(2) RIMS Kôkyûroku Bessatsu B46 (2014), 241247. Rényi‐Parry. germs of. curves. functions. and. associated with real. dynamical zeta algebraic numbers. By. Jean‐Louis VERGER−GAUGRY. *. Abstract. $\beta$>1 be an algebraic number. The relations between the coefficient vector of its minimal polynomial and the digits of the Rényi $\beta$ ‐expansion of unity are investigated in terms of the germ of curve associated with $\beta$ which is constructed from the Salem parametrization, and the Parry Upper function f_{ $\beta$}(z) If $\beta$ is a Parry number, the Parry Upper function f(z) is simply related to the dynamical zeta function $\zeta$_{ $\beta$}(z) of the dynamical system ([0,1], T_{ $\beta$}) where T_{ $\beta$} is the $\beta$ ‐transformation. Using the theory of Puiseux several results on the zeros of f(z) and a classification of $\beta$ \mathrm{s} off Parry numbers are suggested. Let. ,. .. Introduction:. §1. The. Rényi‐Parry. numeration. numeration and inherits the. system. digits. and. [Re] [Pa] [Fr]. algebraicity. uses. $\beta$>1. real number. a. ([0,1], T_{ $\beta$}). properties of the dynamical system. where. ,. base of. as. T_{ $\beta$}. x\rightarrow. :. \{ $\beta$ x\}= $\beta$ x mod1 is the $\beta$ ‐transformation, for instance given by its dynamical zeta function of 1 which $\zeta$_{ $\beta$}(z) [AM] [Bo2] [FLP] [PP] [Po] [V4] or by the Rényi $\beta$ ‐expansion \mathrm{d}_{ $\beta$}(1)=0.t_{1}t_{2}t_{3} controls the language in base $\beta$ [B‐T] [Bl] [Lo]. The analytic function f_{ $\beta$}(z)=-1+\displaystyle \sum_{i\geq 1}t_{i}z^{i} is then fundamental and called the Parry Upper function (at $\beta$ ). When the base of numeration .. $\beta$>1. is. an. algebraic. number. vector of its minimal. a. basic. polynomial. solutions and directions for this. by definition. number is. zeros), are. then called. Perron. a. simple,. and the. is then to find the relations between the coefficient. string of digits (ti).. study. in the. geometrical setting. real number. $\beta$>1. such that. or. numbers, and the collection of Parry. is associated its. \mathrm{d}(1). eventually periodic. Parry. algebraic number, given by. an. question. its minimal. :.. The present. numbers. are. numbers is dense in. polynomial P_{ $\beta$}(X). ,. Parry. infinitely. many. algebraic integers. which. (1, +\infty) [Pa].. $\beta$>1. (ends. curves.. in. assumed to be. Parry polynomial P_{ $\beta$,P}(X)\in P(X)[X] (with. new. A. of germs of. is finite. study gives. a. To. Parry number,. P_{ $\beta$,P}^{*} denoting. its. reciprocal. Received 14. December, 2012, Revised 25 July, 2013, Accepted 16 August, 2013. Subject Classication(s): 11\mathrm{R}06, 30\mathrm{C}15 Key Words: Rényi‐Parry numeration system, Pisot number, Salem number, Puiseux theory, germ of curve, Newton polygon, dynamical zeta function Institut Fourier, Université Joseph Fourier Grenoble I, BP 74‐ Domaine Universitaire, 38402 St‐Martin dHères, France. 2010 Mathematics. *. \mathrm{e} ‐mail:. jlverger@ujf‐grenoble.fr. ©. 2014 Research Institute for Mathematical. Sciences, Kyoto University.. All. rights. reserved..

