arXiv subjects
Publications and source records attributed to Michal Misiurewicz.
We study metric spaces that in some sense thin out at infinity. We define and investigate a measure of sparsity that is a quasi-isometry invariant, and introduce an analogue of topological ends for sparse spaces that is also invariant under quasi-isometries. We study some 51F30examples arising in various contexts.
We estimate from below the topological entropy of the generalized Bunimovich stadium billiards. We do it for long billiard tables, and find the limit of estimates as the length goes to infinity.
For the family of Double Standard Maps $f_{a,b}=2x+a+\frac{b}π \sin2πx \quad\pmod{1}$ we investigate the structure of the space of parameters $a$ when $b=1$ and when $b\in[0,1)$. In the first case the maps have a critical point, but for a set of parameters $E_1$ of positive Lebesgue measure there is an invariant absolutely continuous measure for $f_{a,1}$. In the second case there is an open nonempty set $E_b$ of parameters for which the map $f_{a,b}$ is expanding. We show that as $b\nearrow 1$, the set $E_b$ accumulates on many points of $E_1$ in a regular way from the measure point of view.
We study the dynamics of simple congestion games with two resources where a continuum of agents behaves according to a version of Experience-Weighted Attraction (EWA) algorithm. The dynamics is characterized by two parameters: the (population) intensity of choice $a>0$ capturing the economic rationality of the total population of agents and a discount factor $σ\in [0,1]$ capturing a type of memory loss where past outcomes matter exponentially less than the recent ones. Finally, our system adds a third parameter $b \in (0,1)$, which captures the asymmetry of the cost functions of the two resources. It is the proportion of the agents using the first resource at Nash equilibrium, with $b=1/2$ capturing a symmetric network. Within this simple framework, we show a plethora of bifurcation phenomena where behavioral dynamics destabilize from global convergence to equilibrium, to limit cycles or even (formally proven) chaos as a function of the parameters $a$, $b$ and $σ$. Specifically, we show that for any discount factor $σ$ the system will be destabilized for a sufficiently large intensity of choice $a$. Although for discount factor $σ=0$ almost always (i.e., $b \neq 1/2$) the system will become chaotic, as $σ$ increases the chaotic regime will give place to the attracting periodic orbit of period 2. Therefore, memory loss can simplify game dynamics and make the system predictable. We complement our theoretical analysis with simulations and several bifurcation diagrams that showcase the unyielding complexity of the population dynamics (e.g., attracting periodic orbits of different lengths) even in the simplest possible potential games.
We estimate from below the topological entropy of the Bunimovich stadium billiards. We do it for long billiard tables, and find the limit of estimates as the length goes to infinity.
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 Cesaro attracting on the diagonal, that is, \[ \lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}f^k(x)=b \] for every $x$ in the minimal invariant interval. This solves a known open question whether there exists a nontrivial smooth map other than $x\mapsto axe^{-x}$ with centers of mass of all periodic orbits coinciding. We also study the dependence of the dynamics on the two parameters.
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.
In this paper we study the family of the Lozi maps $L_{a,b} : {\mathbb R}^2 \to {\mathbb R}^2$, $L_{a,b} = (1 + y - a|x|, bx)$, and their strange attractors $Λ_{a,b}$. We introduce the set of kneading sequences for the Lozi map and prove that it determines the symbolic dynamics for that map. We also introduce two other equivalent approaches.
We consider actions of the free semigroup with two generators on the real line, where the generators act as affine maps, one contracting and one expanding, with distinct fixed points. Then every orbit is dense in a half-line, which leads to the question whether it is, in some sense, uniformly distributed. We present answers to this question for various interpretations of the phrase ``uniformly distributed''.
This article is a short introduction to Combinatorial Dynamics for topologists.