秋山, 茂樹

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Title Dynamical norm予想とPisotタイル張り (解析数論と数論諸分野の交流)Author(s)秋山, 茂樹Citation数理解析研究所講究録 (1999), 1091: 241-250Issue Date1999-04URL http://hdl.handle.net/2433/62889RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityDynamical norm Pisot( )Pisot1 Pisot$\beta$ 1 $x$$a_{-i}\in[0, \beta)\cap \mathbb{Z}$ ( )$x= \sum_{0i\geq N}a_{-}i\beta-ia=-N_{0}\beta-N0+a_{-}N0-1\beta-N0-1+\ldots$$x- \sum_{i=N0}a_{-}i\beta^{-i}|<\beta^{-N}$$N\geq N_{0}$ – $x$$\beta$ greedy expansionR\’enyi[18] Parry [15]$\beta$Fin $(\beta)$greedy expansionFin $( \beta)=\mathbb{Z}_{\geq 0}[\frac{1}{\beta}]$ (F)$1/\beta$(F) 11091 1999 241-250 2411. $\beta$ $(1\backslash ^{\backslash }+\sqrt{5})/2$ greedy $0,1$$\beta^{2}=\beta+1$ $100=11$11 $\beta$ greedy expansion1 $=$ 12 $=$ $1.11=10.01$3 – $1.1\cap\rceil \mathrm{J}\perp.\cup\perp$ =10 . $\cup\perp\cap 1$4 $=$ 101.015 $=$ $102.01=101.12=110.02=1000.1001$greedy expansion(F)K.Schmidt [19]Pisot 11 Salem 11 –1 Frougny Solomyak [10] (F)$\beta$ Pisot$\mathbb{Z}_{>0}\subset \mathrm{F}\mathrm{i}\mathrm{n}(\beta)$ $\beta$ Pisot Salem1([4]). $\beta$ $\mathbb{Z}_{\geq 0}\subset \mathrm{F}\mathrm{i}\mathrm{n}(\beta)$ $\beta$ Pisot(F)[10] $\beta$$X^{mm}-a_{m-1}x-1-a_{m}-2x^{m-2}-\ldots-a_{\perp^{x-a_{0}}}$$a_{m-1}\geq a_{m-2}\geq\ldots\geq a_{0}>0$ $\beta$ (F)([11] ) $[1],[3]$ $\beta$ (F)3 2K.Schmidt [1]31 ([4]). $\beta$ 3 Pisot $\beta$ (F)$x^{3}-ax^{2}-bx-l$$a\geq 0$ and $-1\leq b\leq a+1$242Pisot[19][8] [1]2. $\beta$ Pisot (Pisot ) (F)$c$ $c$$c$$c$ 2[19] $c=1$ ( ) 3 $c$1 ([1] example) Pisot(F)2 Pisot2 $c$ Pisot$\beta>1$ – $n$ $\mathbb{Q}(\beta)arrow \mathbb{R}^{n}$$\Phi$ : $\mathbb{Q}(\beta)arrow \mathbb{R}^{n-1}$ trivial$\beta$ (F)1 $\beta$ Pisot Fin $(\beta)$Fin$(\beta)=\mathrm{u}s_{(v}\omega$ (1)$\omega$ $S_{\omega}$ Fin $(\beta)$$\omega$ – $S_{001}$. .001 Fin $(\beta)$S.$S$ Fin $(\beta)$ $.\mathbb{R}_{\geq 0}$$\mathbb{R}$ closure$\mathbb{R}_{\geq 0}=\overline{\mathrm{u}\omega s_{\omega}}$& discrete(1) $\Phi$ $\mathbb{R}^{n-1}$closure2([3]). $\beta$ $n$ Pisot $\Phi(\mathbb{Z}[\beta]_{\geq 0})$ $\mathbb{R}^{n-1}$$\mathbb{R}^{n-1}$$\mathbb{R}^{n-1}=\mathrm{u}\Phi\omega\overline{(S_{\omega})}$243$T_{\omega}=\overline{\Phi(S_{\omega})}$ $\mathcal{T}=\overline{\Phi(S)}$$\Phi(S_{\omega})$ $T_{\omega}$ $T_{\omega}$3([3]). $\beta$ Pisot (F)$T_{\omega}=\overline{\mathrm{I}\mathrm{n}\mathrm{n}(T\omega)}$Inn( $A$$n-1$ Lebesgue$0$ B.Praggastis [15][20] ()substitution tiling– - [3]Thurston [21]$x^{3}-x^{2}-X-1$Rauzy [17] Rauzy Fractal& [12], R.Kenyon [13], R.Kenyon &AVershik [14], B.Solomyak [20], B.Praggastis $[16]_{0}$ Toralautomorphism Markoff partition(substitution) (substitution tiling)1. ?2. ?3. ??()2443 Dynamical Norm$\beta>1$ – $\beta$ greedy expansion1 $1-[\beta]/\beta$ greedy expansion$1-[\beta]/\beta=c_{-2\beta+C_{-}}-23\beta-3+.$ . .$c_{-1}=[\beta]$1.