UNIMODAL INTERVAL MAPS OBTAINED FROM THE MODIFIED CHUA EQUATIONS

1993 ◽  
Vol 03 (02) ◽  
pp. 323-332 ◽  
Author(s):  
MICHAŁ MISIUREWICZ
Keyword(s):  
Periodic Orbit ◽  
Critical Point ◽  
Real Line ◽  
Positive Measure ◽  
Schwarzian Derivative ◽  
Large Range ◽  
Unimodal Maps ◽  
Interval Maps ◽  
Central Piece ◽  
Piecewise Continuous

Following Brown [1992, 1993] we study maps of the real line into itself obtained from the modified Chua equations. We fix our attention on a one-parameter family of such maps, which seems to be typical. For a large range of parameters, invariant intervals exist. In such an invariant interval, the map is piecewise continuous, with most of pieces of continuity mapped in a monotone way onto the whole interval. However, on the central piece there is a critical point. This allows us to find sometimes a smaller invariant interval on which the map is unimodal. In such a way, we get one-parameter families of smooth unimodal maps, very similar to the well-known family of logistic maps x ↦ ax(1−x). We study more closely one of those and show that these maps have negative Schwarzian derivative. This implies the existence of at most one attracting periodic orbit. Moreover, there is a set of parameters of positive measure for which chaos occurs.

Positive exponent in families with flat critical point

1999 ◽  
Vol 19 (3) ◽  
pp. 767-807 ◽  
Author(s):  
HANS THUNBERG
Keyword(s):  
Critical Point ◽  
Finite Order ◽  
Parameter Space ◽  
Positive Measure ◽  
Large Deviation ◽  
Random Variables ◽  
Independent Random Variables ◽  
Unimodal Maps ◽  
Key Points ◽  
Positive Exponent

It is known that in generic, full unimodal families with a critical point of finite order, there exists a set of positive measure in parameter space such that the corresponding maps have chaotic behaviour. In this paper we prove the corresponding statement for certain families of unimodal maps with flat critical point. One of the key-points is a large deviation argument for sums of ‘almost’ independent random variables with only finitely many moments.

Metric properties of non-renormalizableS-unimodal maps. Part I: Induced expansion and invariant measures

1994 ◽  
Vol 14 (4) ◽  
pp. 721-755 ◽  
Author(s):  
Michael Jakobson ◽  
Grzegorz Światek
Keyword(s):  
Critical Point ◽  
Schwarzian Derivative ◽  
Invariant Measures ◽  
Sufficient Conditions ◽  
Limit Set ◽  
Countable Union ◽  
Unimodal Maps ◽  
Unimodal Map ◽  
Starting Condition ◽  
Metric Properties

AbstractFor an arbitrary non-renormalizable unimodal map of the interval,f:I→I, with negative Schwarzian derivative, we construct a related mapFdefined on a countable union of intervals Δ. For each interval Δ,Frestricted to Δ is a diffeomorphism which coincides with some iterate offand whose range is a fixed subinterval ofI. IfFsatisfies conditions of the Folklore Theorem, we callfexpansion inducing. Letcbe a critical point off. Forfsatisfyingf″(c) ≠ 0, we give sufficient conditions for expansion inducing. One of the consequences of expansion inducing is that Milnor's conjecture holds forf: the ω-limit set of Lebesgue almost every point is the interval [f2,f(c)]. An important step in the proof is a starting condition in the box case: if for initial boxes the ratio of their sizes is small enough, then subsequent ratios decrease at least exponentially fast and expansion inducing follows.

scholarly journals Stratification of the space of unimodal interval maps

1983 ◽  
Vol 3 (4) ◽  
pp. 533-539 ◽  
Author(s):  
Louis Block ◽  
David Hart
Keyword(s):  
Periodic Orbit ◽  
Dynamical Property ◽  
Compact Interval ◽  
Unimodal Maps ◽  
Interval Maps ◽  
Orbit Structure

AbstractThe space of ‘quadratic-like’ (unimodal) maps of a compact interval to itself is shown to decompose in a ‘nice’ way (stratify) according to a dynamical property of such maps (the existence of a homoclinic periodic orbit with given period). This decomposition is refined by that discovered by Sarkovskii. Orbit structure and bifurcation properties are also discussed.

scholarly journals AN ALMOST EVERYWHERE VERSION OF SMÍTAL’S ORDER–CHAOS DICHOTOMY FOR INTERVAL MAPS

2008 ◽  
Vol 85 (1) ◽  
pp. 29-50 ◽  
Author(s):  
ALEJO BARRIO BLAYA ◽  
VÍCTOR JIMÉNEZ LÓPEZ
Keyword(s):  
Positive Measure ◽  
Schwarzian Derivative ◽  
Initial Conditions ◽  
Periodic Points ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Almost Everywhere ◽  
The One ◽  
Set Of Points ◽  
Full Lebesgue Measure

