Periods implying almost all periods, trees with snowflakes, and zero entropy maps

Let $X$ be a compact tree, $f$ be a continuous map from $X$ to itself, $End(X)$ be the number of endpoints and $Edg(X)$ be the number of edges of $X$. We show that if $n>1$ has no prime divisors less than $End(X)+1$ and $f$ has a cycle of period $n$, then $f$ has cycles of all periods greater than $2End(X)(n-1)$ and topological entropy $h(f)>0$; so if $p$ is the least prime number greater than $End(X)$ and $f$ has cycles of all periods from 1 to $2End(X)(p-1)$, then $f$ 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 $h(f)>0$ iff there exists $n$ such that $f$ has a cycle of period $mn$ for any $m$. We also define {\it snowflakes} for tree maps and show that $h(f)=0$ iff every cycle of $f$ is a snowflake or iff the period of every cycle of $f$ is of form $2^lm$ where $m\le Edg(X)$ is an odd integer with prime divisors less than $End(X)+1$

Cubic Critical Portraits and Polynomials with Wandering Gaps

Thurston introduced \si_d-invariant laminations (where \si_d(z) coincides with z^d:\ucirc\to \ucirc, $d\ge 2$) and defined \emph{wandering $k$-gons} as sets \T\subset \ucirc such that \si_d^n(\T) consists of $k\ge 3$ distinct points for all $n\ge 0$ and the convex hulls of all the sets \si_d^n(\T) in the plane are pairwise disjoint. He proved that \si_2 has no wandering $k$-gons. Call a lamination with wandering $k$-gons a \emph{WT-lamination}. In a recent paper it was shown that uncountably many cubic WT-laminations, with pairwise non-conjugate induced maps on the corresponding quotient spaces $J$, are realizable as cubic polynomials on their (locally connected) Julia sets. In the present paper we use a new approach to construct cubic WT-laminations with all of the above properties and the extra property that the corresponding wandering branch point of $J$ has a dense orbit in each subarc of $J$ (we call such orbits \emph{condense}), and to show that critical portraits corresponding to such laminations are uncountably dense in the space \A_3 of all cubic critical portraits.Comment: 31 pages, 4 figures; this is the last, third version of the paper which is to appear in Ergodic Theory and Dynamical System

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 $\sim$ on the unit circle invariant under $\sigma_2$). To each lamination we associate a particular {\em geolamination} (the collection $\mathcal{L}_\sim$ of points of the circle and edges of convex hulls of $\sim$-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

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»

Ground increase of cosmic ray intensity on February 16, 1984

The event of February 16, 1984 is one of the two largest ground increases of solar cosmic rays (CR) in the last two cycles of solar activity. This event happended at a decrease of the 21-st cycle against a quiet background. Although at the beginning of 1984 the observed indices of solar activity were higher than those at the end of 1983, the day of February 16 16 may be characterized as very quiet. On that day the geomagnetic perturbance (Sigma F sub p = 14, A sub p = 7) was the lowest in February. After a small Forbush decrease due to the magnetic storm of February 12-13, the CR intensity almost completely recovered by February 16. Thus, the solar particles that came to the Earth on February 16 got into a practically unperturbed magnetosphere, and the variations of secondary CR induced by these particles were not superimposed on any other substantial variations of extraterrestrial or magnetospheric origin