$1= \sum_{i=1}c_{i\beta^{-}=}i.C-1^{C_{-2}}$11 $[0, \beta)\cap \mathbb{Z}$ $c_{-1}c_{-2}\ldots$– $[0, \beta)\cap \mathbb{Z}$1 greedy expansion([15] ) 1$\beta$ simple beta numbercyclic beta number $0$simple beta number cyclic beta number Pisotcyclic beta number $z$ Laurent$\mathcal{G}_{\beta}(z)=1-\sum^{\infty}i=1\frac{c_{-i}}{z^{i}}$$g_{\beta}$ $|z|>1$ $\mathcal{G}_{\beta}(\beta)=0$ $\mathbb{C}$$\beta$ cyclic beta number( Poly\’a –) $\beta$ cyclic beta numberDynamical norm $d(\beta)$2.$d( \beta)=\prod_{g\beta(u)=0}|u|$1(Dynamical Norm ).$d(\beta)=|N\mathbb{Q}(\beta)(\beta)|$245$N_{K}$ $K/\mathbb{Q}$2. $\beta$ $(1+\sqrt{5})/2$ $\beta$ $1=\beta^{-1}+$$\beta^{-2}=.11$ 11[15]1$\mathcal{G}_{\beta}(z)=1-\frac{1}{z}-\frac{1}{z^{2}}$$\mathbb{C}$ $d(\beta)=$$\beta\beta’=1$ $\beta’$ $\beta$3. $\beta$ $x^{3}-x-1$ Pisot$1=\beta^{-2}+\beta^{-3}=.011$ 11 0111 $1=\beta^{-1}+\beta^{-5}=.10001$1$\mathcal{G}_{\beta}(z)=1-\frac{1}{z}-\frac{1}{z^{5}}=\frac{z^{5}-z^{4}-1}{z^{5}}$$(z^{5}-z^{4}-1)=(z^{3}-z-1)(z^{2}-z+1)$ $d(\beta)=1$$\circ$$\mathcal{G}_{\beta}$$\beta$$\beta$ 2 3 Pisot $([4])_{\text{ }}4$Salem ([9])Pisot4([3]). $\beta$ Pisot (F) $\beta$Dynamical Norm$\beta$ Dynamical Norm 1 143 $\langle$ Fin $(\beta)$ $-$–( [3] )2462( ).(Graph directed self affine set)$\{X_{1}, X_{2}, \ldots, X_{m}\}$$X_{i}= \bigcup_{j1}^{i}j=\psi ij(\mathrm{Y}j)$ $\mathrm{Y}_{j}\in\{X_{1}, X_{2}, \ldots, X_{m}\}$$(i=1,2, \ldots, m)$ $\psi_{ij}$$X_{i}$ $X_{j}(j=1,2, \ldots, m)$$\{1, 2, \ldots, m\}$ $Y_{j}=X_{k}$ $iarrow k$4([2]). $\beta$ Pisot $x^{3}-x-1$$\mathcal{T}$ 5$\partial(\mathcal{T})=X_{1}\cup x_{2}\cup X_{3^{\cup}}x_{4^{\cup}}x_{5}$$X_{1}$ $=$ $((\beta’)^{5}X1+(\beta’)^{2})\cup((\beta’)^{4}X1+(\beta’)^{-1})$$X_{2}$ $=$ $((\beta’)^{5}x_{2}+\beta’)\cup((\beta’)^{4}X2+(\beta’)^{-4})$$X_{3}$ $=$ $((\beta’)^{5}x_{3}+\beta’)\cup((\beta’)4X_{3}+(\beta’)^{-2})$$X_{4}$ $=$ $((\beta’)^{5}x_{4}+1)\cup((\beta’)^{4}X4+(\beta’)^{-5})$$X_{5}$ $=$ $((\beta’)^{5}x5+1)\cup((\beta’)4x_{5}+(\beta’)^{-3})$Haussdorff 1.10026. ..$\beta’$ $\beta$ ( )5([5]). $\beta$ 3 Pisot (F)Inn $(T_{\omega})$ $T_{\omega}$ 22473 ( )5\cap $\mathrm{r}2\rceil\llcorner\lrcorner$’encircling method’ –(F) $x^{3}-3_{X^{2}}-4_{X}-1$ Pisot–!–3( ). $\beta$ 3 Pisot (F)$x^{3}-ax^{2}-bx-1$1. $a>2b-4\Leftrightarrow \mathrm{I}\mathrm{n}\mathrm{n}(T)$2. $a=2b-4\Leftrightarrow \mathrm{I}\mathrm{n}\mathrm{n}(T)$3. $a<2b-4\Leftrightarrow \mathrm{I}\mathrm{n}\mathrm{n}(T)$3(cut point)6. $a\leq 2b-4$ Inn $(T)$ $a<2b-4$Pisot 3Dynamical norm248References[1] S.Akiyama, Pisot numbers and greedy algorithm, Number Theory, Dio-phantine, Computational and Algebraic Aspects, Edited by K. Gy\"ory,A. Peth\"o and $\mathrm{V}.\mathrm{T}$ .So’s, 9-21, de Gruyter 1998.[2] S.Akiyama and T.Sadahiro, A self-similar tiling generated by the mini-mal Pisot number, to appear in Proceedings volume of 13th Czech andSlovak Conference on Number Theory, Acta Mathematica et InformaticaUniversitatis Ostraviensis.