1
The union of all residue classes r(m) such that |R/a_{r(m)}R| > |R/c_{r(m)}R| resp. |R/a_{r(m)}R| < |R/c_{r(m)}R|, where R denotes the source, m the modulus and a_{r(m)}, b_{r(m)} and c_{r(m)} the coefficients of f as introduced in the definition of an rcwa mapping.
gap> List([1..3],k->IncreasingOn(T^k)); [ 1(2), 3(4), 3(4) U 1(8) U 6(8) ] gap> List([1..3],k->DecreasingOn(T^k)); [ 0(2), Z \ 3(4), 0(4) U 2(8) U 5(8) ] gap> a := RcwaMapping([[2,0,3],[4,-1,3],[4,1,3]]);; # Collatz' permutation. gap> List([-2..2],k->IncreasingOn(a^k)); [ Z \ 1(8) U 7(8), 0(2), [ ], Z \ 0(3), 1(9) U 4(9) U 5(9) U 8(9) ]
The invariant introduced in . This is (\sum_{i=1}^l a_i \cdot a_{i \mod l + 1})/(\sum_{i=1}^l a_i^2), where l denotes the length of a.
The argument a must be a list of integers. In it is shown that if a is a trajectory of the `original' Collatz mapping n \mapsto (n/2 if n even, 3n+1 if n odd) starting at an odd integer \geq 3 and ending at 1, then the invariant lies in the interval ]9/13,5/7[. gap> C := RcwaMapping([[1,0,2],[3,1,1]]);; gap> List([3,5..49],n->Float(GluckTaylorInvariant(Trajectory(C,n,[1])))); [ 0.701053, 0.696721, 0.708528, 0.707684, 0.706635, 0.695636, 0.711769, 0.699714, 0.707409, 0.693833, 0.710432, 0.706294, 0.714242, 0.699935, 0.714242, 0.705383, 0.706591, 0.698198, 0.712222, 0.714242, 0.709048, 0.69612, 0.714241, 0.701076 ]
Localizations of rcwa mappings of the integers
The rcwa mapping of \Z_{(p)} resp. \Z_{(\pi)} with the same coefficients as the rcwa mapping f of \Z.
The argument p resp. pi must be a prime resp. a set of primes, and the argument f must be an rcwa mapping of \Z whose modulus is a power of p, resp. whose modulus has only prime divisors which lie in pi. gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; # The Collatz mapping. gap> Cycle(LocalizedRcwaMapping(T,2),131/13); [ 131/13, 203/13, 311/13, 473/13, 716/13, 358/13, 179/13, 275/13, 419/13, 635/13, 959/13, 1445/13, 2174/13, 1087/13, 1637/13, 2462/13, 1231/13, 1853/13, 2786/13, 1393/13, 2096/13, 1048/13, 524/13, 262/13 ]
Extracting roots of rcwa mappings
An rcwa mapping g such that g&circum;k=f, provided that such a mapping exists and that there is a method available which can determine it.
gap> Root(ClassTransposition(0,2,1,2),100); <bijective rcwa mapping of Z with modulus 8> gap> last^100 = ClassTransposition(0,2,1,2); true
Special functions for non-bijective mappings
A right inverse of the injective rcwa mapping f, i.e. a mapping g such that f g = 1.
gap> twice := RcwaMapping([[2,0,1]]); Rcwa mapping of Z: n -> 2n gap> twice * RightInverse(twice); IdentityMapping( Integers )
A mapping d such that l d = r d = 1.
The mappings l and r must be injective, and their images must form a partition of their source. gap> twice := RcwaMapping([[2,0,1]]); twiceplus1 := RcwaMapping([[2,1,1]]); Rcwa mapping of Z: n -> 2n Rcwa mapping of Z: n -> 2n + 1 gap> Display(CommonRightInverse(twice,twiceplus1)); Rcwa mapping of Z with modulus 2 n mod 2 | n^f ---------------------------------------+-------------------------------------- 0 | n/2 1 | (n - 1)/2
The image density of the rcwa mapping f.
In the notation introduced in the definition of an rcwa mapping, the image density of an rcwa mapping f is defined by 1/m\frac{1}{m} \sum_{r(m) \in R/mR} |R/c_{r(m)}R|/|R/a_{r(m)}R|. The image density of an injective rcwa mapping is \leq 1, and the image density of a surjective rcwa mapping is \geq 1 (this can be seen easily). Thus in particular the image density of a bijective rcwa mapping is 1. gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; # The Collatz mapping. gap> List( [ T, ClassShift(0,1), RcwaMapping([[2,0,1]]) ], ImageDensity ); [ 4/3, 1, 1/2 ]
InjectiveAsMappingFrom

Given an rcwa mapping f, the function InjectiveAsMappingFrom returns a set S such that the restriction of f to S is injective, and such that the image of S under f is the entire image of f.

gap> InjectiveAsMappingFrom(T); 0(2)
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Probabilistic guesses on the behaviour of trajectories

This section describes some functionality for getting educated guesses on the overall behaviour of the trajectories of a given rcwa mapping. Its contents have deliberately been separated from the documentation of the non-probabilistic functionality related to trajectories of rcwa mappings.

A list of ring elements (see below).
This operation tries to compute the contraction centre of the rcwa mapping f. Assuming its existence this is the unique finite subset S_0 of the source of f on which f induces a permutation and which intersects nontrivially with any trajectory of f. The mapping f is assumed to be contracting, i.e. to have such a contraction centre. As in general contraction centres are likely not computable, the methods for this operation are probabilistic and may return wrong results. The argument maxn is a bound on the starting value and bound is a bound on the elements of the trajectories to be searched. If the limit bound is exceeded, an Info message on Info level 3 of InfoRCWA is given. gap> T := RcwaMapping([[1,0,2],[3,1,2]]);; # The Collatz mapping. gap> S0 := LikelyContractionCentre(T,100,1000); #I Warning: `LikelyContractionCentre' is highly probabilistic. The returned result can only be regarded as a rough guess. See ?LikelyContractionCentre for information on how to improve this guess. [ -136, -91, -82, -68, -61, -55, -41, -37, -34, -25, -17, -10, -7, -5, -1, 0, 1, 2 ]
A floating point value which is intended to be a rough guess on how fast the trajectories of the rcwa mapping f diverge (return value greater than 1) or converge (return value smaller than 1).
Nothing particular is guaranteed. gap> GuessedDivergence(T); #I Warning: GuessedDivergence: no particular return value is guaranteed. 0.866025
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The categories and families of rcwa mappings
true
if f is an rcwa mapping resp. an rcwa mapping of the ring of integers resp. an rcwa mapping of a semilocalization of the ring of integers resp. an rcwa mapping of a polynomial ring in one variable over a finite field, and false otherwise.
Often the same methods can be used for rcwa mappings of the ring of integers and of its semilocalizations. For this reason there is a category IsRcwaMappingOfZOrZ&uscore;pi which is the union of IsRcwaMappingOfZ and IsRcwaMappingOfZ&uscore;pi. IsRcwaMappingOfZOrZ&uscore;pi The internal representation of rcwa mappings is called IsRcwaMappingStandardRep. IsRcwaMappingStandardRep
The family of rcwa mappings of the ring R.