237 research outputs found
Periods implying almost all periods, trees with snowflakes, and zero entropy maps
Let be a compact tree, be a continuous map from to itself,
be the number of endpoints and be the number of edges of .
We show that if has no prime divisors less than and has a
cycle of period , then has cycles of all periods greater than
and topological entropy ; so if is the least prime
number greater than and has cycles of all periods from 1 to
, then has cycles of all periods (this verifies a conjecture
of Misiurewicz for tree maps). Together with the spectral decomposition theorem
for graph maps it implies that iff there exists such that has
a cycle of period for any . We also define {\it snowflakes} for tree
maps and show that iff every cycle of is a snowflake or iff the
period of every cycle of is of form where is an odd
integer with prime divisors less than
The combinatorial Mandelbrot set as the quotient of the space of geolaminations
We interpret the combinatorial Mandelbrot set in terms of \it{quadratic
laminations} (equivalence relations on the unit circle invariant under
). To each lamination we associate a particular {\em geolamination}
(the collection of points of the circle and edges of convex
hulls of -equivalence classes) so that the closure of the set of all of
them is a compact metric space with the Hausdorff metric. Two such
geolaminations are said to be {\em minor equivalent} if their {\em minors}
(images of their longest chords) intersect. We show that the corresponding
quotient space of this topological space is homeomorphic to the boundary of the
combinatorial Mandelbrot set. To each equivalence class of these geolaminations
we associate a unique lamination and its topological polynomial so that this
interpretation can be viewed as a way to endow the space of all quadratic
topological polynomials with a suitable topology.Comment: 28 pages; in the new version a few typos are corrected; to appear in
Contemporary Mathematic
Over-rotation numbers for unimodal maps
We introduce {\it twist unimodal maps} of the interval and describe their
structure. Sufficient conditions for the growth of over-rotation interval in
families of maps are given.Comment: 30 pages; 5 figure
Social Project as a New Method Which Forms Children’s and Juvenile’s Media Literacy
We introduce a new method which forms children’s and juvenile’s media literacy – social projects. We also show the implementation of our social project named «My media safety» using the experience of press-center «VLyceum» of General Educational Municipal Budget school «Chelyabinsk Lyceum № 88».В нашей статье мы знакомим с принципиально новым инструментом развития медиакомпетенций подростков и детей – социальным проектированием. Рассказываем о реализации социального проекта «Моя медиабезопасность» на примере опыта пресс-центра «ВЛицее» МБОУ «Лицей № 88 г. Челябинска»
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