(3) JEAN‐LOUIS VERGER‐Gaugry. 242. polynomial),. as. f_{ $\beta$}(z)=-\displaystyle \frac{1}{$\zeta$_{ $\beta$}(z)}=-\frac{P_{ $\beta$,P}^{*}(z)}{(1-z^{p+1}). (1.1) where. p+1. is the. period length,. where. is the. m. d_{ $\beta$}(1). of. length. if. m. the. between the coefficients of. [Bo2] [Bo3] [V2],. distibution of some. base. algebraic. norm. Conjecture (S. Akiyama) states: that, if $\beta$ N() of $\beta$ satisfies |N( $\beta$)|=|t_{m}-t_{m+p+1}|. P(X). (ti),. and. but their. for instance for. $\beta$ [V1] (cf Akiyama. asymptotic strings of. nonParry. is the natural. followed here. algebraic. numbers. numbers. ,. $\beta$>1 Parry ,. for instance with the. repetitions,. and the Mahler. (ti),. in. with. of the. measure. a. overcomes. nonParry:. general,. zeros, the. Szegó‐Carlson‐Polya. and. observed. this. obscure; and the. are. Theorem. The. difficulty.. approach. It is addressed to. it amounts to write the. unit circle. Parry Upper. which is. noninteger. f(z). function. parametrization G_{ $\beta$}(U, Z)\in \mathbb{C}[[U, Z]] parametrized by the theory of Puiseux [C] to deduce its decomposition as a finite. Taylor. series. then to. use. product of factors. or. the. for. are. Pisot numbers and Salem. some. review). between f(z) and $\zeta$_{ $\beta$}(z). relations. ,. f(z) by introduced in [V4]. was. [P_{ $\beta$}^{*}(z), z-1/ $\beta$]. $\beta$. of. boundary. two‐variable. [AD]. and Kwon. Other relations. .. remains still obscure in. origin the. palindromic motives,. \mathrm{d}_{ $\beta$}(1). Diophantine Approximation questions [AB] [Bu] [Ds] [S]. For. as a. Norm. Dynamical. simple $\beta$. p+1 the period length of. and. preperiod length. numbers. The. .. then the. nonsimple Parry number,. a. and. f_{ $\beta$}(z)=-\displaystyle \frac{1-z^{m} {$\zeta$_{ $\beta$}(z)}=-P_{ $\beta$,P}^{*}(z). (1.2). is. nonsimple $\beta$. the Salem. and the coefficients involved in the formal series and Puiseux series in. them,. relating (t) and the values of the derivatives of the minimal polynomial of $\beta$ The adding of a second variable, most notably introduced differently by Boyd in several articles, is typical .. of studies. moduli of. on. of Puiseux is used for. polynomiality of. curve. U and 1945. (the Rényi‐Parry. article,. algebraic geometry (Lefshetz [Lf],. desingularizing. with the two. polynomial. in. curves. variables). Here, germ of. basic. as a. families of Salem numbers This note is conceived. ingredient. as a. short. §2.. $\beta$>1. \displaystyle \sum_{j=0}^{d}a_{j}X^{j}. polynomial,. be. an. algebraic. its minimal. \mathrm{d}. introduction,. :=\deg $\beta$. number and. assumed. \geq 2,. $\beta$=t_{1}+\displaystyle \sum_{i\geq 2}t_{i}$\beta$^{-i+1} T_{ $\beta$}^{i} :=T_{ $\beta$}(T_{ $\beta$}^{i-1}) i\geq 1, T_{ $\beta$}^{0}:=\mathrm{I}\mathrm{d} ,. gives. equation G_{ $\beta$}. is. a. (i.e. germ. analytic. in. in his. convergent. [BPD]). proofs, to [V5].. without. germ of. curve. Equation. P_{ $\beta$}(X)=a_{d}(X-$\beta$^{(0)})(X-$\beta$^{(1)})\ldots(X-$\beta$^{(d-1)})=. $\beta$=$\beta$^{(0)}, P_{ $\beta$}^{*}(X)=X^{\deg $\beta$}P_{ $\beta$}(1/X) its reciprocal \mathrm{d}_{ $\beta$}(1)=0.t_{1}t_{2}t_{3}\ldots the Rényi $\beta$ ‐expansion of unity,. with .. rise to. parametrization introduced by Salem. with. equivalently and. whose. theory. functions. algebraic. follow. The. M. Pathiaux‐Delefosse. Rényi‐Parry. polynomial,. $\beta$ ). we. [Dl]).. in the so‐called Salem construction for. §2.1. Let. for instance for. the canonical method. the Salem. uses. (M.J. Bertin,. locally,. associated with. curve. in Z ; this method. inthere. curves. Duval. The. ,. :=\lfloor $\beta$ T_{ $\beta$}^{i-1}(1)\rfloor, i\geq 2, \{0, 1, . . . , \lfloor $\beta$\rfloor\} The subrings. t_{1}=\lceil $\beta$-1\rceil=\lfloor $\beta$\rfloor, digits t_{i} belong. to. t_{i}. ..

