\documentclass{rae}
\usepackage{amsmath}
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\title{REAL ANALYSIS AND DYNAMICS: TRAJECTORY OF THE TURNING POINT IS
DENSE FOR A CO-$\sigma$-POROUS SET OF TENT MAPS}
\author{Zolt\'{a}n Buczolich,\thanks{Research supported by the Hungarian
National Foundation of Scientific Research Grant No. 019476 and FKFP
0189/1997.} Department of Analysis, E\"{o}tv\"{o}s Lor\'{a}nd
University, M\'{u}zeum krt. 6-8, H-1088 Budapest, Hungary, e-mail: {\tt
buczo@ludens.elte.hu} }
\markboth{Santa Barbara Symposium -- Z.~Buczolich}
{Santa Barbara Symposium -- Z.~Buczolich}
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\begin{document}\label{zi-conf-ab}
\maketitle
This is a joint work with K. M. Brucks (Milwaukee).
For
$a \in (1,2]$ set $T_a(x) = ax$ for $0 \le x \le \frac{1}{2}$ and $T_a(x) = a(1-x)$ for
$\frac{1}{2} \le x \le 1$. We refer to this family of maps as the family of {\it tent
maps}.
We restrict our attention to the parameters
$a$ from $[\sqrt{2},2]$. If $\sqrt{2} < a^m \le 2$ for some $m \in \{1, 2,
2^2, 2^3, \dots \}$, then the nonwandering set of $T_a$ consists of
$m$ disjoint closed intervals and a finite number of periodic
points [S, p.78].
Moreover, for such $a$ the map $T_a^m$ restricted to
any one of those intervals is a tent map with slope $a^m$, so is
affinely conjugate to $T_{a^m}$. Thus, getting corresponding
results for smaller parameter values is easy. We work with $T_a$
restricted to its core, $[T_a^2(\frac{1}{2}),T_a(\frac{1}{2})]$; the core
is the smallest forward invariant interval containing the turning
point $\frac{1}{2}$. The term trajectory refers to the forward
trajectory.
In [BM] it was proven that for almost every (with respect to Lebesgue measure)
$a \in [\sqrt{2},2]$, the $T_a$ trajectory of the turning point $\frac{1}{2}$
is dense in $[T_a^2(\frac{1}{2}),T_a(\frac{1}{2})]$.
Letting $\cal D$ denote those parameters $a \in [\sqrt{2},2]$ such that the closure of
the trajectory of the turning point under $T_a$ is $[T_a^2(\frac{1}{2}),T_a(\frac{1}{2})]$,
in [BB] we prove:
\medskip
{\bf Theorem}
{\it The set $[\sqrt{2},2] \setminus {\cal D}$ is $\sigma$-
porous.}
\medskip
For a detailed survey of $\sigma$-porosity we refer to
[Z], [R] and the appendix of [T].
The $\sigma$-ideal of $\sigma$-porous sets is a proper
subset of the $\sigma$-ideal of measure zero first category sets.
Therefore our theorem strengthens the result of [BM]. To obtain this
stronger result, a more delicate (refined) study of the kneading
properties of tent maps was necessary. Some of these techniques might be
of independent interest.
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\begin{thebibliography}{BM}
\bibitem[BB]{BB} K. M. Brucks and Z. Buczolich, {\it
Trajectory of the turning point is dense for a co-$\sigma$-porous
set of tent maps}, to appear.
\bibitem[BM] {BM} K. Brucks and M. Misiurewicz,
{\it Trajectory of the turning point is dense for almost all
tent maps,}
ETDS, (1996).
\bibitem [R] {R} D.~L.~Renfro, {\it A Study of Porous and $\sigma$-Porous Sets,}
Pitman Monographs and Surveys in Pure and Applied Mathematics,
Longman Publishers, to appear.
\bibitem[S] {S} S. van Strien, {\it Smooth dynamics on the interval},
New directions in Dynamical Systems, London Math. Soc. Lecture
Note Ser. {\bf 127}, Cambridge Univ. Press, Cambridge - New York,
57-119 (1988)
\bibitem [T]{T} B.~S.~Thomson,
{\it Real Functions,} Lecture Notes in Math. {\bf 1170}, Springer, New York
(1985).
\bibitem[Z]{Z} L.~Zaj\'{\i}\v{c}ek,
{\it Porosity and $\sigma$-porosity,}
Real Anal. Ex. {\bf 13}, 314-347 (1987-88).
\end{thebibliography}
\end{document}