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.

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.

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.

SMOOTH SYMMETRIC AND LORENZ MODELS FOR UNIMODAL MAPS

2003 ◽  
Vol 13 (11) ◽  
pp. 3353-3371 ◽  
Author(s):  
MING-CHIA LI ◽  
MIKHAIL MALKIN
Keyword(s):  
Turning Point ◽  
Invariant Measures ◽  
Structural Instability ◽  
Symmetric Model ◽  
Lorenz Model ◽  
Unimodal Maps ◽  
Unimodal Map ◽  
Ergodic Properties ◽  
Discontinuous Map ◽  
Higher Derivatives

For a given unimodal map F:I→I on the interval I, we consider symmetric unimodal maps (models) so that they are conjugate to F. The question motivated by [Gambaudo & Tresser, 1992] is the following: whether it is possible for symmetric model to preserve smoothness of the initial map F? We construct a symmetric model which is proved to be as smooth as F provided F has a nonflat turning point with sufficient "reserve of local evenness" at the turning point (in terms of one-sided higher derivatives at the turning point, see Definition 2.4 and Theorem 2.7). We also consider from different points of view the relationship between dynamical and ergodic properties of unimodal maps and of symmetric Lorenz maps. In particular, we present a one-to-one correspondence preserving the measure theoretic entropy, between the set of invariant measures of a symmetric unimodal map F and the set of symmetric invariant measures of the Lorenz model of F (Theorem 3.5), where by Lorenz model of F we mean the discontinuous map obtained from F by reversing its decreasing branch. Finally we extend for nonsymmetric unimodal maps, the result of Gambaudo and Tresser [1992] on Ck structural instability of the maps whose rotation interval has irrational end point (answering a question from [Gambaudo & Tresser, 1992]).

scholarly journals Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 80
Author(s):  
Sergey Kryzhevich ◽  
Viktor Avrutin ◽  
Nikita Begun ◽  
Dmitrii Rachinskii ◽  
Khosro Tajbakhsh
Keyword(s):  
Risk Management ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Management Model ◽  
Closing Lemma ◽  
Symbolic Model ◽  
Metric Properties ◽  
Ergodic Measures ◽  
Maximal Invariant

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

scholarly journals Graph directed Markov systems on Hilbert spaces

2009 ◽  
Vol 147 (2) ◽  
pp. 455-488 ◽  
Author(s):  
R. D. MAULDIN ◽  
T. SZAREK ◽  
M. URBAŃSKI
Keyword(s):  
Hausdorff Measure ◽  
Iterated Function System ◽  
Sufficient Conditions ◽  
Topological Pressure ◽  
Limit Set ◽  
Covering Theorem ◽  
Limit Sets ◽  
Markov Systems ◽  
Polish Spaces ◽  
Iterated Function

AbstractWe deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowen's formula (determining the Hausdorff dimension of limit sets in dynamical terms) are introduced and established. We show that, unlike the Euclidean case, the Hausdorff measure of the limit set of a finite iterated function system may vanish. Investigating this issue in greater detail, we introduce the concept of geometrically perfect measures and provide sufficient conditions for geometric perfectness. Geometrical perfectness guarantees the Hausdorff measure of the limit set to be positive. As a by–product of the mainstream of our investigations we prove a 4r–covering theorem for all metric spaces. It enables us to establish appropriate co–Frostman type theorems.

scholarly journals Local and global extrema for functions of several variables

1980 ◽  
Vol 29 (3) ◽  
pp. 362-368 ◽  
Author(s):  
Bruce Calvert ◽  
M. K. Vamanamurthy
Keyword(s):  
Critical Point ◽  
Local Minimum ◽  
Global Minimum ◽  
Sufficient Conditions ◽  
Subject Classification ◽  
General Function ◽  
Several Variables ◽  
Open Question

AbstractLet p: R2 → R be a polynomial with a local minimum at its only critical point. This must give a global minimum if the degree of p is < 5, but not necessarily if the degree is ≥ 5. It is an open question what the result is for cubics and quartics in more variables, except cubics in three variables. Other sufficient conditions for a global minimum of a general function are given.1980 Mathematics subject classification (Amer. Math. Soc.): 26 B 99, 26 C 99.

