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Title:

A little topological counterpart of Birkhoff's ergodic theorem

Description:

For a compact metric space $X$ and a continuous transformation $T: X \to X$ with at least one transitive and recurrent orbit, there is a set $M_0(T)$ of $T$-invariant probability measures on $X$ such that for a comeager set of starting points the set of limit measures is exactly $M_0(T)$.

For a compact metric space $X$ and a continuous transformation $T: X \to X$ with at least one transitive and recurrent orbit, there is a set $M_0(T)$ of $T$-invariant probability measures on $X$ such that for a comeager set of starting points the set of limit measures is exactly $M_0(T)$. Minimize

Publisher:

Mathematical Institute of the Slovak Academy of Sciences

Year of Publication:

2010-06-01T00:00:00Z

Document Type:

article

Language:

English

Subjects:

Birkhoffs ergodic theorem ; Baire category ; topological dynamics ; distribution of sequences ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:S...

Birkhoffs ergodic theorem ; Baire category ; topological dynamics ; distribution of sequences ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q Minimize

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http://www.boku.ac.at/MATH/udt/vol05/no1/92Winkler10-1.pdf

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Title:

Further Baire results on the distribution of subsequences

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This paper presents results about the distribution of subsequences which are typical in the sense of Baire categories. The first main part is concerned with sequences of the type $x_k=n_k\alpha$, $n_1 < n_2 < n_3 < \cdots$, $mod 1$. Improving a result of alбt we show that, if the quotients $q_k = n_{k+1}n_k$ satisfy $q_k \ge 1 + \varepsilon$, th...

This paper presents results about the distribution of subsequences which are typical in the sense of Baire categories. The first main part is concerned with sequences of the type $x_k=n_k\alpha$, $n_1 < n_2 < n_3 < \cdots$, $mod 1$. Improving a result of alбt we show that, if the quotients $q_k = n_{k+1}n_k$ satisfy $q_k \ge 1 + \varepsilon$, then the set of all $\alpha$ such that $(x_k)$ is uniformly distributed is of first Baire category, i.e., for generic $\alpha$ we do not have uniform distribution. Under the stronger assumption $\lim_{k \to \infty} q_k = \infty$ one even has maldistribution for generic $\alpha$, the strongest possible contrast to uniform distribution. Nevertheless, growth conditions on the $n_k$ alone do not suffice to explain various interesting phenomena. In particular, for individual sequences the situation maybe quite diverse: For $n_k=2^k$ there is a set $M$ such that for generic $\alpha$ the set of all limit measures of $(x_k)$ is exactly $M$, while for $n_k=2^k+1$ such an $M$ does not exist. For the rest of the paper we consider appropriately defined Baire spaces $S$ of subsequences. For a fixed well distributed sequence$(x_n)$ we show that there is a set $M$ of measures such that for generic $(n_k) \in S$ the set of limit measures of the subsequence $(x_{n_k})$ is exactly $M$. Minimize

Publisher:

Mathematical Institute of the Slovak Academy of Sciences

Year of Publication:

2007-06-01T00:00:00Z

Document Type:

article

Language:

English

Subjects:

Baire category ; distribution of subsequences ; $n\ ; alpha$-sequences ; well distributed sequences ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-93...

Baire category ; distribution of subsequences ; $n\ ; alpha$-sequences ; well distributed sequences ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q Minimize

DDC:

519 Probabilities & applied mathematics *(computed)*

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http://www.boku.ac.at/MATH/udt/vol02/no1/GoSchWi07.pdf

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Title:

Compactifications, Hartman functions and (weak) almost

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In this paper we investigate Hartman functions on a topological group G. Recall that (ι, C) is a group compactification of G if C is a compact group, ι: G → C is a continuous group homomorphism and ι(G) ⊆ C is dense. A bounded function f: G ↦ → C is a Hartman function if there exists a group compactification (ι, C) and F: C → C such that f = F ◦...