(4) RÉNY1‐PARRY. \mathbb{C}\{U\}[Z]\subset \mathbb{C}\{U, Z\}\subset \mathbb{C}[[U, Z]] polynomiality j\geq 0. ger. n<j). in Z. denote the sets of convergent formal. g=\displaystyle \sum_{n\geq 0,m\geq 0}c_{n,m}U^{n}Z^{m}\in \mathbb{C}[[U, Z]], g=\displaystyle \sum_{n\geq j,m\geq 0}c_{n,m}U^{n}Z^{m} (i.e. with For. .. such that. 243. germs 0F curves and dynamical zeta functions. \mathrm{o}\mathrm{r}\mathrm{d}_{U}g. no nonzero. series, the first. one. with. denotes the greatest inte‐. term. indexed. c_{n,m}U^{n}Z^{m}. by. .. Theorem 2.1.. There exists. a. unique. G_{ $\beta$}(U, Z)\in \mathbb{C}\{U[Z]. such that. G_{ $\beta$}(P_{ $\beta$}^{*}(z), z-\displaystyle \frac{1}{ $\beta$})=f_{ $\beta$}(z) neighbourhood of 1/ $\beta$ four following possibilities. \deg_{Z}G_{ $\beta$}(U, Z)<\deg $\beta$. G_{ $\beta$}(U, Z) decomposes into one : of the (i) either G_{ $\beta$}(U, Z)=U\times e where e=e(U, Z) is a unit in \mathbb{C}\{U, Z\} (ii) or it is equal to e\times W where e=e(U, Z) is a unit in \mathbb{C}\{U, Z\} and (ii‐l) deg_{Z}W(U, Z)=1, \mathrm{o}\mathrm{r}\mathrm{d}_{U}W(U, Z)>1 or (ii‐2) deg_{Z}W(U, Z)=1, \mathrm{o}\mathrm{r}\mathrm{d}_{U}W(U, Z)=1 or (ii‐3) deg_{Z}W(U, Z)>1, \mathrm{o}\mathrm{r}\mathrm{d}_{U}W(U, Z)=1 and W is an irreducible Weierstrass polynomial.. for. in. z. a. with. ,. ,. ,. ,. ,. ,. ,. Denote. G_{ $\beta$}(U, Z) :=b_{d-1}(U)Z^{d-1}+b_{d-2}(U)Z^{d-2}+. (2.1) with. G_{ $\beta$}. b_{j}(U) :=\displaystyle \sum_{r\geq 0}b_{j,r}(U)U^{s} The last three. .. fact that. 1/ $\beta$. is. a. cases come. simple. .. .. :+b_{1}(U)Z+b_{0}(U). possibilities define four types of. from the Weierstrass preparation theorem. of. zero. The four. .. f_{ $\beta$}. .. In the last case, the. unique decomposition of the Weierstrass polynomial W. theory. as a. Newton. applied. of Puiseux. finite. ,. to. applies. polygon. and the. G_{ $\beta$}. provide. to. of. a. product. W=\displaystyle \prod_{ $\xi$}(Z-\sum_{i\geq 0}$\alpha$_{i}$\xi$^{i}U^{\frac{i}{w} ) over. all the Puiseux factors. the Puiseux. forming. a. deduced from the Newton. factors,. \displayst le\sum_{i\geq0}$\alpha$_{i}U^{\frac{i}w} are. fractionary. inator. with. power series for which the. w:=\deg_{Z}W. plane affine. unique conjugacy class. The Puiseux series, involved. ,. with. $\xi$ running. over. in. polygon,. conjugates. exponents. are. \displayst