scholarly journals Parabolic limits of renormalization

2000 ◽  
Vol 20 (1) ◽  
pp. 173-229 ◽  
Author(s):  
BENJAMIN HINKLE
Keyword(s):  
Fixed Point ◽  
Period Doubling ◽  
Unit Interval ◽  
Local Analysis ◽  
Unimodal Maps ◽  
Unimodal Map ◽  
Combinatorial Description ◽  
Combinatorial Rigidity

A unimodal map $f:[0,1] \to [0,1]$ is renormalizable if there is a sub-interval $I \subset [0,1]$ and an $n > 1$ such that $f^n|_I$ is unimodal. The renormalization of $f$ is $f^n|_I$ rescaled to the unit interval.We extend the well-known classification of limits of renormalization of unimodal maps with bounded combinatorics to a classification of the limits of renormalization of unimodal maps with essentially bounded combinatorics. Together with results of Lyubich on the limits of renormalization with essentially unbounded combinatorics, this completes the combinatorial description of limits of renormalization. The techniques are based on the towers of McMullen and on the local analysis around perturbed parabolic points. We define a parabolic tower to be a sequence of unimodal maps related by renormalization or parabolic renormalization. We state and prove the combinatorial rigidity of bi-infinite parabolic towers with complex bounds and essentially bounded combinatorics, which implies the main theorem.As an example we construct a natural unbounded analogue of the period-doubling fixed point of renormalization, called the essentially period-tripling fixed point.

Finite regular invariant measures for Feller processes

1968 ◽  
Vol 5 (1) ◽  
pp. 203-209 ◽  
Author(s):  
V. E. Beneš
Keyword(s):  
Stochastic Process ◽  
Lyapunov Function ◽  
Invariant Measures ◽  
Sufficient Conditions ◽  
Ergodic Averages ◽  
Feller Process ◽  
The Third ◽  
Sequential Compactness ◽  
Finite Invariant Measure ◽  
Necessary And Sufficient

In the study of dynamical systems perturbed by noise, it is important to know whether the stochastic process of interest has a stationary distribution. Four necessary and sufficient conditions are formulated for the existence of a finite invariant measure for a Feller process on a σ-compact metric (state) space. These conditions link together stability notions from several fields. The first uses a Lyapunov function reminiscent of Lagrange stability in differential equations; the second depends on Prokhorov's condition for sequential compactness of measures; the third is a recurrence condition on the ergodic averages of the transition operator; and the fourth is analogous to a condition of Ulam and Oxtoby for the nonstochastic case.

Characterization of isochronous foci for planar analytic differential systems

2005 ◽  
Vol 135 (5) ◽  
pp. 985-998 ◽  
Author(s):  
Jaume Giné ◽  
Maite Grau
Keyword(s):  
Critical Point ◽  
Analytic Functions ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Differential Systems ◽  
Systems Of Differential Equations ◽  
Image Position ◽  
Necessary And Sufficient

We consider the two-dimensional autonomous systems of differential equations of the form where P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.

scholarly journals Order-Invariant Measures on Fixed Causal Sets

2012 ◽  
Vol 21 (3) ◽  
pp. 330-357 ◽  
Author(s):  
GRAHAM BRIGHTWELL ◽  
MALWINA LUCZAK
Keyword(s):  
Linear Extension ◽  
Invariant Measures ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Order Type ◽  
Probability Measures ◽  
Causal Sets ◽  
Natural Extensions ◽  
Countably Infinite ◽  
Order Invariant

A causal set is a countably infinite poset in which every element is above finitely many others; causal sets are exactly the posets that have a linear extension with the order-type of the natural numbers; we call such a linear extension a natural extension. We study probability measures on the set of natural extensions of a causal set, especially those measures having the property of order-invariance: if we condition on the set of the bottom k elements of the natural extension, each feasible ordering among these k elements is equally likely. We give sufficient conditions for the existence and uniqueness of an order-invariant measure on the set of natural extensions of a causal set.

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