523 research outputs found
Milnor’s Conjecture on Monotonicity of Topological Entropy: results and questions
No full text02.08.13 KB. Author has recieved permission from publisher to add the submitted version to Spiral.This note discusses Milnor’s conjecture on monotonicity of entropy and gives a short exposition of the ideas used in its proof which was obtained in joint work with Henk Bruin, see [BvS09]. At the end of this note we explore some related conjectures and questions
Real bounds, ergodicity and negative Schwarzian for multimodal maps
No full textOver the last 20 years, many of the most spectacular results in the field of
dynamical systems dealt specifically with interval and circle maps (or perturbations
and complex extensions of such maps). Primarily, this is because in the
one-dimensional case, much better distortion control can be obtained than for general
dynamical systems. However, many of these spectacular results were obtained
so far only for unimodal maps. The aim of this paper is to provide all the tools for
studying general multimodal maps of an interval or a circle, by obtaining
* real bounds controlling the geometry of domains of certain first return maps,
and providing a new (and we believe much simpler) proof of absense of
wandering intervals;
* provided certain combinatorial conditions are satisfied, large real bounds
implying that certain first return maps are almost linear;
* Koebe distortion controlling the distortion of high iterates of the map, and
negative Schwarzian derivative for certain return maps (showing that the
usual assumption of negative Schwarzian derivative is unnecessary);
* control of distortion of certain first return maps;
* ergodic properties such as sharp bounds for the number of ergodic components
Stable maps are dense in dimensional one
No full textThis is an exposition of Our resent results contained in Kozlovski et al. (Rigidity for real polynomials, preprint, 2003; Density of hyperbolicity, preprint, 2003) and Kozlovski and van Strien (Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, preprint, 2006) where we prove the density of hyperbolicity for one dimensional real maps and non-renomializable complex polynomials. The proofs of these results are very technical, so in this paper we try to show the main ideas on some simplified examples and also give some outlines of the proofs
Milnor's conjecture on monotonicity of topological entropy: Results and questions
No full textThis chapter discusses Milnor's conjecture on monotonicity of entropy and gives a short exposition of the ideas used in its proof. It discusses the history of this conjecture, gives an outline of the proof in the general case, and describes the state of the art in the subject. The proof makes use of an important result by Kozlovski, Shen, and van Strien on the density of hyperbolicity in the space of real polynomial maps, which is a far-reaching generalization of the Thurston Rigidity Theorem. (In the quadratic case, density of hyperbolicity had been proved in studies done by M. Lyubich and J. Graczyk and G. Swiatek.) The chapter concludes with a list of open problems.</p
Hamiltonian flows with random-walk behaviour originating from zero-sum games and fictitious play
No full textIn this paper we introduce Hamiltonian dynamics, inspired by zero-sum games (best response and fictitious play dynamics). The Hamiltonian functions we consider are continuous and piecewise affine (and of a very simple form). It follows that the corresponding Hamiltonian vector fields are discontinuous and multi-valued. Differential equations with discontinuities along a hyperplane are often called 'Filippov systems', and there is a large literature on such systems, see for example (di Bernardo et al 2008 Theory and applications Piecewise-Smooth Dynamical Systems (Applied Mathematical Sciences vol 163) (London: Springer); Kunze 2000 Non-Smooth Dynamical Systems (Lecture Notes in Mathematics vol 1744) (Berlin: Springer); Leine and Nijmeijer 2004 Dynamics and Bifurcations of Non-smooth Mechanical Systems (Lecture Notes in Applied and Computational Mechanics vol 18) (Berlin: Springer)). The special feature of the systems we consider here is that they have discontinuities along a large number of intersecting hyperplanes. Nevertheless, somewhat surprisingly, the flow corresponding to such a vector field exists, is unique and continuous. We believe that these vector fields deserve attention, because it turns out that the resulting dynamics are rather different from those found in more classically defined Hamiltonian dynamics. The vector field is extremely simple: outside codimension-one hyperplanes it is piecewise constant and so the flow t piecewise a translation (without stationary points). Even so, the dynamics can be rather rich and complicated as a detailed study of specific examples show (see for example theorems 7.1 and 7.2 and also (Ostrovski and van Strien 2011 Regular Chaotic Dynf. 16 129–54)). In the last two sections of the paper we give some applications to game theory, and finish with posing a version of the Palis conjecture in the context of the class of non-smooth systems studied in this paper
