scholarly journals Renormalization for Lorenz maps of monotone combinatorial types

2017 ◽  
Vol 39 (1) ◽  
pp. 132-158
Author(s):  
DENIS GAIDASHEV
Keyword(s):  
Critical Point ◽  
A Priori ◽  
Periodic Points ◽  
Unit Interval ◽  
Return Maps ◽  
Lorenz Maps

Lorenz maps are maps of the unit interval with one critical point of order $\unicode[STIX]{x1D70C}>1$ and a discontinuity at that point. They appear as return maps of sections of the geometric Lorenz flow. We construct real a priori bounds for renormalizable Lorenz maps with certain monotone combinatorics and a sufficiently flat critical point, and use these bounds to show existence of periodic points of renormalization, as well as existence of Cantor attractors for dynamics of infinitely renormalizable Lorenz maps.

scholarly journals Physical measures for infinitely renormalizable Lorenz maps

2016 ◽  
Vol 38 (2) ◽  
pp. 717-738 ◽  
Author(s):  
M. MARTENS ◽  
B. WINCKLER
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Crucial Role ◽  
A Priori ◽  
Physical Measure ◽  
Unimodal Maps ◽  
Statistical Behavior ◽  
Physical Measures ◽  
Lorenz Maps

A physical measure on the attractor of a system describes the statistical behavior of typical orbits. An example occurs in unimodal dynamics: namely, all infinitely renormalizable unimodal maps have a physical measure. For Lorenz dynamics, even in the simple case of infinitely renormalizable systems, the existence of physical measures is more delicate. In this article, we construct examples of infinitely renormalizable Lorenz maps which do not have a physical measure. A priori bounds on the geometry play a crucial role in (unimodal) dynamics. There are infinitely renormalizable Lorenz maps which do not have a priori bounds. This phenomenon is related to the position of the critical point of the consecutive renormalizations. The crucial technical ingredient used to obtain these examples without a physical measure is the control of the position of these critical points.

scholarly journals Examples of expanding maps with some special properties

1987 ◽  
Vol 36 (3) ◽  
pp. 469-474 ◽  
Author(s):  
Bau-Sen Du
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Periodic Point ◽  
Topological Entropy ◽  
Real Line ◽  
Periodic Points ◽  
Unit Interval ◽  
Expanding Maps ◽  
The Real ◽  
Piecewise Monotonic

Let I be the unit interval [0, 1] of the real line. For integers k ≥ 1 and n ≥ 2, we construct simple piecewise monotonic expanding maps Fk, n in C0 (I, I) with the following three properties: (1) The positive integer n is an expanding constant for Fk, n for all k; (2) The topological entropy of Fk, n is greater than or equal to log n for all k; (3) Fk, n has periodic points of least period 2k · 3, but no periodic point of least period 2k−1 (2m+1) for any positive integer m. This is in contrast to the fact that there are expanding (but not piecewise monotonic) maps in C0(I, I) with very large expanding constants which have exactly one fixed point, say, at x = 1, but no other periodic point.

scholarly journals Distribution of entanglement in networks of bi-partite full-rank mixed states

2012 ◽  
Vol 12 (5&6) ◽  
pp. 502-534
Author(s):  
G John Lapeyre Jr. ◽  
Sébastien Perseguers ◽  
Maciej Lewenstein ◽  
Antonio Acín
Keyword(s):  
Critical Point ◽  
Quantum Entanglement ◽  
A Priori ◽  
Full Rank ◽  
Entanglement Swapping ◽  
Low Density ◽  
Mixed States ◽  
Initial Application ◽  
Optimal Values ◽  
Entanglement Distribution

We study quantum entanglement distribution on networks with full-rank bi-partite mixed states linking qubits on nodes. In particular, we use entanglement swapping and purification to partially entangle widely separated nodes. The simplest method consists of performing entanglement swappings along the shortest chain of links connecting the two nodes. However, we show that this method may be improved upon by choosing a protocol with a specific ordering of swappings and purifications. A priori, the design that produces optimal improvement is not clear. However, we parameterize the choices and find that the optimal values depend strongly on the desired measure of improvement. As an initial application, we apply the new improved protocols to the Erd\"os--R\'enyi network and obtain results including low density limits and an exact calculation of the average entanglement gained at the critical point.

Homeomorphic restrictions of smooth endomorphisms of an interval

1992 ◽  
Vol 12 (3) ◽  
pp. 429-439 ◽  
Author(s):  
Karen M. Brucks ◽  
Maria Victoria Otero-Espinar ◽  
Charles Tresser
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Periodic Points ◽  
Limit Set ◽  
Asymptotic Dynamics ◽  
Infinite Set

AbstractWe describe the asymptotic dynamics of homeomorphisms obtained as restrictions of generic C2 endomorphisms of an interval with finitely many critical points, all of which are non-flat, and with all periodic points hyperbolic. The ω -limit set of such a restricted endomorphism cannot be infinite, except when the restriction of the endomorphism to the closure of the orbit of some critical point is a minimal homeomorphism of an infinite set.

scholarly journals On a Form of Coordinate Percolation

2008 ◽  
Vol 17 (6) ◽  
pp. 837-845 ◽  
Author(s):  
ELIZABETH R. MOSEMAN ◽  
PETER WINKLER
Keyword(s):  
Critical Exponent ◽  
Real Number ◽  
Critical Point ◽  
Exact Expression ◽  
Unit Interval ◽  
First Derivative ◽  
Infinite Path ◽  
A Site

