Diagonal points having dense orbit

T. K. Subrahmonian Moothathu. "Diagonal points having dense orbit." Colloquium Mathematicae 120.1 (2010): 127-138. <http://eudml.org/doc/286234>.

@article{T2010,
abstract = {Let f: X→ X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ ℕ, $f × f² × ⋯ × f^\{m\}: X^\{m\} → X^\{m\}$ is transitive, (ii) for each m ∈ ℕ, there exists x ∈ X such that the diagonal m-tuple (x,x,...,x) has a dense orbit in $X^\{m\}$ under the action of $f × f² × ⋯ × f^\{m\}$. We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying (ii).},
author = {T. K. Subrahmonian Moothathu},
journal = {Colloquium Mathematicae},
keywords = {recurrence; (weak) mixing; subshift of finite type},
language = {eng},
number = {1},
pages = {127-138},
title = {Diagonal points having dense orbit},
url = {http://eudml.org/doc/286234},
volume = {120},
year = {2010},
}

TY - JOUR
AU - T. K. Subrahmonian Moothathu
TI - Diagonal points having dense orbit
JO - Colloquium Mathematicae
PY - 2010
VL - 120
IS - 1
SP - 127
EP - 138
AB - Let f: X→ X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ ℕ, $f × f² × ⋯ × f^{m}: X^{m} → X^{m}$ is transitive, (ii) for each m ∈ ℕ, there exists x ∈ X such that the diagonal m-tuple (x,x,...,x) has a dense orbit in $X^{m}$ under the action of $f × f² × ⋯ × f^{m}$. We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying (ii).
LA - eng
KW - recurrence; (weak) mixing; subshift of finite type
UR - http://eudml.org/doc/286234
ER -