### Abstract

Let V be a finite set, S be an infinite countable commutative semigroup, {τ_{s}, s ∈ S} be the semigroup of translations in the function space X = V^{S}, A = {A_{n}} be a sequence of finite sets in S, f be a continuous function on X with values in a separable real Banach space B, and let α ∈ B. We introduce in X a "scale metric" generating the product topology. Under some assumptions on f and A, we evaluate the Hausdorff dimension of the set X_{α,f,A} defined by the following formula: X_{α,f,A} = {x : x ∈ X, lim_{n→∞} 1/|A_{n}| ∑_{s∈A(n)} f(τ_{s}x) = α}. It turns out that this dimension does not depend on the choice of a Følner "pointwise averaging" sequence A and is completely specified by the "scale index" of the metric in X. This general model includes the important special cases where S = ℤ^{d} or ℤ_{+}^{d}, d ≥ 1, and the sets A_{n} are infinitely increasing cubes; if B = ℝ^{m} then f(x) = (f_{1}(x), ..., f_{m}(x)), α = (α_{1}, ..., α_{m}), and X_{α,f,A} = {x : x ∈ X, lim_{n→∞} 1/|A_{n}| ∑_{s∈A(n)} f_{1}(τ_{s}x) = = α_{1}, ..., lim _{n→∞} 1/|A_{n}| ∑_{s∈A(n)} f_{m}(τ_{s}x) = α_{m}}. Thus the multifractal analysis of the ergodic averages of several continuous functions is a special case of our results; in particular, in Examples 4 and 5 we generalize the well-known theorems due to Eggleston [3] and Billingsley [1].

Original language | English (US) |
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Pages (from-to) | 535-551 |

Number of pages | 17 |

Journal | Journal of Dynamical and Control Systems |

Volume | 7 |

Issue number | 4 |

DOIs | |

State | Published - Oct 1 2001 |

### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Algebra and Number Theory
- Numerical Analysis
- Control and Optimization

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## Cite this

*Journal of Dynamical and Control Systems*,

*7*(4), 535-551. https://doi.org/10.1023/A:1013158601371