In this paper we investigate Hartman functions on a topological group G. Recall that (ι, C) is a group compactification of G if C is a compact group, ι: G → C is a continuous group homomorphism and ι(G) ⊆ C is dense. A bounded function f: G ↦ → C is a Hartman function if there exists a group compactification (ι, C) and F: C → C such that f = F ◦ ι and F is Riemann integrable, i.e. the set of discontinuities of F is a null set w.r.t. the Haar measure. In particular we determine how large a compactification for a given group G and a Hartman function f: G → C must be, to admit a Riemann integrable representation of f. The connection to (weakly) almost periodic functions is investigated. In order to give a systematic presentation which is self-contained to a reasonable extent, we include several separate sections on the underlying concepts such as finitely additive measures on Boolean set algebras, means on algebras of functions, integration on compact spaces, compactifications of groups and semigroups, the Riemann integral on abstract spaces, invariance of measures and means, continuous extensions of transformations and operations to compactifications, etc. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-14

Source:

http://arxiv.org/pdf/math/0510064v4.pdf

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Document Type:

text

Language:

en

DDC:

512 Algebra *(computed)*

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

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Title:

Compactifications, Hartman functions and (weak) almost periodicity

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-01-15

Source:

http://arxiv.org/pdf/math/0510064v2.pdf

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Document Type:

text

Language:

en

Subjects:

Key words ; Hartman function ; group compactification ; invariant mean ; Riemann integrable

Key words ; Hartman function ; group compactification ; invariant mean ; Riemann integrable Minimize

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Title:

Hartman functions and (weak) almost periodicity

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-01-15

Source:

http://arxiv.org/pdf/math/0510064v1.pdf

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Document Type:

text

Language:

en

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Title:

Robert F. Tichy: 50 Years --- The unreasonable effectiveness of a number theorist

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Mathematical Institute of the Slovak Academy of Sciences

Year of Publication:

2007-09-01T00:00:00Z

Document Type:

article

Language:

English

Subjects:

LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q

LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; DOAJ:Mathematics ; DOAJ:Mathematics and Statistics ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q ; LCC:Mathematics ; LCC:QA1-939 ; LCC:Science ; LCC:Q Minimize

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http://www.boku.ac.at/MATH/udt/vol02/no1/Tichy2-07.pdf

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Title:

FURTHER BAIRE RESULTS ON THE DISTRIBUTION OF SUBSEQUENCES

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Abstract. This paper presents results about the distribution of subsequences which are typical in the sense of Baire. The first main part is concerned with sequences of the type xk = nkα, n1 < n2 < n3 < · · · , mod 1. Improving a result of ˇ Salát we show that, if the quotients qk = nk+1/nk satisfy qk ≥ 1 + ε, then the set of α such that (xk) is...

Abstract. This paper presents results about the distribution of subsequences which are typical in the sense of Baire. The first main part is concerned with sequences of the type xk = nkα, n1 < n2 < n3 < · · · , mod 1. Improving a result of ˇ Salát we show that, if the quotients qk = nk+1/nk satisfy qk ≥ 1 + ε, then the set of α such that (xk) is uniformly distributed is of first Baire category, i.e. for generic α we do not have uniform distribution. Under the stronger assumption limk→ ∞ qk = ∞ one even has maldistribution for generic α, the strongest possible contrast to uniform distribution. Nevertheless, growth conditions on the nk alone do not suffice to explain various interesting phenomena. In particular, for individual sequences the situation maybe quite diverse: For nk = 2k there is a set M such that for generic α the set of all limit measures of (xk) is exactly M, while for nk = 2k + 1 such an M does not exist. For the rest of the paper we consider appropriately defined Baire spaces S of subsequences. For a fixed well distributed sequence (xn) we show that there is a set M of measures such that for generic (nk) ∈ S the set of limit measures of the subsequence (xnk) is exactly M. 1. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-01

Source:

http://arxiv.org/pdf/math/0407295v2.pdf

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Document Type:

text

Language:

en

DDC:

511 General principles of mathematics *(computed)*

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Title:

FURTHER BAIRE RESULTS ON THE DISTRIBUTION OF SUBSEQUENCES

Author:

Description:

Abstract. This paper presents results about the distribution of subsequences which are typical in the sense of Baire. The first part is concerned with sequences of the type xk = nkα, n1 < n2 < n3 < · · ·, mod 1. Improving a result of ˇ Salát we show that, if the quotients qk = nk+1/nk satisfy qk ≥ 1 + ε, then the set of α such that (xk) is unifo...

