## Measures of ǫ-complexity (2008)

### BibTeX

@MISC{08measuresof,

author = {},

title = {Measures of ǫ-complexity},

year = {2008}

}

### OpenURL

### Abstract

We study some measures which are related to the notion of the ǫ-complexity. We prove that measure of ǫ-complexity defined on the base of the notion of ǫ-separability is equivalent to the dual measure that is defined through ǫ-nets.

### Citations

954 |
Introduction to the modern theory of dynamical systems. With a supplementary chapter by Katok and Leonardo
- Hasselblatt, Katok
- 1995
(Show Context)
Citation Context ...sitive eigenvector e with eigenvalue λ > 0 (in our case, in fact, λ > 1). Let P be the set of all lines in R p , generated by non-negative vectors. From the proof of Perron Theorem (see, for example, =-=[8]-=-) ⋂ n∈N M n (P) = {le}, where le is a line, generated by e. Since v(n) > 0 and v(k) = M n v(n + k), one has lv(k) ∈ M n (P) for any n. Hence, v(k) = cke. So, v(n) = λ −n c0e. We have proved the follow... |

348 | Measure theory - Halmos - 1950 |

114 |
Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation
- Furstenberg
- 1967
(Show Context)
Citation Context ...he full shift. Thus, lnCǫ − ln ǫ ≈ ln2 h = = dimH K, − ln1/3 ln λ where dimH K is the Hausdorff dimension of K and λ = 1/3 is the contraction coefficient. We obtained the familiar Furstenberg formula =-=[5]-=-. This example shows that if a subset of a metric space is the result of an inductive procedure governed by a symbolic dynamical system then the ǫ-complexity contains, in fact, an important dynamical ... |

56 |
Topological entropy for noncompact sets
- Bowen
- 1973
(Show Context)
Citation Context ...he phase space (see definition bellow). If one has a dynamical system generated by a continuous map f : X → X where X is a metric space with a distance ρ, one can introduce the sequence of distances (=-=[3]-=-) ρn(x,y) = max 0≤i≤n−1 ρ(fix, f i y), n ∈ N, and study the ǫ-complexity with respect to the distance ρn as a function of “time” n. This function reflects the evolution of instability of orbits in tim... |

25 |
ǫ–entropy and ǫ–capacity of sets in function spaces
- Kolmogorov, Tikhomirov
- 1959
(Show Context)
Citation Context ...plexity in an “abstract” metric space. The main results will be related to the ǫ-complexity defined on the base of the notion of ǫ-separability. The notion was used first by Kolmogorov and Tikhomirov =-=[9]-=- in their study of solutions of PDE and realization of random processes (Shannon suggested to pay attention to this notions in 1949, though). We will also study ǫ-complexities based on the notion of ǫ... |

24 |
Elements of Mathematics. General Topology. Part 1
- Bourbaki
- 1966
(Show Context)
Citation Context ...). It is not difficult to check that bǫ ≤ 2 d (2 d + 1) for a subset of the Euclidean space R d . 22.3 Ultrafilters Now we give some known results and definitions that can be found, for instance, in =-=[6]-=-. Definition 3 A set F ⊂ 2 N is called to be a filter over N iff it satisfies the following conditions: • If A ∈ F and B ∈ F, then A ∩ B ∈ F, • If A ∈ F and A ⊂ B then B ∈ F, • ∅ ̸∈ F. Let an be a seq... |

20 |
Complexity of sequences and dynamical systems, Discrete Math
- FERENCZI
- 1999
(Show Context)
Citation Context ...duction The problems under consideration in this article were originated in the process of study of complexity of behavior of orbits in dynamical systems. While symbolic complexity (see, for instance =-=[4]-=-) deals with symbolic systems and topological complexity ([2]) reflects pure topological features of dynamics, the ǫ-complexity depends essentially on a distance in the phase space (see definition bel... |

12 |
Combinatorial Mathematics,” The Carus
- Ryser
- 1963
(Show Context)
Citation Context ...lt due to the choice of the ultrafilter F. So, we need only to prove Proposition 6; it will be done below. ✷ In the proof of Proposition 6 we will need the Marriage Lemma of P. Hall, see for instance =-=[10]-=-. Lemma 1 For an indexed collections of finite sets F1, F2, . . . , Fk the following conditions are equivalent: x∈Aǫ • there exists an injective function α : {1, 2, ..., k} → k⋃ Fi such that α(i) ∈ Fi... |

3 |
Topological complexity, Ergod. Theory Dyn
- Blanchard, Host, et al.
(Show Context)
Citation Context ... originated in the process of study of complexity of behavior of orbits in dynamical systems. While symbolic complexity (see, for instance [4]) deals with symbolic systems and topological complexity (=-=[2]-=-) reflects pure topological features of dynamics, the ǫ-complexity depends essentially on a distance in the phase space (see definition bellow). If one has a dynamical system generated by a continuous... |

2 |
and G.M.Zaslavsky, Space-time complexity in Hamiltonian dynamics
- Afraimovich
(Show Context)
Citation Context ...ρn(x,y) = max 0≤i≤n−1 ρ(fix, f i y), n ∈ N, and study the ǫ-complexity with respect to the distance ρn as a function of “time” n. This function reflects the evolution of instability of orbits in time =-=[1]-=-. But to study it in details, one needs to know more about general properties of the ǫ-complexity of a metric space (without dynamics). The goal this article is to introduce and study quantities which... |