# symmetries of period-doubling maps[精选推荐pdf]

Symmetries of Period-Doubling MapsLinas Vepstas 12 October 2004 (revised 1 September 2006)AbstractThe concept of self-similarity is central to the notion of a fractal, but the actual symmetry group that generates that self-similarity is rarely named, and it seems fractals are almost never studied according to their symmetries.Yet, in other branches of mathematics and physics, it is well understood that symmetry pro- vides a powerful mechanism for understanding systems. Inthispaper, weidentifythesymmetrygroupofperiod-doublingmapsasbeing a monoid (semigroup) of the modular group PSL(2,Z). To anchor this assertion, we work out an explicit, exactly-solvable fractal curve, the Takagi or Blancmange Curve, as transforming under the three-dimensional representation of the (monoid of the) modular group. By replacing the triangular shape that generates the Blanc-mange curve with a polynomial, we find that the resulting curve transforms under the n+2 dimensional representation of the monoid, where n is the degree of thepolynomial. We also find that the (ill-defined) derivative of the Blancmange curve is essentially the (inverse of the) Cantor function, thus demonstrating the semi- group symmetry on the Cantor Set as well. In fact, any topologically conjugate map will transform under the three-dimensional representation. We then show how all period-doubling maps can demonstrate the monoid symmetry, which is essentially an outcome of the dyadic representation of the monoid. This paper also includes a review of Georges deRham’s 1958 constructionof the Koch snowflake, the Levy C-curve, the Peano space-filling curve and the Minkowski Question Mark function as special cases of a curve with the monoid symmetry. The left-right symmetric Levy C-curve and the left-right symmetric Koch curve are shown each belong to another, inequivalent three dimensional rep- resentation. This paper is part of a set of chapters that explore the relationship between the real numbers, the modular group, and fractals. Its also a somewhat poorly structured, written at least partly as a diary of research results.1Symmetries of Period-Doubling MapsIt has been widely noticed that Farey numbers appear naturally in certain fractals,most famously in the Mandelbrot set. For example figure 1 shows how to count the buds of the Mandelbrot set by the Farey numbers. The reason why the Farey num- bers are appropriate for such counting is somewhat more opaque. However, it can be said that many fractal phenomena, and in particular, period-doubling maps, have an1Figure 1: Mandelbrot Farey NumberingThe above picture is stolen from Robert L. Devaney’s website The Fractal Geome- try of the Mandelbrot Set http://math.bu.edu/DYSYS/FRACGEOM2/node5.html. It demonstrates how to label and count buds on the Mandelbrot set using the Farey frac- tions. That the term “counting” is appropriate follows from the observation that both the sizes and locations of the buds correspond to the location of the fractions on the Farey or Stern-Brocot tree.2infinite binary tree structure that appears naturally. By describing the nature of the self-symmetries of the binary tree, one can effectively describe the nature of the fractal in question.The infinite binary tree is self-similar, in that one can look at the subtree lying under any given node, and see that is isomorphic to the tree as a whole. A given subtree is easily and uniquely located: one may take a walk down the tree, taking either the left Lor the right R branch. Thus, a given node on the tree may be specified by a sequence or string composed of the letters L and R. When L and R are taken to be a certain pair of 2×2 matrices, then resulting set of strings turn out to be equivalent to a certain subset of the modular group PSL(2,Z). The modular group is critically important for a broad swath of number theory, from the theory of elliptic functions to the theory of modular forms. It seems remarkable that the symmetry structure of fractals are connected these topics, although, on closer examination, it turns out that what is remarkable is that the popular literature for the most part has not made this connection. The notable exception seems to be the popular book “Indra’s Pearls” by David Mumford and Caroline Series (need ref). More narrowly, the relationship between number theory and hyperbolic geometry meets in an area of mathematics known as ergodic theory. The group PSL(2,Z) is seen to be a special case of a Fuchsian group, which are generally the discrete subgroups of PSL(2,C). These are implicated in a variety of chaotic dynamical systems, mostnotably in the ergodic or Anosov flow on the tangent space of PSL(2,C). Thus, the relationships between number theory and dynamical systems are known to mathemat-ics; however, this knowledge has not yet escaped fairly narrow confines. In particular, many theorems in the area of dynamical systems are abstractly stated, and make no mention of exactly-solvable