Hyperbolic Measure Research Articles

Toeplitz Operators on Generalized Bergman Spaces

We consider the weighted Bergman spaces $${\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}$$ , where we set $${d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}$$ , with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.

On the distribution of periodic orbits

Let f : $M\to M$ be a $C^{1+\varepsilon}$-map on a smooth Riemannian manifold $M$ and let $\Lambda\subset M$ be a compact $f$-invariant locally maximal set. In this paper we obtain several results concerning the distribution of theperiodic orbits of $f|_\Lambda$. These results are non-invertible and, in particular, non-uniformly hyperbolic versions of well-known results by Bowen, Ruelle, and others in the case of hyperbolic diffeomorphisms. We show that the topological pressure Ptop$(\varphi)$ can be computed by the values of the potential $\varphi$ on the expanding periodic orbits and also that every hyperbolic ergodic invariant measure is well-approximated by expanding periodic orbits. Moreover, we provethat certain equilibrium states are Bowen measures. Finally, we derive a large deviation resultfor the periodic orbits whose time averages are apart from the space average of a given hyperbolic invariant measure.

Quasi-nearly subharmonic functions in locally uniformly homogeneous spaces

We define nonnegative quasi-nearly subharmonic functions on so called locally uniformly homogeneous spaces. We point out that this function class is rather general. It includes quasi-nearly subharmonic (thus also subharmonic, quasisubharmonic and nearly subharmonic) functions on domains of Euclidean spaces \({{\mathbb{R}}^n}\), n ≥ 2. In addition, quasi-nearly subharmonic functions with respect to various measures on domains of \({{\mathbb{R}}^n}\), n ≥ 2, are included. As examples we list the cases of the hyperbolic measure on the unit ball Bn of \({{\mathbb{R}}^n}\), the \({{\mathcal{M}}}\)-invariant measure on the unit ball B2n of \({{\mathbb{C}}^n}\), n ≥ 1, and the quasihyperbolic measure on any domain \({D\subset {\mathbb{R}}^n}\), \({D\ne {\mathbb{R}}^n}\). Moreover, we show that if u is a quasi-nearly subharmonic function on a locally uniformly homogeneous space and the space satisfies a mild additional condition, then also up is quasi-nearly subharmonic for all p > 0.

Hyperbolic measures, moments and coefficients. Algebra on hyperbolic functions

With convolutions, we determine the Fourier transform of u n [ sinh ( u ) ] n when n is a positive integer. Studying the expansion and taking the Fourier transform of [ u [ sinh ( u ) ] ] d [ u tanh ( u ) ] n when n and d are strictly positive integers, we obtain some polynomials and new probability densities related to them.

Approximation properties on invariant measure and Oseledec splitting in non-uniformly hyperbolic systems

We prove that each invariant measure in a non-uniformly hyperbolic system can be approximated by atomic measures on hyperbolic periodic orbits. This contributes to our main result that the mean angle (Definition 1.10), independence number (Definition 1.6) and Oseledec splitting for an ergodic hyperbolic measure with simple spectrum can be approximated by those for atomic measures on hyperbolic periodic orbits, respectively. Combining this result with the approximation property of Lyapunov exponents by Wang and Sun, 2005 (Theorem 1.9), we strengthen Katok's closing lemma (1980) by presenting more extensive information not only about the state system but also its linearization. In the present paper, we also study an ergodic theorem and a variational principle for mean angle, independence number and Liao's style number (Definition 1.3) which are bases for discussing the approximation properties in the main result.

Equality of pressures for diffeomorphisms preserving hyperbolic measures

For a diffeomorphism which preserves a hyperbolic measure the potential $${\varphi^{u} = - \log|{\rm Jac}\,df|_{E^u}|}$$ is studied. Various types of pressure of $${\varphi^u}$$ are introduced. It is shown that these pressures satisfy a corresponding variational principle.

