scholarly journals Gibbs measures for foliated bundles with negatively curved leaves

2016 ◽  
Vol 38 (4) ◽  
pp. 1238-1288 ◽  
Author(s):  
SÉBASTIEN ALVAREZ
Keyword(s):  
Invariant Measure ◽  
Gibbs Measure ◽  
Geodesic Flow ◽  
Gibbs Measures ◽  
Holonomy Group ◽  
Bijective Correspondence ◽  
Weighted Averages ◽  
Negatively Curved ◽  
Harmonic Measures ◽  
Foliated Bundle

In this paper we develop a notion of Gibbs measure for the geodesic flow tangent to a foliated bundle over a compact negatively curved base. We also develop a notion of$F$-harmonic measure and prove that there exists a natural bijective correspondence between these two concepts. For projective foliated bundles with$\mathbb{C}\mathbb{P}^{1}$-fibers without transverse invariant measure, we show the uniqueness of these measures for any Hölder potential on the base. In that case we also prove that$F$-harmonic measures are realized as weighted limits of large balls tangent to the leaves and that their conditional measures on the fibers are limits of weighted averages on the orbits of the holonomy group.

scholarly journals Harmonic measures and the foliated geodesic flow for foliations with negatively curved leaves

2014 ◽  
Vol 36 (2) ◽  
pp. 355-374 ◽  
Author(s):  
SÉBASTIEN ALVAREZ
Keyword(s):  
Gibbs Measure ◽  
Geodesic Flow ◽  
Bijective Correspondence ◽  
Negatively Curved ◽  
Harmonic Measures

In this paper we define a notion of Gibbs measure for the geodesic flow tangent to a foliation with negatively curved leaves and associated to a particular potential $H$. We prove that there is a canonical bijective correspondence between these measures and Garnett’s harmonic measures.

scholarly journals Generic points of shift-invariant measures in the countable symbolic space

2018 ◽  
Vol 166 (2) ◽  
pp. 381-413
Author(s):  
AI–HUA FAN ◽  
MING–TIAN LI ◽  
JI–HUA MA
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Invariant Measure ◽  
Gibbs Measure ◽  
Relative Entropy ◽  
Invariant Measures ◽  
Gibbs Measures ◽  
Symbolic Space ◽  
Entropy Dimension ◽  
Generic Points

AbstractWe are concerned with sets of generic points for shift-invariant measures in the countable symbolic space. We measure the sizes of the sets by the Billingsley-Hausdorff dimensions defined by Gibbs measures. It is shown that the dimension of such a set is given by a variational principle involving the convergence exponent of the Gibbs measure and the relative entropy dimension of the Gibbs measure with respect to the invariant measure. This variational principle is different from that of the case of finite symbols, where the convergent exponent is zero and is not involved. An application is given to a class of expanding interval dynamical systems.

scholarly journals ON INHOMOGENEOUS p-ADIC POTTS MODEL ON A CAYLEY TREE

2005 ◽  
Vol 08 (02) ◽  
pp. 277-290 ◽  
Author(s):  
FARRUKH MUKHAMEDOV ◽  
UTKIR ROZIKOV
Keyword(s):  
Gibbs Measure ◽  
Potts Model ◽  
Cayley Tree ◽  
Nearest Neighbor ◽  
Gibbs Measures ◽  
Nearest Neighbors ◽  
Translation Invariant

We consider a nearest-neighbor inhomogeneous p-adic Potts (with q≥2 spin values) model on the Cayley tree of order k≥1. The inhomogeneity means that the interaction Jxy couplings depend on nearest-neighbors points x, y of the Cayley tree. We study (p-adic) Gibbs measures of the model. We show that (i) if q∉pℕ then there is unique Gibbs measure for any k≥1 and ∀ Jxy with | Jxy |< p-1/(p -1). (ii) For q∈p ℕ, p≥3 one can choose Jxy and k≥1 such that there exist at least two Gibbs measures which are translation-invariant.