Let ai,bi, i = 0, 1, 2, . . . be drawn uniformly and independently from the unit interval, and let t be a fixed real number. Let a site (i, j) ∈ $\N^2$ be open if ai + bj ≤ t, and closed otherwise. We obtain a simple, exact expression for the probability Θ(t) that there is an infinite path (oriented or not) of open sites, containing the origin. Θ(t) is continuous and has continuous first derivative except at the critical point (t=1), near which it has critical exponent (3 − $\sqrt{5}$)/2.

scholarly journals Second type of criticality in the brain uncovers rich multiple-neuron dynamics

2019 ◽  
Vol 116 (26) ◽  
pp. 13051-13060 ◽  
Author(s):  
David Dahmen ◽  
Sonja Grün ◽  
Markus Diesmann ◽  
Moritz Helias
Keyword(s):  
Motor Cortex ◽  
Critical Point ◽  
High Performance ◽  
A Priori ◽  
Population Level ◽  
Macaque Monkey ◽  
Computational Performance ◽  
Brain Signals ◽  
Large Numbers ◽  
Large Dispersion

Cortical networks that have been found to operate close to a critical point exhibit joint activations of large numbers of neurons. However, in motor cortex of the awake macaque monkey, we observe very different dynamics: massively parallel recordings of 155 single-neuron spiking activities show weak fluctuations on the population level. This a priori suggests that motor cortex operates in a noncritical regime, which in models, has been found to be suboptimal for computational performance. However, here, we show the opposite: The large dispersion of correlations across neurons is the signature of a second critical regime. This regime exhibits a rich dynamical repertoire hidden from macroscopic brain signals but essential for high performance in such concepts as reservoir computing. An analytical link between the eigenvalue spectrum of the dynamics, the heterogeneity of connectivity, and the dispersion of correlations allows us to assess the closeness to the critical point.

MAP DYNAMICS OF REPRODUCTION

1995 ◽  
Vol 05 (02) ◽  
pp. 381-396 ◽  
Author(s):  
PAUL E. PHILLIPSON ◽  
PETER SCHUSTER
Keyword(s):  
Critical Point ◽  
Causal Relationship ◽  
Crucial Role ◽  
Control Parameter ◽  
Original Structure ◽  
Alternative Mechanism ◽  
Point Mapping ◽  
One Dimensional ◽  
Return Maps ◽  
Holistic Process

One-dimensional return maps characterized by one critical point produce a universal sequence of periodic orbits, the Metropolis, Stein & Stein (MSS) sequence. Iterates of such maps can be regarded as an object whose structure is defined by the MSS sequence. The two critical point mapping [Formula: see text] as the control parameter b varies between [Formula: see text] and 1 is shown to display evolution from a single MSS structure to two identical MSS structures, suggestive of reproduction. The process of reproduction connotes a causal relationship whereby an original structure provides the blueprint for a copy. Here causality is replaced by an omniscient instruction, the mapping, and reproduction is achieved by progressively changing the instruction through variation of the control parameter. Bistability plays a crucial role in map dynamics of reproduction, which is viewed as a holistic process providing an alternative mechanism to digital replication.

REMARK ON ESTIMATE OF A POTENTIAL IN TERMS OF EIGENVALUES OF THE STURM–LIOUVILLE OPERATOR

2008 ◽  
Vol 22 (23) ◽  
pp. 2177-2180
Author(s):  
EVGENY KOROTYAEV
Keyword(s):  
Boundary Conditions ◽  
Liouville Operator ◽  
A Priori Estimates ◽  
Mixed Boundary ◽  
A Priori ◽  
Mixed Boundary Conditions ◽  
Unit Interval ◽  
Priori Estimates ◽  
Sturm Liouville ◽  
Neumann Eigenvalues

We consider the Sturm–Liouville operator on the unit interval. We obtain two-sided a priori estimates of potential in terms of Dirichlet and Neumann eigenvalues and eigenvalues for 2 types of mixed boundary conditions.

scholarly journals Quadratic maps with a periodic critical point of period 2

2016 ◽  
Vol 13 (06) ◽  
pp. 1393-1417
Author(s):  
Jung Kyu Canci ◽  
Solomon Vishkautsan
Keyword(s):  
Critical Point ◽  
Projective Line ◽  
Periodic Points ◽  
Complete Classification ◽  
Quadratic Polynomials ◽  
Quadratic Maps ◽  
Period 2 ◽  
Classification Of Graphs

We provide a complete classification of possible graphs of rational preperiodic points of endomorphisms of the projective line of degree 2 defined over the rationals with a rational periodic critical point of period 2, under the assumption that these maps have no periodic points of period at least 7. We explain how this extends results of Poonen on quadratic polynomials. We show that there are exactly 13 possible graphs, and that such maps have at most nine rational preperiodic points. We provide data related to the analogous classification of graphs of endomorphisms of degree 2 with a rational periodic critical point of period 3 or 4.

scholarly journals Periodic points and chaotic functions in the unit interval

1978 ◽  
Vol 18 (2) ◽  
pp. 255-265 ◽  
Author(s):  
G.J. Butler ◽  
G. Pianigiani
Keyword(s):  
Dense Subset ◽  
Periodic Points ◽  
Unit Interval ◽  
Chaotic Behaviour ◽  
Open Dense Subset

It is shown that the set of chaotic self-maps of the unit interval contains an open dense subset of the space of all continuous self-maps of the unit interval. Other aspects of chaotic behaviour are also considered together with some illustrative examples.

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