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14
On Khintchine exponents and Lyapunov exponents of continued fractions
, 2008
"... Assume that x ∈ [0,1) admits its continued fraction expansion x = [a1(x), a2(x), · · ·]. The Khintchine exponent γ(x) of x is defined by 1 Pn γ(x): = lim n→ ∞ n j=1 log aj(x) when the limit exists. Khintchine spectrum dim Eξ is fully studied, where Eξ: = {x ∈ [0, 1) : γ(x) = ξ} (ξ ≥ 0) and dim d ..."
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Cited by 15 (8 self)
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Assume that x ∈ [0,1) admits its continued fraction expansion x = [a1(x), a2(x), · · ·]. The Khintchine exponent γ(x) of x is defined by 1 Pn γ(x): = lim n→ ∞ n j=1 log aj(x) when the limit exists. Khintchine spectrum dim Eξ is fully studied, where Eξ: = {x ∈ [0, 1) : γ(x) = ξ} (ξ ≥ 0) and dim denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum dim Eξ, as function of ξ ∈ [0,+∞), is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by γϕ 1 Pn (x): = lim n→ ∞ ϕ(n) j=1 log aj(x) are also studied, where ϕ(n) tends to the infinity faster than n does. Under some regular conditions on ϕ, it is proved that the fast Khintchine spectrum dim({x ∈ [0,1] : γϕ (x) = ξ}) is a constant function. Our method also works for other spectra like the Lyapunov spectrum and the fast Lyapunov spectrum.
Calculating Hausdorff Dimension Of Julia Sets And Kleinian Limit Sets
 Amer. J. Math
"... We present a new algorithm for efficiently computing the Hausdorff dimension of sets X invariant under conformal expanding dynamical systems. By locating the periodic points of period up to N , we construct approximations s N which converge to dim(X) superexponentially fast in N . This method can b ..."
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Cited by 13 (2 self)
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We present a new algorithm for efficiently computing the Hausdorff dimension of sets X invariant under conformal expanding dynamical systems. By locating the periodic points of period up to N , we construct approximations s N which converge to dim(X) superexponentially fast in N . This method can be used to give rigorous estimates for important examples, including hyperbolic Julia sets and limit sets of Schottky and quasifuchsian groups.
Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions
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FROM APOLLONIUS TO ZAREMBA: LOCALGLOBAL PHENOMENA IN THIN ORBITS
"... Abstract. We discuss a number of natural problems in arithmetic, arising in completely unrelated settings, which turn out to have a common formulation involving “thin ” orbits. These include the localglobal problem for integral Apollonian gaskets and Zaremba’s Conjecture on finite continued fractio ..."
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Abstract. We discuss a number of natural problems in arithmetic, arising in completely unrelated settings, which turn out to have a common formulation involving “thin ” orbits. These include the localglobal problem for integral Apollonian gaskets and Zaremba’s Conjecture on finite continued fractions with absolutely bounded partial quotients. Though these problems could have been posed by the ancient Greeks, recent progress comes from a pleasant synthesis of modern techniques from a variety of fields, including harmonic analysis, algebra, geometry, combinatorics, and dynamics. We describe the problems, partial progress, and some of the tools alluded to above.
Computation of a Class of Continued Fraction Constants
"... We describe a class of algorithms which compute in polynomial – time important constants related to the Euclidean Dynamical System. Our algorithms are based on a method which has been previously introduced by Daudé Flajolet and Vallée in [10] and further used in [13, 32]. However, the authors did no ..."
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Cited by 4 (0 self)
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We describe a class of algorithms which compute in polynomial – time important constants related to the Euclidean Dynamical System. Our algorithms are based on a method which has been previously introduced by Daudé Flajolet and Vallée in [10] and further used in [13, 32]. However, the authors did not prove the correctness of the algorithm and did not provide any complexity bound. Here, we describe a general framework where the DFV–method leads to a proven polynomial–time algorithm that computes ”spectral constants ” relative to a class of Dynamical Systems. These constants are closely related to eigenvalues of the transfer operator. Since it acts on an infinite–dimensional space, exact spectral computations are almost always impossible, and are replaced by (proven) numerical approximations. The transfer operator can be viewed as an infinite matrix M = (Mi,j)1≤i,j< ∞ which is the limit (in some precise sense) of the sequence of truncated matrices Mn: = (Mi,j)1≤i,j<n of order n where exact computations are possible. Using results of [1], we prove that each isolated eigenvalue λ of M is a limit of a sequence λn ∈ SpMn, with exponential speed. Then, coming back to the Euclidean Dynamical System, we compute (in polynomial time) three important constants which play a central rôle in the Euclidean algorithm: (i) the GaussKuzminWirsing constant related to the speed of convergence of the continued fraction algorithm to its limit density; (ii) the Hensley constant which occurs in the leading term of the variance of the number of steps of the Euclid algorithm; (iii) the Hausdorff dimension of the Cantor sets relative to constrained continued fraction expansions.
On Transfer Operators for Continued Fractions with Restricted Digits
, 2001
"... For I N, let I denote those numbers in the unit interval whose continued fraction digits all lie in I. Dene the corresponding transfer operator L I; f(z) = P n2I 1 n+z f 1 n+z for Re() > max(0; I ), where Re() = I is the abscissa of convergence of the series P n2I n . Wh ..."
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For I N, let I denote those numbers in the unit interval whose continued fraction digits all lie in I. Dene the corresponding transfer operator L I; f(z) = P n2I 1 n+z f 1 n+z for Re() > max(0; I ), where Re() = I is the abscissa of convergence of the series P n2I n . When acting on a certain Hilbert space H I; , we show that the operator L I; is conjugate to an integral operator K I; . If furthermore is real, then K I; is selfadjoint, so that L I; : H I; ! H I; has purely real spectrum. It is proved that L I; also has purely real spectrum when acting on various Hilbert or Banach spaces of holomorphic functions, on the nuclear space C ! [0; 1], and on the Frechet space C 1 [0; 1]. The analytic properties of the map 7! L I; are investigated. For certain alphabets I of an arithmetic nature (eg. I = fprimesg, I = fsquaresg, I an arithmetic progression, I the set of sums of two squares) it is shown that 7! L I; admits an analytic continuation beyond the halfplane Re() > I .
Etudes sur la récurrence de certains systèmes dynamiques topologiques et arithmétiques
, 2008
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VARIATIONS OF HAUSDORFF DIMENSION IN THE EXPONENTIAL FAMILY
"... Abstract. In this paper we deal with the following family of exponential maps (fλ: z ↦ → λ(e z − 1))λ∈[1,+∞). Denoting d(λ) the hyperbolic dimension of fλ. It is proved in [Ur,Zd 1] that the function λ ↦ → d(λ) is real analytic in (1, +∞), and in [Ur,Zd 2] that it is continuous in [1, +∞). In this p ..."
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Abstract. In this paper we deal with the following family of exponential maps (fλ: z ↦ → λ(e z − 1))λ∈[1,+∞). Denoting d(λ) the hyperbolic dimension of fλ. It is proved in [Ur,Zd 1] that the function λ ↦ → d(λ) is real analytic in (1, +∞), and in [Ur,Zd 2] that it is continuous in [1, +∞). In this paper we prove that this map is C 1 on [1, +∞), with d ′ (1 +) = 0. Moreover we prove that 8 d ′ (1 + ε) ∼ −ε 2d(1)−2 if d(1) < 3 2, d ′ (1 + ε)  � −ε log ε if d(1) = 3 2, d ′ (1 + ε)  � ε if d(1)> 3 2. In particular, if d(1) < 3, then there exists λ0> 1 such that d(λ) < d(1)