[3] S.Akiyama, Self affine tiling and Pisot numeration system, to appear in’Number Theory and its Applications’, ed. by K. Gy\"ory and S. Kane-mitsu, Kluwer[4] S.Akiyama, Cubic Pisot units with finite beta expansions, preprint.[5] S.Akiyama, Pisot numeration system and self affine tiling (In Japanese),Surikaisekikenkyusho Kokyuroku No. 1060 (1998), 34-40.[6] S.Akiyama, On the boundary of self affine tilings generated by Pisotnumbers, in preparation.[7] $\mathrm{M}.\mathrm{J}$.Bertin, A. Decomp-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse and $\mathrm{J}.\mathrm{P}$ . Schreiber, Pisot and Salem numbers, Birkh\"auser Ver-lag, Basel-Boston-Berlin 1992[8] A.Bertrand, D\’evelopment en base de Pisot et r\’epartition modulo 1,C.R.Acad.Sc., Paris 385 (1977) 419-421[9] $\mathrm{D}.\mathrm{W}$ . Boyd, Salem numbers of degree four have periodic expansions,Th\’eorie des nombres, Number Theory, Walter de Gruyter, Berlin, NewYork, 57-64, 1989[10] C.Rougny and B.Solomyak, Finite beta-expansions, Ergod. Th. and Dy-nam. Sys. 12 (1992) 713-723[11] M.Hollander, Linear numeration systems, finite beta expansions, anddiscrete spectrum of sub.stitution dynamical $\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t},\mathrm{e}\mathrm{m}\mathrm{S}$ , thesis of Washing-ton Univ.[12] S.Ito and M.Kimura, On Rauzy Fractal, Japan J. Indust. Appl. Math.8 (1991) 461-486249[13] R.Kenyon, The construction of self-similar tilings, Geom. Rnct. Anal.6 (1996), no. 3, 471-488.[14] R.Kenyon and A.Vershik, Arithmetic construction of sofic partitions ofhyperbolic toral automorphisms. Ergodic Theory Dynam. Systems 18(1998), no. 2, 357–372.[15] W.Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci.Hungar. 11 (1960) 269-278[16] B.Praggastis, Markov partition for hyperbolic toral automorphism,Ph.D.Thesis, Univ. of Washington, 1992.[17] G.Rauzy, Nombres Alg\’ebriques et substitutions, Bull. Soc. France 110(1982) 147-178.[18] A.R\’enyi, Representations for real numbers and their ergodic properties.Acta Math. Acad. Sci. Hungar 8 (1957) 477-493.[19] K.Schmidt, On periodic expansions of Pisot numbers and Salem num-bers, Bull. London Math. Soc. 12 (1980) 269-278[20] B.Solomyak, Dynamics of self-similar tilings. Ergodic Theory Dynam.Systems 17 (1997), no. 3, 695-738.[21] $\mathrm{W}.\mathrm{P}$ .Thurston, Groups, Tilings and Finite state automata, AMS Collo-quium lectures, 1989.Shigeki AkiyamaDepartment of Mathematics, Faculty of ScienceNiigata University,Ikarashi-28050, Niigata950-2181, JAPAN$\mathrm{E}$-mail: akiyama@math.sc.niigata-u.ac.jphttp: $//\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{g}.\mathrm{g}\mathrm{e}$ .niigata-u. $\mathrm{a}\mathrm{c}.\mathrm{j}\mathrm{p}/\sim \mathrm{a}\mathrm{k}\mathrm{i}\mathrm{y}\mathrm{a}\mathrm{m}\mathrm{a}$250

Dynamical norm予想とPisotタイル張り (解析数論と数論諸分野の交流)

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Dynamical norm予想とPisotタイル張り (解析数論と数論諸分野の交流)

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