Invariance of Picard Dimensions Under Basic Perturbations

The Picard dimension dim μ of a signed Radon measure μ on the punc- tured closed unit ball 0 < |x| 1 in the d-dimensional euclidean space with d 2 is the cardinal number of the set of extremal rays of the cone of positive continuous distributional solutions u of the Schrodinger equation (−� + μ)u = 0 on the punc- tured open unit ball 0 < |x| < 1 with vanishing boundary values on the unit sphere |x |= 1. If the Green function of the above equation on 0 < |x| < 1 characterized as the minimal positive continuous distributional solution of (−� + μ)u = δy ,t he Dirac measure supported by the point y, exists for every y in 0 < |x| < 1 ,t henμ is referred to as being hyperbolic on 0 < |x| < 1. A basic perturbation γ is a radial Radon measure which is both positive and absolutely continuous with respect to the d-dimensional Lebesgue measure dx whose Radon-Nikodym density d γ( x)/dx is bounded by a positive constant multiple of |x| −2 . The purpose of this paper is to show that the Picard dimensions of hyperbolic radial Radon measures μ are invariant under basic perturbations γ : dim(μ + γ) = dim μ. Three applications of this invariance are also given.

Ergodic Measures of SRB Attractors

In the stochastic context, an invariant set is decomposed into the union of ergodic basins, and each of basin possesses the fractal structure determined by ergodic measures. This paper is to show that when a hyperbolic SRB measure is mixing, the set of measures with zero entropy and the set of measures with positive entropy but without SRB are both dense on the set of all invariant measures on the closure of the ergodic basin in the Pesin set, and moreover that in the set of invariant measures as above a measure with ergodicity and SRB exists uniquely.

Hyperbolicity, heterodimensional cycles and Lyapunov exponents for partially hyperbolic dynamics

We prove a dichotomy of C2 partially hyperbolic sets with one-dimensional center direction admitting no zero Lyapunov exponents, either hyperbolicity over the supports of ergodic measures or approximation by a heterodimensional cycle. This provides a partial result to the C1 Palis Conjecture that claims a dichotomy, hyperbolicity or homoclinic bifurcations in a dense subset of the space of C1 diffeomorphisms. Moreover, a theorem of Mane applied in the proof is modified to have an additional property concerning the Hausdorff distance between a periodic orbit and the support of a hyperbolic ergodic measure.

Dirichlet problem at infinity on Gromov hyperbolic metric measure spaces

We study the asymptotic Dirichlet problem for p-harmonic functions in a very general setting of Gromov hyperbolic metric measure spaces.

Vanishing of the first reduced cohomology with values in an L^p-representation

We prove that the first reduced cohomology with values in a mixing L p -representation, 1<p<∞, vanishes for a class of amenable groups including connected amenable Lie groups. In particular this solves for this class of amenable groups a conjecture of Gromov saying that every finitely generated amenable group has no first reduced ℓ p -cohomology. As a byproduct, we prove a conjecture by Pansu. Namely, the first reduced L p -cohomology on homogeneous, closed at infinity, Riemannian manifolds vanishes. We also prove that a Gromov hyperbolic geodesic metric measure space with bounded geometry admitting a bi-Lipschitz embedded 3-regular tree has non-trivial first reduced L p -cohomology for large enough p. Combining our results with those of Pansu, we characterize Gromov hyperbolic homogeneous manifolds: these are the ones having non-zero first reduced L p -cohomology for some 1<p<∞.

Dimension of hyperbolic measures of random diffeomorphisms

We consider dynamics of compositions of stationary random C 2 diffeomorphisms. We will prove that the sample measures of an ergodic hyperbolic invariant measure of the system are exact dimensional. This is an extension to random diffeomorphisms of the main result of Barreira, Pesin and Schmeling (1999), which proves the Eckmann-Ruelle dimension conjecture for a deterministic diffeomorphism.

On hyperbolic measures and periodic orbits

We prove that if a diffeomorphism on a compact manifold preserves a nonatomic ergodic hyperbolic Borel probability measure, then there exists a hyperbolic periodic point such that the closure of its unstable manifold has positive measure. Moreover, the support of the measure is contained in the closure of all such hyperbolic periodic points. We also show that if an ergodic hyperbolic probability measure does not locally maximize entropy in the space of invariant ergodic hyperbolic measures, then there exist hyperbolic periodic points that satisfy a multiplicative asymptotic growth and are uniformly distributed with respect to this measure.