The dynamics of complex box mappings
No full text88 pages, 18 figuresIn holomorphic dynamics, complex box mappings arise as first return maps to well-chosen domains. They are a generalization of polynomial-like mapping, where the domain of the return map can have infinitely many components. They turned out to be extremely useful in tackling diverse problems. The purpose of this paper is: -To illustrate some pathologies that can occur when a complex box mapping is not induced by a globally defined map and when its domain has infinitely many components, and to give conditions to avoid these issues. -To show that once one has a box mapping for a rational map, these conditions can be assumed to hold in a very natural setting. Thus we call such complex box mappings dynamically natural. -Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques: (*)the Enhanced Nest by Kozlovski-Shen-van Strien; (*)the Covering Lemma by Kahn-Lyubich; (*)the QC-Criterion, the Spreading Principle. The purpose of this paper is to make these tools more accessible so that they can be used as a 'black box', so one does not have to redo the proofs in new settings. -To give an intuitive, but also rather detailed, outline of the proof of the following results by Kozlovski-van Strien for non-renormalizable dynamically natural box mappings: (*)puzzle pieces shrink to points; (*)topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate. -We prove the fundamental ergodic properties for dynamically natural box mappings. This leads to some necessary conditions for when such a box mapping supports a measurable invariant line field on its filled Julia set. These mappings are the analogues of Lattes maps in this setting. -We prove a version of Mane's Theorem for complex box mappings concerning expansion along orbits of points that avoid a neighborhood of the set of critical points
Rational maps with real multipliers
No full textLet f be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever J(f) belongs to a smooth curve, it also belongs to a circle. Then we discuss rational functions whose Julia sets belong to a circle
Fictitious play in 3 x 3 games : chaos and dithering behaviour
No full textIn the 60's Shapley provided an example of a two player fictitious game with periodic
behaviour. In this game, player A aims to copy B's behaviour and player B aims to play one
ahead of player A. In this paper we continue to study a family of games which generalize
Shapley's example by introducing an external parameter, and prove that there exists an
abundance of periodic and chaotic behavior with players dithering between different strategies. The reason for all this, is that there exists a periodic orbit (consisting of playing mixed
strategies) which is of 'jitter type': such an orbit is neither attracting, repelling or of saddle
type as nearby orbits jitter closer and further away from it in a manner which is reminiscent
of a random walk motion. We prove that this behaviour holds for an open set of games
Invariant Measures Exist Without a Growth Condition
No full textGiven a non-flat S-unimodal interval map f, we show that there exists C which only depends on the order of the critical point c such that if \Df(n) (f (c))\ greater than or equal to C for all n sufficiently large, then f admits an absolutely continuous invariant probability measure (acip). As part of the proof we show that if the quotients of successive intervals of the principal nest of f are sufficiently small, then f admits an acip. As a special case, any S-unimodal map with critical order l <2 + ε having no central returns possesses an acip. These results imply that the summability assumptions in the theorems of Nowicki & van Strien [21] and Martens & Nowicki [17] can be weakened considerably
Hub dynamics in heterogeneous networks with noise
No full textIn this thesis we study the dynamics of expanding circle maps on networked systems. In particular, we focus our attention on those networks in which the underlying structure is given by a heterogeneous graph, that is, few nodes in the networks are massively connected, while the number of connections for the remaining nodes is significantly smaller.
We show that for the massively connected nodes, the deviation between a mean field reduction and the true dynamics is a random fluctuation that converges to a normal distribution in the large system size limit.Open Acces
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