cases. Conversely, many areas of concrete number theory fail to emphasize or even point out the highly fractal nature of the relationships. Thus, it seems appropriate to work out in greater detail some concrete connections betweenfractals, the infinite binary tree, and the modular group.The focus of this paper is to elucidate how the infinite binary tree, the set of of all strings generated by two letters L and R, and the set of period doubling fractals are related, and how their self-similarities can be precisely described. This is done in two parts. An earlier paper (need ref to my Minkowski question mark paper) deals with two-dimensional matrix representations, enumerating the elements of these sets, and questions about measure theory and the relationship to the Cantor set. This paper begins with a brief review of those results. It is then followed by a development of thethree-dimensional matrix representation for a very specific fractal, the Takagi curve, and then by various generalizations to broader cases.The first to be explored are the three-dimensional representations. It is shown thatthere are an uncountably infinite number of these, and that they may be labelled by a real or complex number. They are all inequivalent, in that there is no similarity trans- form that takes one into another. These occur naturally as the set of self-symmetries of the a family of fractal curves studied by Teiji Takagi in 1903[?], and further devel- oped by G.H. Hardy in 1916 and by others. These now bear the name Takagi curve or blancmange curve. The Takagi curve is constructed from the iterated tent map. The three-dimensional representation can thus be shown to apply to any iterated map that is topologically con-3jugate to the tent map. The class of such curves is large, and contains many interesting specimens, such as the Logistic map of unit height. It is this observation that leads to a conjecture that essentially all period-doubling maps transform under the three- dimensional representation. In practice, however, there is a big difference between proving that a map is conjugate to the tent map, and giving explicit expression to the conjugating function. As an example, we show how one very simple iterated map is conjugate to the tent map, but the conjugating function is the Minkowski question markfunction. Thus, finding the conjugating function is a non-trivial exercise. The above introduction singled out some special matrices for L and R. One may ask what happens if one picks some arbitrary matrices for L and R, or even a pair of arbitrary functions, and then iterates on them by constructing all possible strings containing L and R. This question was explored by Georges de Rham in 1957[?]. Hedemonstrates that under certain broad conditions for the fixed points of L and R, any such generalized pair, when iterated, will produce a continuous but non-differentiable curve. A variety of classic fractal curves arise in this construction, including the Kochsnowflake, the Levy C-curve, and the Peano space-filling curve. The fact that all ofthese curves can be identified with the set of strings in two letters implies that the general considerations about the self-similarity of binary trees can be applied to these cases2Two-dimensional Representation BasicsThis section reviews the basics of the two-dimensional matrix representations of theself-similarities of the infinite binary tree. It is an abbreviated presentation of the ma- terial in xxx (ref my paper on Minkowski question mark), and is here only to provide context and notation for the rest of this document. A critical property of the Farey numbers is that they can be arranged on a binarytree, known as the Farey tree or Stern-Brocot tree, as shown in figure2. One may navigate to a particular location in the binary tree by starting at the root node (located at 1/1), and making a sequence of left and right moves. The value of the Farey fraction labelling that node can be obtained by performing a simple matrix calculation. LetL =?10 11¶ and R =?11 01¶ (1)be the “left” and “right” matrices. A sequence of moves, such as RLLLRRL, results in a matrix?ab cd¶ (2)with integer entries a,b,c,d. The value of the Farey number located at the correspond- ing node is a+b c+d This is the simplest example of what I’ll call a “matrix representation” associated with a binary tree. This particular representation is two-dimensional (it deals with 2×24Figure 2: Farey TreeAn illustration of the Farey tree or Stern-Brocot tree, showing the Farey numbers ar- ranged on a binary tree.5matrices), and will be called the “Farey” or “continued fraction” representation, to distinguish it from other possible two-dimensional representations. The matrix given above is unimodular: that is, its determinant is one. This can be easily seen because the determinant of L and R is one, and the determinant of a product of matrices is equal to the product of the determinants. The entries of the matrix are always integers, and thus its easily seen that the matrix belongs to the group of unimodular 2×2 matrices over the integers SL(2,Z). Given that the Farey fraction is given by the ratio, so that ?