Pisot expansion in self-inducing systems
Shigeki Akiyama
1. Feb 2009: Number Theory and Ergodic Theory (Kanazawa)
Open problems for youngsters
A Pisot number is the real root > 1 of a polynomial xd − cd−1 xd−1 − · · · − c0 withci ∈ Z, whose other roots are strictly within the unit circle. If c0 = ±1, then it is calleda Pisot unit. Let (X, B, µ, T ) be a measure theoretical dynamical system. For any subset
Y ∈ B of positive measure, an induced system (Y, B0 , µ0 , T 0 ) is canonically deﬁned by theﬁrst return map T 0 (x) = T m(x) (x) ∈ Y . The induced system may behave quite diﬀerently
from the original. But in cases, there is an expansive aﬃne map π such that (Y, B0 , µ0 , T 0 )and (X, B, µ, T ) are conjugate through π. Then we say that it has a self-inducing structure.
An eigenvalue of the aﬃne map (expansion constant) in a self-inducing system becomes a
Pisot number, moreover a Pisot unit, in many important examples.
1. Substitutive dynamical system. This is introduced as the simplest self-inducingsystem. Let σ be a primitive substitution on the monoid {1, . . . , k}∗ with a ﬁxed pointx ∈ {1, . . . , k}N , i.e., σ(x) = x. Let s is the shift s(x1 x2 . . . ) = x2 x3 . . . . Then theclosure of s-orbits of x gives a dynamical system (Xσ , σ) which is strictly ergodic.
Pisot conjecture reads if σ is an irreducible Pisot substitution, then (Xσ , σ) has purely
discrete spectrum. There are two known ways to access this problem. One is to studydirectly the spectral measure by observing pattern repetitions (see [7], Solomyak [4]).
Another is to construct a geometric realization called Rauzy fractal, which is conjugate
to the translation of a torus (Arnoux-Ito [3]).
2. Irrational rotation.
Let X = R/Z = [0, 1) and T (x) = x + α where α ∈ [0, 1) \ Q. Then (X, T ) is strictly
ergodic. Standard successive induced systems are given as irrational rotations acting
on smaller intervals of lengths αn . We can compute αn by the continued fraction of α.
By Legendre theorem, one of the induced systems of (X, T ) has self-inducing structuresiﬀ α is quadratic irrational. Gauss theorem tells that α is reduced iﬀ the system (X, T )itself has a self-inducing structure. The expansion constant is a quadratic Pisot unit.
See Yasutomi [9] and [2] for self-inducing coding of irrational rotation.
3. Self-similar tiling.
Tiling dynamical system is a natural generalization of substitutive system. Let Ti (i =1, . . . k) be protiles (closure ∪of its interior) in R2 and Ji be the set discrete translations
2such that R is covered by Ti + Ji without overlaps. Tiling dynamical system T isdeﬁned by the orbit closure of the tiling by translation R2 with a suitable topology. If
we assume that T is self-similar (self-inducing), then (T , R2 ) is strictly ergodic. Thissystem is never mixing but weak mixing is possible. It is known than the system is not
weakly mixing (i.e. having a non-trivial point spectrum) iﬀ the expansion constantis a complex Pisot number. Under certain coincidence conditions, (T , R2 ) becomes
purely discrete (see [8]). This coincidence is considered to be close to Pisot property.
4. Piecewise isometry.
This problem arose from discretized process of rational rotation. Let ζp be a p-th
root of unity. Consider a piecewise isometry T : x → ζp x discontinuously acts on thefundamental parallelogram X = [0, 1] − ζ p [0, 1]. If the image by T goes out of X thenit is pulled back into X through real direction. Clearly the 2-dim Lebesgue measureis invariant by T . This gives a highly non-ergodic system and it is conjectured that
almost all points in X is periodic. We can observe self-inducing structures when p ≤ 10and p = 12. Expansion constants are Pisot units (see [6, 1]), and become cubic for
p = 7 and 9.
My questions are rather obscure and possibly belong to a kind of a folklore.
1. Need more examples. Related questions.
We wish to have more example of self-inducing structures. What about interval exchange transforms or diﬀerent continued fraction algorithms ? It is nice to ﬁnd such
systems related to Diophantine approximation.
2. Why do Pisot numbers appear in expansion constants?
In above interesting examples, the expansion constant is a Pisot number and evenmore, a Pisot unit. We do not know the essential reason but a hint is found in selfsimilar tilings. In this case, we have a special distribution of powers of eigenvalues ofassociated isometry. We do not know much on tiling with inﬁnite rotational symmetry.
Its spectrum is conjectured to be mixed.
3. Number theoretical applications?
If the system has self-inducing structure, then we can naturally associate a number
theoretical expansion (multiplicative coding). This means the system has both additiveand multiplicative structures (c.f. Kamae’s number system [5]). In the best case, thisgives an algebraic way to construct a natural extension of a given number theoreticalalgorithm. Under this, we can study (eventually and purely) periodic orbits in detail.
In this manner, we wish to create tight connection between ergodic theory and number
theory. A sample question: what is the additive coding of Rosen continued fraction?

References[1] S. Akiyama, H. Brunotte, A. Pethő, and W. Steiner, Periodicity of certain piecewiseaﬃne planar maps, Tsukuba J. Math. 32 (2008), no. 1, 1–55.
[2] S. Akiyama and M. Shirasaka, Recursively renewable words and coding of irrationalrotations, J. Math. Soc. Japan 59 (2007), no. 4, 1199–1234.
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Simon Stevin 8 (2001), no. 2, 181–207.
[4] M. Hollander and B. Solomyak, Two-symbol Pisot substitutions have pure discrete spectrum, Ergodic Theory Dynam. Systems 23 (2003), no. 2, 533–540.
[5] T. Kamae, Numeration systems as dynamical systems—introduction, Dynamics &stochastics, IMS Lecture Notes Monogr. Ser., vol. 48, Inst. Math. Statist., Beachwood,OH, 2006, pp. 198–211.
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[7] M. Queﬀélec, Substitution dynamical systems—Spectral analysis, Lecture Notes in Mathematics, vol. 1294, Springer-Verlag, Berlin, 1987.
[8] B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems 17(1997), no. 3, 695–738.
[9] S. Yasutomi, On sturmian sequences which are invariant under some substitutions, Number theory and its applications (Kyoto, 1997), Kluwer Acad. Publ., Dordrecht, 1999,
pp. 347–373.