Browsing by Author "Misiurewicz, Michał"
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Item Coarse entropy of metric spaces(Springer, 2024) Geller, William; Misiurewicz, Michał; Sawicki, Damian; Mathematical Sciences, School of ScienceCoarse geometry studies metric spaces on the large scale. The recently introduced notion of coarse entropy is a tool to study dynamics from the coarse point of view. We prove that all isometries of a given metric space have the same coarse entropy and that this value is a coarse invariant. We call this value the coarse entropy of the space and investigate its connections with other properties of the space. We prove that it can only be either zero or infinity, and although for many spaces this dichotomy coincides with the subexponential-exponential growth dichotomy, there is no relation between coarse entropy and volume growth more generally. We completely characterise this dichotomy for spaces with bounded geometry and for quasi-geodesic spaces. As an application, we provide an example where coarse entropy yields an obstruction for a coarse embedding, where such an embedding is not precluded by considerations of volume growth.Item Constant slope maps on the extended real line(Cambridge, 2017) Misiurewicz, Michał; Roth, Samuel; Mathematical Sciences, School of ScienceFor a transitive countably piecewise monotone Markov interval map we consider the question of whether there exists a conjugate map of constant slope. The answer varies depending on whether the map is continuous or only piecewise continuous, whether it is mixing or not, what slope we consider and whether the conjugate map is defined on a bounded interval, half-line or the whole real line (with the infinities included).Item Entropy locking(2018) Cosper, David; Misiurewicz, Michał; Mathematical Sciences, School of ScienceWe prove that in certain one-parameter families of piecewise continuous piecewise linear interval maps with two laps, topological entropy stays constant as the parameter varies. The proof is simple and applies to a large set of families.Item Family of chaotic maps from game theory(Taylor & Francis, 2021) Chotibut, Thiparat; Falniowski, Fryderyk; Misiurewicz, Michał; Piliouras, Georgios; Mathematical Sciences, School of ScienceFrom 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.Item Farey–Lorenz Permutations for Interval Maps(World Scientific, 2018-02) Geller, William; Misiurewicz, Michał; Mathematical Sciences, School of ScienceLorenz-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.Item Follow-the-Regularized-Leader Routes to Chaos in Routing Games(Proceedings of Machine Learning Research, 2021) Bielawski, Jakub; Chotibut, Thiparat; Falniowski, Fryderyk; Kosiorowski, Grzegorz; Misiurewicz, Michał; Piliouras, Georgios; Mathematical Sciences, School of ScienceWe study the emergence of chaotic behavior of Follow-the-Regularized Leader (FoReL) dynamics in games. We focus on the effects of increasing the population size or the scale of costs in congestion games, and generalize recent results on unstable, chaotic behaviors in the Multiplicative Weights Update dynamics to a much larger class of FoReL dynamics. We establish that, even in simple linear non-atomic congestion games with two parallel links and \emph{any} fixed learning rate, unless the game is fully symmetric, increasing the population size or the scale of costs causes learning dynamics to becomes unstable and eventually chaotic, in the sense of Li-Yorke and positive topological entropy. Furthermore, we prove the existence of novel non-standard phenomena such as the coexistence of stable Nash equilibria and chaos in the same game. We also observe the simultaneous creation of a chaotic attractor as another chaotic attractor gets destroyed. Lastly, although FoReL dynamics can be strange and non-equilibrating, we prove that the time average still converges to an \emph{exact} equilibrium for any choice of learning rate and any scale of costs.Item A fresh look at the notion of normality(SNS, 2020-12) Bergelson, Vitaly; Downarowicz, Tomasz; Misiurewicz, Michał; Mathematical Sciences, School of ScienceLet G be a countable cancellative amenable semigroup and let (Fn) be a (left) Følner sequence in G. We introduce the notion of an (Fn)-normal element of {0,1}G. When G = (N,+) and Fn={1,2,...,n}, the (Fn)-normality coincides with the classical notion. We prove that: ∙ If (Fn) is a Følner sequence in G, such that for every α∈(0,1) we have ∑nα|Fn|<∞, then almost every x∈{0,1}G is (Fn)-normal. ∙ For any Følner sequence (Fn) in G, there exists an Cham\-per\-nowne-like (Fn)-normal set. ∙ There is a natural class of "nice" Følner sequences in (N,×). There exists a Champernowne-like set which is (Fn)-normal for every nice Følner \sq. ∙ Let A⊂N be a classical normal set. Then, for any Følner sequence (Kn) in (N,×) there exists a set E of (Kn)-density 1, such that for any finite subset {n1,n2,…,nk}⊂E, the intersection A/n1∩A/n2∩…∩A/nk has positive upper density in (N,+). As a consequence, A contains arbitrarily long geometric progressions, and, more generally, arbitrarily long "geo-arithmetic" configurations of the form {a(b+ic)j,0≤i,j≤k}. ∙ For any Følner \sq\ (Fn) in (N,+) there exist uncountably many (Fn)-normal Liouville numbers. ∙ For any nice Følner sequence (Fn) in (N,×) there exist uncountably many (Fn)-normal Liouville numbers.Item Lozi-like maps(AIMS, 2018-06) Misiurewicz, Michał; Štimac, Sonja; Mathematical Sciences, School of ScienceWe 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.Item Periodic points of latitudinal maps of the m-dimensional sphere(AIMS, 2016-11) Graff, Grzegorz; Misiurewicz, Michał; Nowak-Przygodzki, Piotr; Department of Mathematical Sciences, School of ScienceLet f be a smooth self-map of the m-dimensional sphere Sm. Under the assumption that f preserves latitudinal foliations with the fibres S1, 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.Item Random Interval Homeomorphisms(2014) Alsedà, Lluís; Misiurewicz, Michał; Department of Mathematical Sciences, School of ScienceWe investigate homeomorphisms of a compact interval, applied randomly. We consider this system as a skew product with the two-sided Bernoulli shift in the base. If on the open interval there is a metric in which almost all maps are contractions, then (with mild additional assumptions) there exists a global pullback attractor, which is a graph of a function from the base to the fiber. It is also a forward attractor. However, the value of this function depends only on the past, so when we take the one-sided shift in the base, it disappears. We illustrate those phenomena on an example, where there are two piecewise linear homeomorphisms, one moving points to the right and the other one to the left.