−a−b −c−d¶ =?ab cd¶?−10 0−1¶gives the same fraction, we can see that the relevant matrices in fact belong to the projective group PSL(2,Z). This group is known as the “Modular Group”, and it is central to many areas of number theory and in particular the theory of elliptic integrals and modular forms.An alternate labelling of an infinite binary tree is with the “dyadic rationals”, ra- tional numbers of the form m/2nwith m an odd integer, and n a non-negative integer. This is very simply the tree with rows0 11 1 1 2 1 43 4 1 83 85 87 8This tree, the dyadic rational tree, also has a matrix representation for converting a sequence of left-right moves to the value of the dyadic fraction located at the resulting node. The left and right dyadic matrices areLD=?10 01 2¶ and RD=?10 1 21 2¶ (3)and these generate what will be called the two-dimensional “dyadic representation” of moves on the binary tree. The value of the dyadic fraction after a sequence of moves is 2c+d where c and d are the entries in the matrix shown in 2. The dyadic matrix that results from a sequence of left and right moves is clearly not unimodular. However, insofar as it labels a set of moves on the binary tree, it is isomorphic to the matrices arising in the Farey representation. There is a (unique)matrix corresponding to each node in the tree, and these can be trivially identified with one another. The representations are not, however “equivalent”: there does not exist a similarity matrix S that caries the one to the other. That is, the equationsSLD= LCSand SRD= RCS6are satisfied only by S = 0. Here, we wrote LCand RCfor the L and R of equation 1, in order to distinguish them from the dyadic L and R. These two representations are developed and examined in greater detail in an- other paper (ref my paper on Minkowski question mark), and the details will not be reproduced here; the focus of this paper are the three-dimensional and the higher- dimensional representations. However, there are several important points that must be noted, in order to avoid confusion. Most important is to understand that the moves on the binary tree are not isomorphic to the modular group. Although it can be shown that LCand RCgenerate all of PSL(2,Z), the non-negative powers do not. The non- negative powers generate only what is called a “monoid” or a “semigroup”, and not a full group, because the inverse elements are missing. The monoid does not contain the elements L−1or R−1or any strings containing these elements. Although it may at first seem seem that the negative powers correspond to upward walks on the binary tree, this is not so. The meaning of the negative powers in terms of their action on the tree cannot be made consistent or even complete. Thus, while the monoid can be formally extended to a full group by adjoining the formal inverses, the resulting group no longer describes motion on a binary tree. That is, the monoid acts on the binary tree; but it is impossible to extend this action to an action of the full group. Another critical aspect make note of is that the action of the monoid is an action onacertainsetofintervalsofthe realnumberline. When theinfinitebinarytreeislabelled either with the dyadic fractions or with the Farey fractions, it may be shown that any fraction occurring on the left is always strictly less then another fraction on the right. Thus, the binary tree extending under any given node has distinct and unique (Cauchy)limits to the left and to the right, thus defining an interval of the real number line. The subtree under any given node is clearly isomorphic to the tree as a whole. Finally, one may note that the dyadic rationals are dense in the reals, as are the Farey fractions. In fact, every rational shows up somewhere on the Farey tree. These properties make it clear that the act of navigating to any particular sub-node of the binary tree is the sameas a map from one interval (say the unit interval) to a certain specific subinterval. It is not possible to navigate to just any interval whatsoever; these are constrained. A detailed discussion of which subintervals are allowed, and how to enumerate them, is given elsewhere (cf. my Minkowski question mark paper). Roughly, though, it may be said that the allowed intervals may be enumerated as Q×N. It will become apparent, in the text of this paper, that the “allowed” intervals are precisely those for which a period-doubling fractal is self-similar.2.1Numerical representationsThe set of strings in two letters is clearly isomorphic to the binary numbers, in that any given string in L and R can be converted to a string on 0 and 1, which may then be taken to be a binary number. Fractions such as 1/3, as well as irrational numbers correspondto strings of infinite length. A convenient alternate notation is in terms of continued fractions. One