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### Open Access Journals

DCDS-B

It is well known that a continuous piecewise monotone
interval map with positive topological entropy is semiconjugate to a
map of a constant slope and the same entropy, and if it is
additionally transitive then this semiconjugacy is actually a
conjugacy. We generalize this result to
piecewise continuous piecewise monotone interval maps, and
as a consequence, get it also for piecewise monotone graph maps.
We show that assigning to a continuous transitive piecewise
monotone map of positive entropy a map of constant slope conjugate to
it defines an operator, and show that this operator is not continuous.

DCDS

Let $T:[0,1]\to [0,1]$ be a piecewise differentiable piecewise
monotone map, and let $r>1$. It is well known that if
$|T'|\le r$ (respectively $|T'|\ge r$) then $h_{t o p}(T)\le$ log $r$ (respectively $h_{t o p}(T)\ge$ log $r$). We show that if additionally
$|T'| < r $ (respectively $ |T'| > r $) on some subinterval and $T$ is
topologically transitive then the inequalities for the entropy are
strict. We also give examples that the assumption of piecewise
monotonicity is essential, even if $T$ is continuous.
In one class of examples the dynamical dimension of the whole interval
can be made arbitrarily small.

DCDS

About 5 years ago, Dai, Zhou and Geng proved the following
result. If $X$ is a metric compact space and $f:X\to X$ a
Lipschitz continuous map, then the Hausdorff dimension of
$X$ is bounded from below by the topological entropy of
$f$ divided by the logarithm of its Lipschitz constant. We
show that this is a simple consequence of a 30 years old
Bowen's definition of topological entropy for noncompact
sets. Moreover, a modification of this definition provides
a new insight into the entropy of subshifts of finite
type.

DCDS

We investigate a class of homeomorphisms of a cylinder, with all
trajectories convergent to the cylinder base and one fixed point in
the base. Let A be a nonempty finite or countable family of
sets, each of which can be a priori an $\omega$-limit set. Then there
is a homeomorphism from our class, for which A is the family of
all $\omega$-limit sets.

keywords:
$\omega$-limit set.

DCDS

We study $C^2$-structural stability of interval maps with
negative Schwarzian. It turns out that for a dense set of
maps critical points either have trajectories
attracted to attracting periodic orbits or are
persistently recurrent. It follows that for any
structurally stable unimodal map the $\omega$-limit set
of the critical point is minimal.

DCDS

We define a broad class of piecewise smooth plane homeomorphisms which have properties similar to the properties of Lozi maps, including the existence of a hyperbolic attractor. We call those maps *Lozi-like*. For those maps one can apply our previous results on kneading theory for Lozi maps. We show a strong numerical evidence that there exist Lozi-like maps that have kneading sequences different than those of Lozi maps.

DCDS

Let $f$ be a smooth self-map of the $m$-dimensional sphere $S^m$.
Under the assumption that $f$ preserves latitudinal foliations
with the fibres $S^1$, we estimate from below the number of fixed
points of the iterates of $f$. The paper generalizes the results
obtained by Pugh and Shub and by Misiurewicz.

DCDS

We investigate a family of maps that arises from a model in economics
and game theory. It has some features similar to renormalization and
some similar to intermittency. In a one-parameter family of maps in
dimension 2, when the parameter goes to 0, the maps converge to the
identity. Nevertheless, after a linear rescaling of both space and
time, we get maps with attracting invariant closed curves. As the
parameter goes to 0, those curves converge in a strong sense to a
certain circle. We call those phenomena microdynamics. The model can
be also understood as a family of discrete time approximations to a
Brown-von Neumann differential equation.

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