Rational dynamical zeta functions for birational transformations

Abstract

We propose a conjecture for the exact expression of the unweighted dynamical zeta function for a family of birational transformations of two variables, depending on two parameters. This conjectured function is a simple rational expression with integer coefficients. This yields an algebraic value for the topological entropy. Furthermore, the generating function for the Arnold complexity is also conjectured to be a rational expression with integer coefficients with the same singularities as for the dynamical zeta function. This leads, at least in this example, to an equality between the Arnold complexity and the exponential of the topological entropy. We also give a semi-numerical method to effectively compute the Arnold complexity.

Introduction

To study the complexity of continuous, or discrete, dynamical systems, a large number of concepts have been introduced [1], [2]. A non-exhaustive list includes the Kolmogorov–Sinai metric entropy [3], [4], the Adler–Konheim–McAndrew topological entropy [5], the Arnold complexity [6], the Lyapounov characteristic exponents, the various fractal dimensions, [7], [8] the Feigenbaum's numbers of period-doubling cascades [9], [10], etc. Many authors have tried to study and discuss the relations between these various notions in an abstract framework [11], [12]. Inequalities have been shown, for instance the metric entropy is bounded by the topological entropy, let us also mention the Kaplan–Yorke relation [13], [14]. Furthermore, many specific dynamical systems have been introduced enabling to see these notions at work. Some of the most popular are the Lorentz system [15], the baker map [16], the logistic map [17], the Henon map [18]. Each of these systems has been useful to understand and exemplify the previous complexity measures.

Here, we introduce another two-parameter family of mapping of two variables, originating from lattice statistical mechanics, for which much can be said. In particular, we will conjecture an exact algebraic value for the exponential of the topological entropy and for the asymptotic of the Arnold complexity. Furthermore, these two measures of complexity will be found to be equal for all the values of the two parameters, generic or not (the notion of genericity is explained below). A fundamental distinction must be made between the previously mentioned complexity measures according to their invariance under certain classes of transformations. One should distinguish, at least, two different sets of complexity measures, the ones which are invariant under the larger classes of variables transformations, like the topological entropy or the Arnold complexity [6], and the other measures of complexity which also have invariance properties, but under a “less large” set of transformations, and are therefore more sensitive to the details of the mapping (for instance they will depend on the metric).

We now introduce the following two parameters family of birational transformations $\text{k}{}_{\mathrm{<mtext>\alpha ,</mtext><mspace\; sp="0.16"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>\epsilon </mtext>}}$:

$$\text{u}{}_{\mathrm{n+1}}\text{=1−u}{}_{\mathrm{n}}\text{+u}{}_{\mathrm{n}}\text{/v}{}_{\mathrm{n}}\text{,}\text{v}{}_{\mathrm{n+1}}\text{=ε+v}{}_{\mathrm{n}}\text{−v}{}_{\mathrm{n}}\text{/u}{}_{\mathrm{n}}\text{+α·(1−u}{}_{\mathrm{n}}\text{+u}{}_{\mathrm{n}}\text{/v}{}_{\mathrm{n}}\text{)}$$

which can also be written projectively:

$$\text{u}{}_{\mathrm{n+1}}\text{=(v}{}_{\mathrm{n}}\text{t}{}_{\mathrm{n}}\text{−u}{}_{\mathrm{n}}\text{v}{}_{\mathrm{n}}\text{+u}{}_{\mathrm{n}}\text{t}{}_{\mathrm{n}}\text{)·u}{}_{\mathrm{n}}\text{,}\text{v}{}_{\mathrm{n+1}}\text{=ε·u}{}_{\mathrm{n}}\text{·v}{}_{\mathrm{n}}\text{·t}{}_{\mathrm{n}}\text{+(u}{}_{\mathrm{n}}\text{−t}{}_{\mathrm{n}}\text{)·v}{}_{\mathrm{n}}{}^2\text{+α·(v}{}_{\mathrm{n}}\text{t}{}_{\mathrm{n}}\text{−u}{}_{\mathrm{n}}\text{v}{}_{\mathrm{n}}\text{+u}{}_{\mathrm{n}}\text{t}{}_{\mathrm{n}}\text{)·u}{}_{\mathrm{n}}\text{,}\text{t}{}_{\mathrm{n+1}}\text{=u}{}_{\mathrm{n}}\text{·v}{}_{\mathrm{n}}\text{·t}{}_{\mathrm{n}}\text{.}$$

As far as complexity calculations are concerned, the α=0 case is singled out. In that case, it is convenient to use a change of variables (see Appendix A) to get the very simple form k_ε:

$$\text{y}{}_{\mathrm{n+1}}\text{=z}{}_{\mathrm{n}}\text{+1−ε}\text{,}\text{z}{}_{\mathrm{n+1}}\text{=y}{}_{\mathrm{n}}\text{·}\text{z}{}_{\mathrm{n}}\text{−ε}\text{z}{}_{\mathrm{n}}\text{+1}$$

or on its homogeneous counterpart:

$$\text{y}{}_{\mathrm{n+1}}\text{=(z}{}_{\mathrm{n}}\text{+t}{}_{\mathrm{n}}\text{−ε·t}{}_{\mathrm{n}}\text{)·(z}{}_{\mathrm{n}}\text{+t}{}_{\mathrm{n}}\text{)}\text{,}\text{z}{}_{\mathrm{n+1}}\text{=y}{}_{\mathrm{n}}\text{·(z}{}_{\mathrm{n}}\text{−ε·t}{}_{\mathrm{n}}\text{)}\text{,}\text{t}{}_{\mathrm{n+1}}\text{=t}{}_{\mathrm{n}}\text{·(z}{}_{\mathrm{n}}\text{+t}{}_{\mathrm{n}}\text{)}\text{.}$$

These transformations derive from a transformation acting on a q×q matrices M [20]:

$$\text{K}{}_{\mathrm{q}}\text{=t∘I,}$$

where t permutes the entries $\text{M}{}_{\mathrm{<mtext>1,</mtext><mspace\; sp="0.16"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>2</mtext>}}$ with $\text{M}{}_{\mathrm{<mtext>3,</mtext><mspace\; sp="0.16"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>2</mtext>}}$ and I is the homogeneous inverse: I(M)=det(M)·M⁻¹. Transformations of this type, generated by the composition of permutations of the entries and matrix inverse, naturally emerge in the analysis of lattice statistical mechanics symmetries [19].

These transformations turn out to provide a set of examples for which various conjectures can be made. This is the aim of this paper which is organized as follows: in the first part of the paper we exactly compute the growth of the complexity of the first successive iterations (degree of the successive expressions). From these integers, we conjecture various algebraic values for the complexity. Different cases, corresponding respectively to α=0 and α≠0, are distinguished in two subsections. The results of these sections are confirmed by a semi-numerical method we introduce. In the second part of the paper we address the problem of evaluating another measure of the complexity, namely the topological entropy. This is done computing formally the first terms of the expansion of the generating function of the number of fixed points. This leads us to conjecture rational expressions for these generating functions. The same singled out (α,ε)-values as for the complexity growth appear, and are separately analyzed also in two subsections. The last section is devoted to a discussion about a possible “diffeomorphism of the torus” interpretation for the rationality of the generating functions we conjecture.