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37
On the topological pressure of random bundle transformations in subadditive case
 J. Math. Anal. Appl
"... Abstract. In this paper, we define the topological pressure for subadditive potentials via separated sets in random dynamical systems and we give a proof of the relativized variational principle for the topological pressure. Key words and phrases Variational principle; Topological pressure; entropy ..."
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Abstract. In this paper, we define the topological pressure for subadditive potentials via separated sets in random dynamical systems and we give a proof of the relativized variational principle for the topological pressure. Key words and phrases Variational principle; Topological pressure; entropy 1
Hyperbolicity And Recurrence In Dynamical Systems: A Survey Of Recent Results
, 2002
"... We discuss selected topics of current research interest in the theory of dynamical systems, with emphasis on dimension theory, multifractal analysis, and quantitative recurrence. The topics include the quantitative versus the qualitative behavior of Poincare recurrence, the product structure of inva ..."
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We discuss selected topics of current research interest in the theory of dynamical systems, with emphasis on dimension theory, multifractal analysis, and quantitative recurrence. The topics include the quantitative versus the qualitative behavior of Poincare recurrence, the product structure of invariant measures and return times, the dimension of invariant sets and invariant measures, the complexity of the level sets of local quantities from the point of view of Hausdorff dimension, and the conditional variational principles as well as their applications to problems in number theory.
FRACTAL WEYL LAWS FOR ASYMPTOTICALLY HYPERBOLIC MANIFOLDS
"... Abstract. For asymptotically hyperbolic manifolds with hyperbolic trapped sets we prove a fractal upper bound on the number of resonances near the essential spectrum, with power determined by the dimension of the trapped set. This covers the case of general convex cocompact quotients (including the ..."
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Abstract. For asymptotically hyperbolic manifolds with hyperbolic trapped sets we prove a fractal upper bound on the number of resonances near the essential spectrum, with power determined by the dimension of the trapped set. This covers the case of general convex cocompact quotients (including the case of connected trapped sets) where our result implies a bound on the number of zeros of the Selberg zeta function in disks of arbitrary size along the imaginary axis. Although no sharp fractal lower bounds are known, the case of quasifuchsian groups, included here, is most likely to provide them. Let M = Γ\Hn, n ≥ 2, be a convex cocompact quotient of hyperbolic space, i.e. a conformally compact manifold of constant negative curvature. Let δΓ ∈ [0, n − 1) be the Hausdorff dimension of its limit set, which by Patterson–Sullivan theory equals the abscissa of convergence of its Poincaré series [Pa, Su79]. Let ZΓ(s) be the Selberg zeta function: ZΓ(s) = exp − ∑ ∞ ∑ 1 e
Length distortion and the Hausdorff dimension of limit sets
 Amer. J. Math
"... Abstract. Let Γ be a convex cocompact quasiFuchsian Kleinian group. We define the distortion function along geodesic rays lying on the boundary of the convex hull of the limit set, where each ray is pointing in a randomly chosen direction. The distortion function measures the ratio of the intrinsi ..."
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Abstract. Let Γ be a convex cocompact quasiFuchsian Kleinian group. We define the distortion function along geodesic rays lying on the boundary of the convex hull of the limit set, where each ray is pointing in a randomly chosen direction. The distortion function measures the ratio of the intrinsic to extrinsic metrics, and is defined asymptotically as the length of the ray goes to infinity. Our main result is that the distortion function is both almost everywhere constant and bounded above by the Hausdorff dimension of the limit set of Γ. As a consequence, we are able to provide a geometric proof of the following result of Bowen: If the limit set of Γ is not a round circle, then the Hausdorff dimension of the limit set is strictly greater than one. The proofs are developed from results in PattersonSullivan theory and ergodic theory. 1. Introduction. The
DIMENSION ESTIMATES IN SMOOTH DYNAMICS: A SURVEY OF RECENT RESULTS
"... Abstract. We survey recent results in the dimension theory of dynamical systems, with emphasis on the study of repellers and hyperbolic sets of smooth dynamics. We discuss the most preeminent results in the area as well as the main difficulties in developing a general theory. Despite many interesti ..."
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Abstract. We survey recent results in the dimension theory of dynamical systems, with emphasis on the study of repellers and hyperbolic sets of smooth dynamics. We discuss the most preeminent results in the area as well as the main difficulties in developing a general theory. Despite many interesting and nontrivial developments, only the case of conformal dynamics is completely understood. On the other hand, the study of the dimension of invariant sets of nonconformal maps unveiled several new phenomena, but it still lacks today a satisfactory general approach. Indeed, we have only a complete understanding of a few classes of invariant sets of nonconformal maps satisfying certain simplifying assumptions. For example, the assumptions may ensure that there is a clear separation between different Lyapunov directions or that numbertheoretical properties do not influence the dimension. Contents
Dimension of Cantor Sets with Complicated Geometry
 Equadi 95 Proceedings
, 1998
"... We study the dimension of a broad class of Cantor sets, constructed with a decreasing sequence of closed sets that may satisfy no asymptotic behavior. They are modeled by arbitrary symbolic dynamics, and the geometry of the construction may depend on all the symbolic past. ..."
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We study the dimension of a broad class of Cantor sets, constructed with a decreasing sequence of closed sets that may satisfy no asymptotic behavior. They are modeled by arbitrary symbolic dynamics, and the geometry of the construction may depend on all the symbolic past.
The Hausdorff dimension of average conformal repellers under random perturbation
, 909
"... Abstract. We prove that the Hausdorff dimension of an average conformal repeller is stable under random perturbations. Our perturbation model uses the notion of a bundle random dynamical system. system Key words and phrases Hausdorff dimension, topological pressure, random dynamical 1 Introduction. ..."
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Abstract. We prove that the Hausdorff dimension of an average conformal repeller is stable under random perturbations. Our perturbation model uses the notion of a bundle random dynamical system. system Key words and phrases Hausdorff dimension, topological pressure, random dynamical 1 Introduction. In the dimension theory of dynamical system, only the Hausdorff dimension of invariant sets of conformal dynamical system is well understood. Since the work of Bowen, who 0
Pressures for Asymptotically Subadditive Potentials Under a Mistake Function
, 2010
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Finer Geometric Rigidity Of Limit Sets Of Conformal Ifs
, 2001
"... We consider infinite conformal iterated function systems in the phase space IR d with d 3. Let J be the limit set of such a system. Under a mild technical assumption which is always satis ed if the system is finite, we prove that either the Hausdorff dimension of J exceeds the topological dimension ..."
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We consider infinite conformal iterated function systems in the phase space IR d with d 3. Let J be the limit set of such a system. Under a mild technical assumption which is always satis ed if the system is finite, we prove that either the Hausdorff dimension of J exceeds the topological dimension k of the closure of J or else the closure of J is a proper compact subset of either a geometric sphere or an affine subspace of dimension k. A similar dichotomy holds for conformal expanding repellers.