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37
Sets of "NonTypical" Points Have Full Topological Entropy and Full Hausdorff Dimension
 ISRAEL J. MATH
, 2000
"... For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This ..."
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Cited by 77 (18 self)
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For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute a nontrivial mathematical application of this theory.
A nonadditive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems
 Ergodic Theory Dynam. Systems
, 1996
"... A nonadditive version of the thermodynamic formalism is developed. This allows us to obtain lower and upper bounds for the dimension of a broad class of Cantorlike sets. These are constructed with a decreasing sequence of closed sets that may satisfy no asymptotic behavior. Moreover, they are cod ..."
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Cited by 68 (17 self)
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A nonadditive version of the thermodynamic formalism is developed. This allows us to obtain lower and upper bounds for the dimension of a broad class of Cantorlike sets. These are constructed with a decreasing sequence of closed sets that may satisfy no asymptotic behavior. Moreover, they are coded by an arbitrary symbolic dynamics, and the geometry of the construction may depend on all the symbolic past. Applications include estimates of dimension for hyperbolic sets of maps that need not be differentiable.
Rigidity for quasiMöbius group actions
 J. Differential Geom
"... Abstract. If a group acts by uniformly quasiMöbius homeomorphisms on a compact Ahlfors nregular space of topological dimension n such that the induced action on the space of distinct triples is cocompact, then the action is quasisymmetrically conjugate to an action on the standard nsphere by Möb ..."
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Cited by 33 (9 self)
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Abstract. If a group acts by uniformly quasiMöbius homeomorphisms on a compact Ahlfors nregular space of topological dimension n such that the induced action on the space of distinct triples is cocompact, then the action is quasisymmetrically conjugate to an action on the standard nsphere by Möbius transformations. 1.
Wave 0trace and length spectrum on convex cocompact hyperbolic manifolds
 Comm. Anal. Geom
"... Abstract. For convex cocompact hyperbolic quotients Γ\H n+1 we obtain a formula relating the 0trace of the wave operator with the resonances and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. Then, by writing this 0trace with the length s ..."
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Cited by 25 (12 self)
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Abstract. For convex cocompact hyperbolic quotients Γ\H n+1 we obtain a formula relating the 0trace of the wave operator with the resonances and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. Then, by writing this 0trace with the length spectrum, we prove precise asymptotics of the number of closed geodesics with an effective, exponentially small error term when the dimension of the limit set of Γ is greater than n 2. 1.
ON THE DIFFERENTIAL FORM SPECTRUM OF HYPERBOLIC MANIFOLDS
, 2003
"... Abstract. We give a lower bound for the bottom of the L 2 differential form spectrum on hyperbolic manifolds, generalizing thus a wellknown result due to Sullivan and Corlette in the function case. Our method is based on the study of the resolvent associated with the Hodgede Rham Laplacian and lea ..."
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Cited by 15 (3 self)
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Abstract. We give a lower bound for the bottom of the L 2 differential form spectrum on hyperbolic manifolds, generalizing thus a wellknown result due to Sullivan and Corlette in the function case. Our method is based on the study of the resolvent associated with the Hodgede Rham Laplacian and leads to applications for the (co)homology and topology of certain classes of hyperbolic manifolds. 1.
Computing The Dimension Of Dynamically Defined Sets I: E2 and Bounded . . .
"... We present a powerful approach to computing the Hausdorff dimension of certain conformally selfsimilar sets. We illustrate this method for the dimension dim H (E 2 ) of the set E 2 , consisting of those real numbers whose continued fraction expansions contain only the digits 1 or 2. A very striking ..."
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Cited by 14 (3 self)
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We present a powerful approach to computing the Hausdorff dimension of certain conformally selfsimilar sets. We illustrate this method for the dimension dim H (E 2 ) of the set E 2 , consisting of those real numbers whose continued fraction expansions contain only the digits 1 or 2. A very striking feature of this method is that the successive approximations converge to dim(E 2 ) at a superexponential rate.
Dimension Estimates In Nonconformal Hyperbolic Dynamics
"... We establish sharp lower and upper estimates for the Hausdorff and box dimensions of invariant sets of maps that need not be conformal nor differentiable. The estimates are given in terms of the geometry of the invariant sets. Our approach is based on the thermodynamic formalism, and uses its nonadd ..."
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Cited by 10 (5 self)
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We establish sharp lower and upper estimates for the Hausdorff and box dimensions of invariant sets of maps that need not be conformal nor differentiable. The estimates are given in terms of the geometry of the invariant sets. Our approach is based on the thermodynamic formalism, and uses its nonadditive version. As a byproduct of our approach we give a new characterization of the lower and upper box dimensions.
Almost additive thermodynamic formalism: some recent developments
"... This is a survey on recent developments concerning a thermodynamic formalism for almost additive sequences of functions. While the nonadditive thermodynamic formalism applies to much more general sequences, at the present stage of the theory there are no general results concerning for example a var ..."
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Cited by 8 (2 self)
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This is a survey on recent developments concerning a thermodynamic formalism for almost additive sequences of functions. While the nonadditive thermodynamic formalism applies to much more general sequences, at the present stage of the theory there are no general results concerning for example a variational principle for the topological pressure or the existence of equilibrium or Gibbs measures, at least without further very restrictive assumptions. On the other hand, in the case of almost additive sequences it is possible to establish a variational principle and to discuss the existence and uniqueness of equilibrium and Gibbs measures, among several other results. After presenting in a selfcontained manner the foundations of the theory, the survey includes the description of three applications of the almost additive thermodynamic formalism: a multifractal analysis of Lyapunov exponents for a class of nonconformal repellers; a conditional variational principle for limits of almost additive sequences; and the study of dimension spectra that consider simultaneously limits into the future and into the past.
ETA INVARIANT AND SELBERG ZETA FUNCTION OF ODD TYPE OVER CONVEX COCOMPACT HYPERBOLIC MANIFOLDS
"... Abstract. We show meromorphic extension and give a complete description of the divisors of a Selberg zeta function of odd type Zo Γ,Σ (λ) associated to the spinor bundle Σ on an odd dimensional convex cocompact hyperbolic manifold Γ\H2n+1. As a byproduct we do a full analysis of the spectral and sc ..."
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Cited by 7 (2 self)
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Abstract. We show meromorphic extension and give a complete description of the divisors of a Selberg zeta function of odd type Zo Γ,Σ (λ) associated to the spinor bundle Σ on an odd dimensional convex cocompact hyperbolic manifold Γ\H2n+1. As a byproduct we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We show that there is a natural eta invariant η(D) associated to the Dirac operator D over a convex cocompact hyperbolic manifold Γ\H2n+1 and that exp(πiη(D)) = Zo Γ,Σ (0), thus extending Millson’s formula to this setting. Under some assumption on the exponent of convergence of Poincaré series for the group Γ, we also define an eta invariant for the odd signature operator, and we show that for Schottky 3dimensional hyperbolic manifolds it gives the argument of a holomorphic function which appears in the Zograf factorization formula relating two natural Kähler potentials for WeilPetersson metric on Schottky space. 1.