Conformal and harmonic measures on laminations associated with rational maps

Introduction Affine and hyperbolic laminations Measures and currents on laminations Laminations associated with rational maps Measures on laminations associated with rational maps Appendix A. Laminations associated with Kleinian groups List of notations Bibliography.

On the asymptotic distribution of zeros of modular forms

We study the distribution of zeros of holomorphic modular forms. Assuming the Generalized Riemann Hypothesis we show that the zeros of Hecke eigenforms for the modular group become equidistributed with respect to the hyperbolic measure on the modular domain as the weight grows.

Periodic probability measures are dense in the set of invariant measures

We show that if a mixing diffeomorphism of a compact manifold preserves an ergodic hyperbolic probability measure, then the measures supported by hyperbolic periodic points are dense in the set of invariant measures. This is a generalization of the result shown by Sigmund.

Hyperbolic efficiency and return to the dollar

This paper, after establishing the relations between hyperbolic graph measure of technical efficiency and the radial measures of technical efficiency, shows that the dual, cost and revenue interpretation of the hyperbolic efficiency measure is related to Georgescu-Roegen's notion of “return to the dollar” [N. Georgescu-Roegen, in: Koopmans, T. (Ed.), Activity Analysis of Production and Allocation, Wiley, New York, 1951, pp. 98–115]. Once this relation is established, it leads to a derivation of an allocative efficiency index, which measures the price distortions using data on observed costs and revenues without requiring information on prices.

Product structure of Poincaré recurrence

We provide new non-trivial quantitative information on the behavior of Poincare recurrence. In particular we establish the almost everywhere coincidence of the recurrence rate and of the pointwise dimension for a large class of repellers, including repellers without finite Markov partitions.Using this information, we are able to show that for locally maximal hyperbolic sets the recurrence rate possesses a certain local product structure, which closely imitates the product structure provided by the families of local stable and unstable manifolds, as well as the almost product structure of hyperbolic measures.

Graph efficiency and productivity measures: an application to US agriculture

Hyperbolic measures of efficiency and productivity change with respect to a graph representation of production technology allow researchers to consider output and input dimensions simultaneously in measuring producer performance. Hyperbolic efficiency measures have been proposed, but empirical implementation has not followed, either in efficiency analysis or in productivity analysis. The objectives of this paper are to define hyperbolic performance measures on a graph representation of production technology, to motivate their use by stating some of their advantages over their radial counterparts, and to introduce a direct formulation to calculate them making use of Data Envelopment Analysis techniques. The ideas are illustrated by calculating hyperbolic efficiency and Malmquist productivity indexes for a US agricultural panel data set.

MULTIFRACTAL SPECTRUM AND THERMODYNAMICAL FORMALISM OF THE FAREY TREE

Let (Ω, μ) be a set of real numbers to which we associate a measure μ. Let α≥0, let Ωα={x∈Ω/α(x)=α}, where α is the concentration index defined by Halsey et al. [1986]. Let fH(α) be the Hausdorff dimension of Ωα. Let fL(α) be the Legendre spectrum of Ω, as defined in [Riedi &amp; Mandelbrot, 1998]; and fC(α) the classical computational spectrum of Ω, defined in [Halsey et al., 1986]. The task of comparing fH, fCand fLfor different measures μ was tackled by several authors [Cawley &amp; Mauldin, 1992; Mandelbrot &amp; Riedi, 1997; Riedi &amp; Mandelbrot, 1998] working, mainly, on self-similar measures μ. The Farey tree partition in the unit segment induces a probability measure μ on an universal class of fractal sets Ω that occur in physics and other disciplines. This measure μ is the Hyperbolic measure μℍ, fundamentally different from any self-similar one. In this paper we compare fH, fCand fLfor μℍ.