DCDS-B
Semiconjugacy to a map of a constant slope
Lluís Alsedà Michał Misiurewicz
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.
keywords: interval Markov maps topological entropy semiconjugacy to a map of constant slope Piecewise monotonotone maps measure of maximal entropy.
DCDS
Strict inequalities for the entropy of transitive piecewise monotone maps
Michał Misiurewicz Peter Raith
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.
keywords: transitivity. piecewise monotone maps Interval maps topological entropy
DCDS
On Bowen's definition of topological entropy
Michał Misiurewicz
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.
keywords: Hausdorff dimension. Topological entropy
DCDS
Omega-limit sets for spiral maps
Bruce Kitchens Michał Misiurewicz
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
Dense set of negative Schwarzian maps whose critical points have minimal limit sets
Alexander Blokh Michał Misiurewicz
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.
keywords: negative Schwarzian Interval maps structural stability persistent recurrence.
DCDS
Lozi-like maps
Michał Misiurewicz Sonja Štimac

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.

keywords: Lozi map Lozi-like map attractor symbolic dynamics kneading theory
DCDS
Periodic points of latitudinal maps of the $m$-dimensional sphere
Grzegorz Graff Michał Misiurewicz Piotr Nowak-Przygodzki
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.
keywords: topological degree Periodic points smooth maps.
DCDS
Microdynamics for Nash maps
William Geller Bruce Kitchens Michał Misiurewicz
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.
keywords: Scaling Invariant curve.

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