le\sum_{i\geq0}$\alpha$_{i}$\xi$^{i}U^{\frac{i}w} rational. the wth‐roots of. integers. unity.. The. with. common. denom‐. defines. polynomial G_{ $\beta$}. a. curve. C_{ $\beta$}:=\displaystyle \{(U, Z)\in \mathbb{C}^{2}|G_{ $\beta$}(U, Z)= \sum_{m,n\underline{>}0}A_{m,n}U^{n}Z^{m}=0\} with coefficients the first. A_{m,n}. projection. in. U , and. ficients. \mathbb{Q}( $\beta$) along U ‐plane).. field extension of. map, of \mathb {C}. §2.2. Let. a. (i.e.. \tex{∪. the. ,. with. Perron‐Frobenius operator and. a. ramified. covering $\pi$_{ $\beta$}. :. C_{ $\beta$}\rightar ow \mathbb{C},. Eigenvalues. \mathbb{C}((U)) :=\displaystyle \bigcup_{n\in \mathbb{N}^{*} U^{-n}\mathbb{C}[[U]] be the field of formal Laurent series of the variable denote \mathbb{C}((U))^{*} :=\displaystyle \bigcup_{m\in \mathbb{N}^{*} \mathbb{C}( U^{1/m}) the field of Laurent‐Puiseux series with coef‐ in \mathb {C} := The ring \mathbb{C}[[U]]^{*} \displaystyle \bigcup_{m\in \mathbb{N}^{*} \mathbb{C}[ U^{1/m}] of the Puiseux series contains the ..

(5) JEAN‐LOUIS VERGER‐Gaugry. 244. \mathbb{C}[[U]] By the Theorem of Puiseux [C] \mathbb{C}((U))^{*} is algebraically closed, and any polynomial in \mathbb{C}[[U]][X] has at least one X ‐root in \mathbb{C}[[U]]^{*} By the change of ori‐ gin, with \overline{f_{ $\beta$} (Z) :=f(z) and \overline{P_{ $\beta$}^{*} (Z) :=P_{ $\beta$}^{*}(z) we obtain new coefficients vectors. Denoting ring of formal. series. .. .. ,. \mathrm{K}_ $\beta$}^{\mathcal{G} the smallest Galois extension containing \mathrm{K}_{$\beta$}, \displaystyle \overline{f_{ $\beta$} (Z)=\sum_{j\geq 1}$\lambda$_{j}Z^{j}. \mathrm{K}_{ $\beta$}:=\mathbb{Q}( $\beta$). ,. $\lambda$_{j}=$\lambda$_{j}( $\beta$). :=\displaystyle \sum_{q\geq 0}t_{j+q}\left(\begin{ar ay}{l} j+q\ j \end{ar ay}\right)(\frac{1}{ $\beta$})^{q},. and. by. $\gam a$_{q}=\displaystyle\sum_{j=q}^{d}a_{d-j}\left(\begin{ar ay}{l j\ q \end{ar ay}\right)(\frac{1}{$\beta$})^{j-q}\in\mathrm{K}_{$\beta$}. .. j\geq 1. The d\times d square matrix M , with coefficients in an. operator. Eigenspaces of. (\mathbb{C}[ U] ^{*})^{d}. the vector space. on. dimension one, with. .. \mathbb{Q}($\gamma$_{1}^{\pm 1}, $\gamma$_{2}^{\pm 1}, \ldots, $\gamma$_{d}^{\pm 1})[U]=\mathrm{K}_{ $\beta$}[U] This vector space. priori Puiseux. a. series. as. Theorem 