AbstractWe prove that iff:I=[0,1]→Iis aC3-map with negative Schwarzian derivative, nonflat critical points and without wild attractors, then exactly one of the following alternatives must occur: (i) R(f) has full Lebesgue measureλ; (ii) bothS(f) andScramb(f) have positive measure. HereR(f),S(f), andScramb(f) respectively stand for the set of approximately periodic points off, the set of sensitive points to the initial conditions off, and the two-dimensional set of points (x,y) such that {x,y} is a scrambled set forf. Also, we show that iffis piecewise monotone and has no wandering intervals, then eitherλ(R(f))=1 orλ(S(f))>0, and provide examples of mapsg,hof this type satisfyingS(g)=S(h)=Isuch that, on the one hand,λ(R(g))=0 andλ2(Scramb(g))=0 , and, on the other hand,λ(R(h))=1 .

scholarly journals Constant slope maps on the extended real line

2017 ◽  
Vol 38 (8) ◽  
pp. 3145-3169 ◽  
Author(s):  
MICHAŁ MISIUREWICZ ◽  
SAMUEL ROTH
Keyword(s):  
Real Line ◽  
Half Line ◽  
Piecewise Monotone ◽  
Bounded Interval ◽  
Interval Map ◽  
Piecewise Continuous ◽  
Constant Slope

For 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).

Adding machines and wild attractors

1997 ◽  
Vol 17 (6) ◽  
pp. 1267-1287 ◽  
Author(s):  
HENK BRUIN ◽  
GERHARD KELLER ◽  
MATTHIAS ST. PIERRE
Keyword(s):  
Dynamical System ◽  
Critical Point ◽  
Cantor Set ◽  
Limit Set ◽  
Unimodal Maps ◽  
Irrational Rotations ◽  
Circle Rotation ◽  
Omega Limit Set ◽  
Adding Machines

We investigate the dynamics of unimodal maps $f$ of the interval restricted to the omega limit set $X$ of the critical point for cases where $X$ is a Cantor set. In particular, many cases where $X$ is a measure attractor of $f$ are included. We give two classes of examples of such maps, both generalizing unimodal Fibonacci maps [LM, BKNS]. In all cases $f_{|X}$ is a continuous factor of a generalized odometer (an adding machine-like dynamical system), and at the same time $f_{|X}$ factors onto an irrational circle rotation. In some of the examples we obtain irrational rotations on more complicated groups as factors.

scholarly journals Non-accessible critical points of Cremer polynomials

2000 ◽  
Vol 20 (5) ◽  
pp. 1391-1403 ◽  
Author(s):  
JAN KIWI
Keyword(s):  
Periodic Orbit ◽  
Critical Point ◽  
Critical Points ◽  
Julia Set ◽  
Complex Polynomial

It is shown that a polynomial with a Cremer periodic orbit has a non-accessible critical point in its Julia set provided that the Cremer periodic orbit is approximated by small cycles. Also, this paper contains a new proof of the Douady–Shishikura inequality for the number of non-repelling cycles of a complex polynomial.

An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps

1994 ◽  
Vol 14 (4) ◽  
pp. 621-632 ◽  
Author(s):  
V. Baladi ◽  
D. Ruelle
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Constant Weight ◽  
Piecewise Constant ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Interval Map ◽  
Piecewise Continuous

AbstractWe consider a piecewise continuous, piecewise monotone interval map and a piecewise constant weight. With these data we associate a weighted kneading matrix which generalizes the Milnor—Thurston matrix. We show that the determinant of this matrix is related to a natural weighted zeta function.

Fibonacci maps re(aℓ)visited

1995 ◽  
Vol 15 (1) ◽  
pp. 99-120 ◽  
Author(s):  
Gerhard Keller ◽  
Tomasz Nowicki
Keyword(s):  
Invariant Measure ◽  
Critical Point ◽  
Schwarzian Derivative ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractWe prove that unimodal Fibonacci maps with negative Schwarzian derivative and a critical point of order ℓ have a finite absolutely continuous invariant measure if ℓ ∈ (1 ℓ1) where ℓ1 is some number strictly greater than 2. This extends results of Lyubich and Milnor for the case ℓ = 2.

Entropy of snakes and the restricted variational principle

1992 ◽  
Vol 12 (4) ◽  
pp. 791-802
Author(s):  
M. Misiurewicz ◽  
J. Tolosa
Keyword(s):  
Variational Principle ◽  
Periodic Orbit ◽  
Topological Entropy ◽  
Periodic Orbits ◽  
Symmetric Case ◽  
Easy Method ◽  
Interval Maps ◽  
Interval Map ◽  
Ergodic Measures

AbstractFor interval maps, we define the entropy of a periodic orbit as the smallest topological entropy of a continuous interval map having this orbit. We consider the problem of computing the limit entropy of longer and longer periodic orbits with the same ‘pattern’ repeated over and over (one example of such orbits is what we call ‘snakes’). We get an answer in the form of a variational principle, where the supremum of metric entropies is taken only over those ergodic measures for which the integral of a certain function is zero. In a symmetric case, this gives a very easy method of computing this limit entropy. We briefly discuss applications to topological entropy of countable chains.

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