Browsing by Subject "interval maps"

Now showing 1 - 4 of 4
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    Family of chaotic maps from game theory
    (Taylor & Francis, 2021) Chotibut, Thiparat; Falniowski, Fryderyk; Misiurewicz, Michał; Piliouras, Georgios; Mathematical Sciences, School of Science
    From a two-agent, two-strategy congestion game where both agents apply the multiplicative weights update algorithm, we obtain a two-parameter family of maps of the unit square to itself. Interesting dynamics arise on the invariant diagonal, on which a two-parameter family of bimodal interval maps exhibits periodic orbits and chaos. While the fixed point b corresponding to a Nash equilibrium of such map f is usually repelling, it is globally Cesàro attracting on the diagonal, that is, limn→∞1n∑n−1k=0fk(x)=b for every x∈(0,1). This solves a known open question whether there exists a ‘natural’ nontrivial smooth map other than x↦axe−x with centres of mass of all periodic orbits coinciding. We also study the dependence of the dynamics on the two parameters.
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    Farey–Lorenz Permutations for Interval Maps
    (World Scientific, 2018-02) Geller, William; Misiurewicz, Michał; Mathematical Sciences, School of Science
    Lorenz-like maps arise in models of neuron activity, among other places. Motivated by questions about the pattern of neuron firing in such a model, we study periodic orbits and their itineraries for Lorenz-like maps with nondegenerate rotation intervals. We characterize such orbits for the simplest such case and gain substantial information about the general case.
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    Spaces of transitive interval maps
    (2015-10) Kolyada, Sergiĭ; Misiurewicz, Michał; Snoha, L’ubomír
    On a compact real interval, the spaces of all transitive maps, all piecewise monotone transitive maps and all piecewise linear transitive maps are considered with the uniform metric. It is proved that they are contractible and uniformly locally arcwise connected. Then the spaces of all piecewise monotone transitive maps with given number of pieces as well as various unions of such spaces are considered and their connectedness properties are studied.
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    Special α-limit sets
    (American Mathematical Society, 2020) Kolyada, Sergiǐ; Misiurewicz, Michał; Snoha, L’ubomír
    We investigate the notion of the special α-limit set of a point. For a continuous selfmap of a compact metric space, it is defined as the union of the sets of accumulation points over all backward branches of the map. The main question is whether a special α-limit set has to be closed. We show that it is not the case in general. It is unknown even whether a special α-limit set has to be Borel or at least analytic (it is in general an uncountable union of closed sets). However, we answer this question affirmatively for interval maps for which the set of all periodic points is closed. We also give examples showing how those sets may look like and we provide some conjectures and a problem.