2.2.. one. e_{M}\in \mathrm{K}_{ $\beta$}[[U]][X] a unit polynomial, with $\sigma$_{1}\in \mathrm{K}_{ $\beta$}[ U] , ,. The other X ‐roots $\sigma$_{2}, $\sigma$_{3} ,. distinct,. have the. .. .. respective. :;. $\sigma$_{d}\in \mathbb{C}[[U]]^{*} of F. ,. as. direct. of d. sum. F=F(U, X)=. polynomial. and. W_{M}(U, X) :=(X-$\sigma$_{1}(U)). e_{M}=(X-$\sigma$_{2})(X-$\sigma$_{3})(\ldots)(X-$\sigma$_{d}). are. such that. derivatives. of. the. ,. are. constant terms. c_{0,j}:=$\sigma$_{j}(0)=\displaystyle \frac{1}{$\beta$^{(j-1)} -\frac{1}{ $\beta$} $\sigma$_{j}(U)\in\underline{\mathrm{K} _{ $\beta$}^{\mathcal{G} [ U]. polynomial. P_{ $\beta$}^{*}(X). ,. with. at c_{0,j} ,. for. coefficients. 2\leq j\leq d,. in the. algebra. over. \mathb {Q} generated by. the. as. $\sigma$_{j}(U)=c_{0,j}+\displayst le\frac{1}\overline{P_ $\beta$}^{*}(c_{0,j})\prime}U-\frac{\overline{P_ $\beta$}^{*}(c_{0,j})\prime\prime}{2(\overline{P_ $\beta$}^{*}(c_{0,j})^{3}\prime}U^{2}+\ldots. (2.5). §3. Let .. since. is the matrix. $\sigma$_{1}(U)=(\displaystyle\frac{1}{$\gam a$_{1} )U-(\frac{$\gam a$_{2} {$\gam a$_{1}^{3} )U^{2}+(\frac{2$\gam a$_{2}^{2}-$\gam a$_{1}$\gam a$_{3} {$\gam a$_{1}^{5} )U^{3}+\ldots. (2.4). $\beta$. a. ,. of M_{U} lies in the maximal ideal. e_{M}\times W_{M} with. as. Weierstrass. corresponding. and. into. With the above‐mentioned notations, the characteristic. uniquely decomposed. the. with. F(U, X)=X^{d}+\displaystyle \frac{$\gamma$_{d-1} {$\gamma$_{d} X^{d-1}+\ldots+\frac{$\gamma$_{2} {$\gamma$_{d} X^{2}+\frac{$\gamma$_{1} {$\gamma$_{d} X-\frac{1}{$\gamma$_{d} U. (2.3) is. In. splits. Eigenvalues,. fact, only Eigenvalue \det(X\mathrm{I}\mathrm{d}-M_{U})\in \mathrm{K}_{ $\beta$}[U][X] even to belongs U\mathbb{C}[[U]]^{*} (this Eigenvalue U\mathrm{K}_{ $\beta$}[ U] ). .. ,. Let. M=_{U}:\left(bgin{ary}l 0& \cdots&0\frac{U}$\gma_{d}\ 1&0\cdots&0-\frac{$gma_{1}$\gam _{d}\ 0&1 \ 0& 1 &-\frac{$gma_{d-1}$\gam _{d} \en{ary}\ight). (2.2). of. \overline{P_{ $\beta$}^{*} (Z)=Z($\gamma$_{1}+$\gamma$_{2}Z+\ldots+$\gamma$_{d}Z^{d-1}). and. ,. with. let. us. us. turn to the. consider. Main Theorems. explicit computation of. {}^{t}(0p_{j,1}p_{j,2::}.. the. p_{j,d-1} ), j\geq 1. ,. Rényi‐Parry. germ of. curve. associated with. the last column vector of the matrix. $\gamma$_{d}^{j}M_{0}^{j}..

