### 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) |
---|---|

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 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

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

### Cite this

*Journal of Dynamical and Control Systems*,

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

}

*Journal of Dynamical and Control Systems*, vol. 7, no. 4, pp. 535-551. https://doi.org/10.1023/A:1013158601371

**Multifractal analysis of ergodic averages : A generalization of eggleston's theorem.** / Tempelman, A. A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Multifractal analysis of ergodic averages

T2 - A generalization of eggleston's theorem

AU - Tempelman, A. A.

PY - 2001/10/1

Y1 - 2001/10/1

N2 - Let V be a finite set, S be an infinite countable commutative semigroup, {τs, s ∈ S} be the semigroup of translations in the function space X = VS, A = {An} 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 Xα,f,A defined by the following formula: Xα,f,A = {x : x ∈ X, limn→∞ 1/|An| ∑s∈A(n) f(τsx) = 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 = ℤd or ℤ+d, d ≥ 1, and the sets An are infinitely increasing cubes; if B = ℝm then f(x) = (f1(x), ..., fm(x)), α = (α1, ..., αm), and Xα,f,A = {x : x ∈ X, limn→∞ 1/|An| ∑s∈A(n) f1(τsx) = = α1, ..., lim n→∞ 1/|An| ∑s∈A(n) fm(τsx) = αm}. 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].

AB - Let V be a finite set, S be an infinite countable commutative semigroup, {τs, s ∈ S} be the semigroup of translations in the function space X = VS, A = {An} 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 Xα,f,A defined by the following formula: Xα,f,A = {x : x ∈ X, limn→∞ 1/|An| ∑s∈A(n) f(τsx) = 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 = ℤd or ℤ+d, d ≥ 1, and the sets An are infinitely increasing cubes; if B = ℝm then f(x) = (f1(x), ..., fm(x)), α = (α1, ..., αm), and Xα,f,A = {x : x ∈ X, limn→∞ 1/|An| ∑s∈A(n) f1(τsx) = = α1, ..., lim n→∞ 1/|An| ∑s∈A(n) fm(τsx) = αm}. 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].

UR - http://www.scopus.com/inward/record.url?scp=0035497207&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035497207&partnerID=8YFLogxK

U2 - 10.1023/A:1013158601371

DO - 10.1023/A:1013158601371

M3 - Article

AN - SCOPUS:0035497207

VL - 7

SP - 535

EP - 551

JO - Journal of Dynamical and Control Systems

JF - Journal of Dynamical and Control Systems

SN - 1079-2724

IS - 4

ER -