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## Colloquium Mathematicum

2011 | 125 | 1 | 55-81

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

EN

### Abstrakty

EN
In this continuation of the preceding paper (Part I), we consider a sequence $(Fₙ)_{n≥0}$ of i.i.d. random Lipschitz mappings 𝖷 → 𝖷, where 𝖷 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) $Xₙ^{x} = Fₙ ∘ ... ∘ F₁(x)$ starting at x ∈ 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 theorem and a hyperbolic extension of the space 𝖷 as well as the process $(Xₙ^{x})$.
The results are applied to a class of examples, namely, the reflected affine stochastic recursion given by $X₀^{x} = x ≥ 0$ and $Xₙ^{x} = |AₙX_{n-1}^{x} - Bₙ|$, where (Aₙ,Bₙ) is a sequence of two-dimensional i.i.d. random variables with values in ℝ⁺⁎ × ℝ⁺⁎.

55-81

wydano
2011

### Twórcy

autor
• Laboratoire de Mathématiques et Physique Théorique, Université François Rabelais Tours, Fédération Denis Poisson - CNRS, Parc de Grandmont, 37200 Tours, France
autor
• Institut für Mathematische Strukturtheorie, (Math C), Technische Universität Graz, Steyrergasse 30, A-8010 Graz, Austria