(6) RÉNY1‐PARRY. convention. By. of. homogeneous,. p_{j,0}=0. put:. we. degree j. for all. ,. j\geq 1. polynomials p_{j,i}\in \mathbb{Z}[$\gamma$_{1}, $\gamma$_{2}, . . . , $\gamma$_{d}] d-1,. The. .. for i=1 , 2,. satisfy,. and. ,. 245. germs 0F curves and dynamical zeta functions. .. .. :,. are. p_{1,i}=-$\gamma$_{i},. Pj+1,i=- $\gamma$ iPj,d-1+ $\gamma$ dPj,i-1 j\geq 1. Theorem 3.1. are:. coefficients of the Rényi‐Parry. The constant. b_{j,0}=$\lambda$_{j-1}+\displaystyle\sum_{q\geqd}$\lambda$_{q}\frac{p_{q-d+1,j-1} {$\gam a$_{d}^{q-d} The other coefficients. The. (ii‐3). of. conjugated. by. the. these one. of Newton. irreducible. conjugated. curves cross. transports. in. as. Case. (i). an. expressed. is. a. mations,. it. an. means. algebraic. if all the coefficients. b_{j,0}. The other. cases. .. conjugation. to 0 , is are. a. if. which is not. possible. and. a. Parry number,. classication of. $\beta$ being. to. plane. is. irreducibles. the. analytic. a. Parry number,. (ii‐1), (ii‐2), (ii‐3). and this. case can. algebraic. number. $\beta$>1 of ,. de‐. 1\leq j\leq d-1.. some. sense,. by. to 0 , then the eventual. periodicity. numbers. only if::. asymptotic density of in. nonParry. b_{j,0}.. for all. ,. weak,. equal. over. When. Another consequence of the. number and that the constant coefficients. obtained: in this case, eventual. suggests. and. that the number and. summations, when equal. (2.1).. curve. (Galois‐) conjugates of 1/ $\beta$.. number. b_{j,0}=0. germ of. following.. exactly corresponds. Parry. geometrical steming. intersection with this. The. With the above‐mentioned notations, the. (assumed \geq 2 ),. Since 0 is. .. beta‐conjugates of $\beta$. in terms of the constant coefficients. Theorem 4.1. gree d. beta‐conjugate of $\beta$. a. to the. origin (U=0, z=1/ $\beta$) parametrized. decomposition of the. algebraic number,. does not cancel at the. in Theorem 2.1. of the. ,. in Theorem 2.1 is the is. 2.1, corresponds. equation U=0 then their. inverse is called. If $\beta$>1. \leq j\leq d-1.. deduced from the derivatives of the characteristic. neighbourhood. a. Diophantine Approximation. §4.. given by (2.1). .. onto the collection of the. Theorem 3.2.. be. curves. are. for all1. ,. in Theorem. polygon,. the U ‐plane in \mathb {C}^{2} of. decomposition of G_{ $\beta$}. function f(z). $\sigma$_{j}(U). ,. Puiseux series involved in the. point, for which the. curves. b_{j,r}, j\geq 0, r\geq 1. F and the formal series. case. curve. and. b_{0,0}=0. polynomial. of. germ. the. b_{j,0}. are. missing. easy Liouville. periodicity. given by. sum‐. terms in these. arguments. So that. of the sequence. (t). is. is forced.. in Theorem 2.1. to attribute the rational number. w_{ $\beta$}:=1-\displaystyle \frac{ $\delta$}{d}\in[0, 1]. correspond. to weaker. arguments.. It.

(7) JEAN‐LOUIS VERGER‐Gaugry. 246. to the. G_{ $\beta$} (it. associated with. 1\leq m< $\delta$) degree, small,. or. in. The rational. .. equal. to 1. not. are. certainly The. Let. .. $\delta$\in\{1, 2, . . :; d-1\}. integer w_{ $\beta$}. $\beta$. is the. degree. of the Weierstrass. b_{ $\delta$,0}\neq 0. greatest integer \leq d-1 such that. is at. is close to 0 if. large departure say that. By convention,. strong enough of the set. G_{ $\beta$}. admits. off the set of. w_{ $\beta$}=0. if and. to force the eventual. a. with. Weierstrass. Parry numbers,. only. if. periodicity. $\beta$. is. a. polynomial. b_{m,0}=0. polynomial i.e. with. Parry. of. for. high. degrees. $\delta$. number. These. ot the sequence of. { w_{ $\beta$}| $\beta$>1 nonParry algebraic number} probably. the subset of it formed when. number w_{ $\beta$} may be used. $\beta$>1. where. digits (ti),. possible ordering/correlation of (ti).. a. topology. particular. ,. is the. and close to 1 if. conditions