Abstract. This paper presents results about the distribution of subsequences which are typical in the sense of Baire. The first part is concerned with sequences of the type xk = nkα, n1 < n2 < n3 < · · ·, mod 1. Improving a result of ˇ Salát we show that, if the quotients qk = nk+1/nk satisfy qk ≥ 1 + ε, then the set of α such that (xk) is uniformly distributed is of first Baire category, i.e. for generic α we do not have uniform distribution. Under the stronger assumption limk→ ∞ qk = ∞ one even has maldistribution for generic α, the strongest possible contrast to uniform distribution. The second part reverses the point of view by considering appropriately defined Baire spaces S of subsequences. For a fixed well distributed sequence (xn) we show that there is a set M of measures such that for generic (nk) ∈ S the set of limit measures of the subsequence (xnk) is exactly M. 1. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-10-31

Source:

http://arxiv.org/pdf/math/0407295v1.pdf

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text

Language:

en

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Title:

FURTHER BAIRE RESULTS ON THE DISTRIBUTION OF SUBSEQUENCES

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Description:

Dedicated to Professor Robert F. Tichy on the occasion of his 50th birthday Abstract. This paper presents results about the distribution of subsequences which are typical in the sense of Baire categories. The first main part is concerned with sequences of the type xk = nkα, n1 < n2 < n3 < · · · , mod 1. Improving a result of ˇ Salát we show that...

Dedicated to Professor Robert F. Tichy on the occasion of his 50th birthday Abstract. This paper presents results about the distribution of subsequences which are typical in the sense of Baire categories. The first main part is concerned with sequences of the type xk = nkα, n1 < n2 < n3 < · · · , mod 1. Improving a result of ˇ Salát we show that, if the quotients qk = nk+1/nk satisfy qk ≥ 1 + ε, then the set of all α such that (xk) is uniformly distributed is of first Baire category, i.e. for generic α we do not have uniform distribution. Under the stronger assumption limk→ ∞ qk = ∞ one even has maldistribution for generic α, the strongest possible contrast to uniform distribution. Nevertheless, growth conditions on the nk alone do not suffice to explain various interesting phenomena. In particular, for individual sequences the situation maybe quite diverse: For nk = 2k there is a set M such that for generic α the set of all limit measures of (xk) is exactly M, while for nk = 2k + 1 such an M does not exist. For the rest of the paper we consider appropriately defined Baire spaces S of subsequences. For a fixed well distributed sequence (xn) we show that there is a set M of measures such that for generic (nk) ∈ S the set of limit measures of the subsequence (xnk) is exactly M. 1. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-10-31

Source:

http://arxiv.org/pdf/math/0407295v3.pdf

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Document Type:

text

Language:

en

DDC:

511 General principles of mathematics *(computed)*

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Title:

Compactifications, Hartman functions and (weak) almost periodicity

Author:

Description:

In this paper we investigate Hartman functions on a topological group G. Recall that (ι, C) is a group compactification of G if C is a compact group, ι: G → C is a continuous group homomorphism and ι(G) ⊆ C is dense. A bounded function f: G ↦ → C is a Hartman function if there exists a group compactification (ι, C) and F: C → C such that f = F ◦...

In this paper we investigate Hartman functions on a topological group G. Recall that (ι, C) is a group compactification of G if C is a compact group, ι: G → C is a continuous group homomorphism and ι(G) ⊆ C is dense. A bounded function f: G ↦ → C is a Hartman function if there exists a group compactification (ι, C) and F: C → C such that f = F ◦ ι and F is Riemann integrable, i.e. the set of discontinuities of F is a null set w.r.t. the Haar measure. In particular we determine how large a compactification for a given group G and a Hartman function f: G → C must be, to admit a Riemann integrable representation of f. The connection to (weakly) almost periodic functions is investigated. In order to give a systematic presentation which is self-contained to a reasonable extent, we include several separate sections on the underlying concepts such as finitely additive measures on Boolean set algebras, means on algebras of functions, integration on compact spaces, compactifications of groups and semigroups, the Riemann integral on abstract spaces, invariance of measures and means, continuous extensions of transformations and operations to compactifications, etc. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-13

Source:

http://arxiv.org/pdf/math/0510064v3.pdf

http://arxiv.org/pdf/math/0510064v3.pdf Minimize

Document Type:

text

Language:

en

Subjects:

Key words ; Hartman function ; group compactification ; invariant mean ; Riemann integrable

Key words ; Hartman function ; group compactification ; invariant mean ; Riemann integrable Minimize

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