Invariant Measures and Minimal Sets of Horospherical Flows.
Invariant measures and the compactness of the domain
We consider piecewise monotonic and expanding transformations τ of a real interval (not necessarily bounded) into itself with countable number of points of discontinuity of τ’ and with some conditions on the variation which need not be a bounded function (although it is bounded on any compact interval). We prove that such transformations have absolutely continuous invariant measures. This result generalizes all previous “bounded variation” existence theorems.
Invariant measures exist under a summability condition for unimodal maps.
Invariant measures for a diffeomorphism which expands the leaves of a foliation
Invariant measures for Burgers equation with stochastic forcing.
Invariant measures for Chebyshev maps.
Invariant measures for nonexpansive Markov operators on Polish spaces [Book]
Invariant measures for piecewise convex transformations of an interval
We investigate the existence and ergodic properties of absolutely continuous invariant measures for a class of piecewise monotone and convex self-maps of the unit interval. Our assumption entails a type of average convexity which strictly generalizes the case of individual branches being convex, as investigated by Lasota and Yorke (1982). Along with existence, we identify tractable conditions for the invariant measure to be unique and such that the system has exponential decay of correlations on...
Invariant measures for position dependent random maps with continuous random parameters
We consider a family of transformations with a random parameter and study a random dynamical system in which one transformation is randomly selected from the family and applied on each iteration. The parameter space may be of cardinality continuum. Further, the selection of the transformation need not be independent of the position in the state space. We show the existence of absolutely continuous invariant measures for random maps on an interval under some conditions.
Invariant measures for random dynamical systems [Book]
Invariant measures for the defocusing Nonlinear Schrödinger equation
We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schrödinger equations on the disc of the plane . We also prove an estimate giving some intuition to what may happen in dimensions.
Invariant measures for the stable foliation on negatively curved periodic manifolds
We classify reversible measures for the stable foliation on manifolds which are infinite covers of compact negatively curved manifolds. We extend the known results from hyperbolic surfaces to varying curvature and to all dimensions.
Invariant Measures of C...-Diffeomorphisms of Markov Type in R...
Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds
Invariant measures related with randomly connected Poisson driven differential equations
We consider the stochastic differential equation (1) for t ≥ 0 with the initial condition u(0) = x₀. We give sufficient conditions for the existence of an invariant measure for the semigroup corresponding to (1). We show that the existence of an invariant measure for a Markov operator P corresponding to the change of measures from jump to jump implies the existence of an invariant measure for the semigroup describing the evolution of measures along trajectories and vice versa.
Invariant scrambled sets and maximal distributional chaos
For the full shift (Σ₂,σ) on two symbols, we construct an invariant distributionally ϵ-scrambled set for all 0 < ϵ < diam Σ₂ in which each point is transitive, but not weakly almost periodic.
Invariant sets and connecting orbits for nonlinear evolution equations at resonance
We study the problem of existence of orbits connecting stationary points for the nonlinear heat and strongly damped wave equations being at resonance at infinity. The main difficulty lies in the fact that the problems may have no solutions for general nonlinearity. To address this question we introduce geometrical assumptions for the nonlinear term and use them to prove index formulas expressing the Conley index of associated semiflows. We also prove that the geometrical assumptions are generalizations...
Invariant Sets of Anosovs Diffeomorphisms.
Invariant sets with zero measure and full Hausdorff dimension.
Invariant two-forms for geodesic flows.