but. $\beta$. number. algebraic. u/v. be. a. as a. $\beta$. runs over a. classifying parameter. rational number in. would have to be characterized. It is. defined. neighbourhood on. [0 1). Interestingly, ,. just. known that. $\Lambda$_{0}. of. unity.. the set of the real the set. is dense in. merits. attention,. The rational. algebraic. numbers. $\Lambda$_{u/v}:=\{ $\beta$>1|w_{ $\beta$}=u/v\} (1, +\infty) [Pa].. References. [AB]. B. Adamczewski and Y.. god.. [AD]. Th.. Dynam. Sys.. 27. Bugeaud, Dynamics foor $\beta$ ‐shiftts and Diophantine approximation,. (2007),. 16951711.. S. Akiyama and D.Y. Kwon Constructions. (2008),. Monatsh. Math. 155. [AM] [B‐T]. Er‐. of. Pisot and Salem numbers with. flat palindromes,. 265275.. Mazur, On periodic points, Annals of Math. 81 (1965), 8299. BerthÉ, P. Liardet and J. Thuswaldner, Dynamical directions. M. Artin and B.. G.. Barat,. V.. Ann. Inst. Fourier 56. (2006),. in. numeration,. 19872092.. [BPD] M.J. Bertin and M. PATHIAUX‐DELEFOSSE, Conjecture de Lehmer et petits nombres de Salem. (Lehmers conjecture and small Salem numbers) Queens Papers in Pure and Applied Mathematics, 81, Kingston: Queens University (1989). [Bl] F. Blanchard, $\beta$ ‐expansions and Symbolic Dynamics, Theoret. Comput. Sci. 65 (1989), 131141. [Bo2] D.W. Boyd, On beta expansions for Pisot numbers, Math. Comp. 65 (1996), 841860. [Bo3] D.W. Boyd, The beta expansions for Salem numbers, in Organic Mathematics, Canad. Math. Soc. Conf. Proc. 20 (1997), A.M.S., Providence, RI, 117131. [Bu] Y. Bugeaud, On the $\beta$ ‐expansion of an algebraic number in an algebraic base $\beta$ Integers 9 (2009), ,. A20, 215‐226. [C] E. CASAS‐ALVERO, Singularities of Plane Curves, Cambridge Univerity [Ds] A. Dubickas, On $\beta$ ‐expansions of unity for rational and transcendental 61. (2011),. Press. (2000).. numbers. $\beta$. ,. Math. Slovaca. 705‐716.. [Dl] [FLP]. Duval, Rational Puiseux expansions, Compositio Mathematica 70 (1989), 119154. L. Flatto, J.C. Lagarias and B. Poonen, The zeta function of the beta‐transfo rmation, Ergod. Th. Dynam. Sys. 14 (1994), 237266. [Fr] CH. Frougny, Number Representation and Finite Automata, London Math. Soc. Lecture Note Ser. D.. 279. [Lf] [Lo] [Pa]. (2000),. 207228.. Lefshetz, Algebraic Geometry, Princeton University Press, (1953). M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, (2003). W. Parry, On the $\beta$ ‐expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401‐. S.. 416.. [PP]. W. Parry and M.. Pollicott,. [Po] [Re]. Zeta. functions. and the. periodic orbit. structure. of hyperbolic dy‐. namics, Astérisque (1990), M. Pollicott, Dynamical zeta functions, Integers 11B (2011). A. RÉNYI, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. 187—188. Hungar.. 8. (1957),. 477493.. 1268..

(8) RÉNY1‐PARRY. 247. germs 0F curves and dynamical zeta functions. Barat, V. BerthÉ and A. Siegel, Boundary of central tiles associated with Pisot beta‐substitution and purely periodic expansions, Monatsh. Math. 155 (2008), 377‐419. [V1] J.‐L. VERGER‐GAUGRY, On gaps in Rényi $\beta$ ‐expansions of unity for $\beta$> 1 an algebraic number,. [S]. S.. Akiyama,. G.. Ann. Inst. Fourier 56. [V2]. J.‐L. 155. (2006),. VERGER‐GAUGRY,. (2008),. 25652579.. On the. dichotomy of Perron. numbers and. beta‐conjugates,. Monatsh. Math.. 277299.. VERGER‐GAUGRY, Unifo rm distribution of the Galois conjugates and beta‐conjugates of a [V3] Parry number and the dichotomy of Perron numbers, Uniform Distribution Theory J. 3 (2008), J.‐L.. 157190.. [V4]. J.‐L.. VERGER‐GAUGRY, Beta‐conjugates of. 11B, (2011). [V5] J.‐L. VERGER‐GAUGRY, On. (2012).. real. algebraic. the Puiseux fa ctors. of. numbers. the germ. of. as. Puiseux. curve. of. a. expansions, Integers. real. algebraic number,.

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