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Stable rank and real rank of compact transformation group C*-algebras

Robert J. Archbold, Eberhard Kaniuth (2006)

Studia Mathematica

Let (G,X) be a transformation group, where X is a locally compact Hausdorff space and G is a compact group. We investigate the stable rank and the real rank of the transformation group C*-algebra C₀(X)⋊ G. Explicit formulae are given in the case where X and G are second countable and X is locally of finite G-orbit type. As a consequence, we calculate the ranks of the group C*-algebra C*(ℝⁿ ⋊ G), where G is a connected closed subgroup of SO(n) acting on ℝⁿ by rotation.

Statistical properties of unimodal maps

Artur Avila, Carlos Gustavo Moreira (2005)

Publications Mathématiques de l'IHÉS

We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition...

Statistical stability of geometric Lorenz attractors

José F. Alves, Mohammad Soufi (2014)

Fundamenta Mathematicae

We consider the robust family of geometric Lorenz attractors. These attractors are chaotic, in the sense that they are transitive and have sensitive dependence on initial conditions. Moreover, they support SRB measures whose ergodic basins cover a full Lebesgue measure subset of points in the topological basin of attraction. Here we prove that the SRB measures depend continuously on the dynamics in the weak* topology.

Stochastic Banach principle in operator algebras

Genady Ya. Grabarnik, Laura Shwartz (2007)

Studia Mathematica

The classical Banach principle is an essential tool for the investigation of ergodic properties of Cesàro subsequences. The aim of this work is to extend the Banach principle to the case of stochastic convergence in operator algebras. We start by establishing a sufficient condition for stochastic convergence (stochastic Banach principle). Then we prove stochastic convergence for bounded Besicovitch sequences, and as a consequence for uniform subsequences.

Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings

Marc Peigné, Wolfgang Woess (2011)

Colloquium Mathematicae

In this continuation of the preceding paper (Part I), we consider a sequence ( F ₙ ) n ≥ 0 of i.i.d. random Lipschitz mappings → , where is a proper metric space. We investigate existence and uniqueness of invariant measures, as well as recurrence and ergodicity of the induced stochastic dynamical system (SDS) X ₙ x = F ₙ ∘ . . . ∘ F ₁ ( x ) starting at x ∈ . The main results concern the case when the associated Lipschitz constants are log-centered. Principal tools are local contractivity, as considered in detail in Part I, the Chacon-Ornstein...

Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity

Marc Peigné, Wolfgang Woess (2011)

Colloquium Mathematicae

Consider a proper metric space and a sequence ( F ₙ ) n ≥ 0 of i.i.d. random continuous mappings → . It induces the stochastic dynamical system (SDS) X ₙ x = F ₙ ∘ . . . ∘ F ₁ ( x ) starting at x ∈ . In this and the subsequent paper, we study existence and uniqueness of invariant measures, as well as recurrence and ergodicity of this process. In the present first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying stochastic...

Stochastic effects on biodiversity in cyclic coevolutionary dynamics

Tobias Reichenbach, Mauro Mobilia, Erwin Frey (2008)

Banach Center Publications

Finite-size fluctuations arising in the dynamics of competing populations may have dramatic influence on their fate. As an example, in this article, we investigate a model of three species which dominate each other in a cyclic manner. Although the deterministic approach predicts (neutrally) stable coexistence of all species, for any finite population size, the intrinsic stochasticity unavoidably causes the eventual extinction of two of them.

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