## Relative complexity of random walks in random sceneries.(English)Zbl 1258.37004

Let $${(X, \mathcal{B}, m, T)}$$ be a probability preserving transformation and $$P$$ be a $$T$$-generator. For $${n\geq1,\;\varepsilon>0}$$ and a $$T$$-invariant sub-$$\sigma$$-algebra $$\mathcal{C}\subseteq\mathcal{B}$$, the author define the following random variable $K_{\mathcal{C}}(P, n, \varepsilon):=\min\left\{\#F:\, F\subset X,\, m\left(\bigcup\limits_{z\in F}B(n, P, z, \varepsilon)\,||\,\mathcal{C}\right)>1-\varepsilon\right\},$ where ${B(n, P, z, \varepsilon)=\bigcup_{a\in P_n:\, \overline{d}^{(P)}_n(a, P_n(z))\leq\varepsilon}a},\;P_n(z)$ is defined by $z\in P_n(z)\in {P_n=\bigvee_{j=0}^{n-1}T^{-j}P},\;\overline{d}^{(P)}_n$ is a Hamming metric on $$P_n,$$ and $$m(\cdot||\,\mathcal{C})$$ denotes conditional measure with respect to $$\mathcal{C}.$$ Such a family of random variables is called the “relative complexity’of $$T$$ with respect to $$P$$ given $$\mathcal{C}$$”. At the beginning of paper the author develop the theory of relative complexity also including such notions as “$$\mathcal{C}$$-complexity sequence” and “relative entropy dimentions”. These notions are relativized versions of those in [S. Ferenczi, Isr. J. Math. 100, 189–207 (1997; Zbl 1095.28510); A. Katok and J.-P. Thouvenot, Ann. Inst. Henri Poincaré, Probab. Stat. 33, No. 3, 323–338 (1997; Zbl 0884.60009); S. Ferenczi and K. K. Park, Discrete Contin. Dyn. Syst. 17, No. 1, 133–141 (2007; Zbl 1128.37004)].
In the second part of paper, the author applies this theory to a concrete probability preserving transformation which is an $$\alpha$$-stable ($$\alpha\in (1,2]$$) random walk on ergodic random scenery. In particular, it is obtained invariants for relative isomorphism of these.

### MSC:

 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory 37A05 Dynamical aspects of measure-preserving transformations 37A50 Dynamical systems and their relations with probability theory and stochastic processes 60F05 Central limit and other weak theorems 60F17 Functional limit theorems; invariance principles

### Citations:

Zbl 1095.28510; Zbl 0884.60009; Zbl 1128.37004
Full Text:

### References:

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