Conditional measure and flip invariance of Bowen-Margulis and harmonic measures on manifolds of negative curvature

1995 ◽  
Vol 15 (4) ◽  
pp. 807-811 ◽  
Author(s):  
Chengbo Yue
Keyword(s):  
Harmonic Measure ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Conditional Measure ◽  
Rigidity Result ◽  
Harmonic Measures

AbstractKifer and Ledrappier have asked whether the harmonic measures {νx} on manifolds of negative curvature are equivalent to the conditional measures of the harmonic measure v of the geodesic flow associated with the fibration {SxM}x∈M. We settle this question with a rigidity result. We also clear up the same problem concerning the Patterson-Sullivan measure and the Bowen–Margulis measure.

scholarly journals Escape rates for Gibbs measures

2011 ◽  
Vol 32 (3) ◽  
pp. 961-988 ◽  
Author(s):  
ANDREW FERGUSON ◽  
MARK POLLICOTT
Keyword(s):  
Hausdorff Dimension ◽  
Asymptotic Behaviour ◽  
Gibbs Measure ◽  
Gibbs Measures ◽  
Small Hole ◽  
Escape Rate ◽  
Conformal Repeller

AbstractIn this paper we study the asymptotic behaviour of the escape rate of a Gibbs measure supported on a conformal repeller through a small hole. There are additional applications to the convergence of the Hausdorff dimension of the survivor set.

scholarly journals Periodic Gibbs measures for models with uncountable set of spin values on a Cayley tree

2015 ◽  
Vol 18 (01) ◽  
pp. 1550006 ◽  
Author(s):  
U. A. Rozikov ◽  
F. H. Haydarov
Keyword(s):  
Gibbs Measure ◽  
Cayley Tree ◽  
Nearest Neighbor ◽  
Gibbs Measures ◽  
Sufficient Condition ◽  
Translation Invariant ◽  
Periodic Gibbs Measure

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. We show that periodic Gibbs measures are either translation-invariant or periodic with period two. We describe two-periodic Gibbs measures of the model. For k = 1 we show that there is no any periodic Gibbs measure. In case k ≥ 2 we get a sufficient condition on Hamiltonian of the model with uncountable set of spin values under which the model has no periodic Gibbs measure. We construct several models which have at least two periodic Gibbs measures.

scholarly journals Dynamics of the geodesic flow of a foliation

1988 ◽  
Vol 8 (4) ◽  
pp. 637-650 ◽  
Author(s):  
Paweł G. Walczak
Keyword(s):  
Riemannian Manifold ◽  
Lyapunov Exponents ◽  
Geodesic Flow ◽  
Smooth Measure ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Foliated Riemannian Manifold

AbstractThe geodesic flow of a foliated Riemannian manifold (M, F) is studied. The invariance of some smooth measure is established under some geometrical conditions on F. The Lyapunov exponents and the entropy of this flow are estimated. As an application, the non-existence of foliations with ‘short’ second fundamental tensors is obtained on compact negatively curved manifolds.

scholarly journals Recurrence rates for observations of flows

2011 ◽  
Vol 32 (5) ◽  
pp. 1727-1751 ◽  
Author(s):  
JÉRÔME ROUSSEAU
Keyword(s):  
Lower Bound ◽  
Invariant Measure ◽  
Upper Bound ◽  
Geodesic Flow ◽  
Suspension Flow ◽  
Local Dimension ◽  
Anosov Flow ◽  
Recurrence Rates ◽  
Forward Measure ◽  
Push Forward

AbstractWe study Poincaré recurrence for flows and observations of flows. For Anosov flow, we prove that the recurrence rates are linked to the local dimension of the invariant measure. More generally, we give for the recurrence rates for the observations an upper bound depending on the push-forward measure. When the flow is metrically isomorphic to a suspension flow for which the dynamic on the base is rapidly mixing, we prove the existence of a lower bound for the recurrence rates for the observations. We apply these results to the geodesic flow and we compute the recurrence rates for a particular observation of the geodesic flow, i.e. the projection on the manifold.

scholarly journals Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés

1989 ◽  
Vol 9 (3) ◽  
pp. 433-453 ◽  
Author(s):  
Y. Guivarc'h
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Compact Manifolds ◽  
Constant Negative Curvature ◽  
Skew Products ◽  
Ergodic Properties ◽  
Anosov Systems

AbstractWe study the ergodic properties of a class of dynamical systems with infinite invariant measure. This class contains skew-products of Anosov systems with ℝd. The results are applied to theKproperty of skew-products and also to the ergodicity of the geodesic flow on abelian coverings of compact manifolds with constant negative curvature.

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