scholarly journals Unimodal expanding maps of the interval

1988 ◽  
Vol 38 (1) ◽  
pp. 125-130 ◽  
Author(s):  
Bau-Sen Du
Keyword(s):  
Periodic Point ◽  
Positive Zero ◽  
Expanding Maps ◽  
Expanding Map ◽  
Piecewise Monotonic ◽  
Period 2

Let I = [0, 1] and let f be an unimodal expanding map in C0(I, I). If f has an expanding constant for some integers m ≧ 0 and n ≧ 1, where λn is the unique positive zero of the polynomial x2n+1 − 2x2n−1 −1, then we show that f has a periodic point of period 2m(2n + 1). The converse of the above result is trivially false. The condition in the above result is the best possible in the sense that we cannot have the same conclusion if the number λn is replaced by any smaller positive number and the generalisation of the above result to arbitrary piecewise monotonic expanding maps in C0(I, I) is not possible.

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 Piecewise expanding maps on the plane with singular ergodic properties

2000 ◽  
Vol 20 (6) ◽  
pp. 1851-1857 ◽  
Author(s):  
MASATO TSUJII
Keyword(s):  
Empirical Measure ◽  
Expanding Maps ◽  
Ergodic Properties ◽  
Expanding Map ◽  
Open Set ◽  
Point Measure ◽  
Piecewise Expanding Maps

For $1\le r<\infty$, we construct a piecewise $C^{r}$ expanding map $F:D\to D$ on the domain $D=(0,1)\times (-1,1)\subset\mathbb{R}^{2}$ with the following property: there exists an open set $B$ in $D$ such that the diameter of $F^{n}(B)$ converges to $0$ as $n\to\infty$ and the empirical measure $n^{-1}\sum_{k=0}^{n-1}\delta_{F^{k}(x)}$ converges to the point measure $\delta_{p}$ at $p=(0,0)$ as $n \to\infty$ for any point $x\in B$.

Non-ergodicity for C1 expanding maps and g-measures

1996 ◽  
Vol 16 (3) ◽  
pp. 531-543 ◽  
Author(s):  
Anthony N. Quasf
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Probability Measures ◽  
Continuous Invariant Measure ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Expanding Map ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractWe introduce a procedure for finding C1 Lebesgue measure-preserving maps of the circle isomorphic to one-sided shifts equipped with certain invariant probability measures. We use this to construct a C1 expanding map of the circle which preserves Lebesgue measure, but for which Lebesgue measure is non-ergodic (that is there is more than one absolutely continuous invariant measure). This is in contrast with results for C1+e maps. We also show that this example answers in the negative a question of Keane's on uniqueness of g-measures, which in turn is based on a question raised by an incomplete proof of Karlin's dating back to 1953.

scholarly journals Transfer operators for ultradifferentiable expanding maps of the circle

2020 ◽  
pp. 1-20 ◽  
Author(s):  
MALO JÉZÉQUEL
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Nuclear Power ◽  
Transfer Operator ◽  
Singular Values ◽  
Expanding Maps ◽  
Smooth Functions ◽  
Transfer Operators ◽  
Expanding Map

Given a ${\mathcal{C}}^{\infty }$ expanding map $T$ of the circle, we construct a Hilbert space ${\mathcal{H}}$ of smooth functions on which the transfer operator ${\mathcal{L}}$ associated to $T$ acts as a compact operator. This result is made quantitative (in terms of singular values of the operator ${\mathcal{L}}$ acting on ${\mathcal{H}}$ ) using the language of Denjoy–Carleman classes. Moreover, the nuclear power decomposition of Baladi and Tsujii can be performed on the space ${\mathcal{H}}$ , providing a bound on the growth of the dynamical determinant associated to ${\mathcal{L}}$ .

COUPLED-EXPANDING MAPS AND MATRIX SHIFTS

2013 ◽  
Vol 23 (01) ◽  
pp. 1350013 ◽  
Author(s):  
MARCIN KULCZYCKI ◽  
PIOTR OPROCHA
Keyword(s):  
Transition Matrix ◽  
Sufficient Conditions ◽  
Expanding Maps ◽  
Expanding Map

For an irreducible transition matrix A of size m × m, which is not a permutation, a map f : X → X is said to be strictly A-coupled-expanding if there are nonempty sets V1,…, Vm ⊂ X such that the distance between any two of them is positive and f(Vi) ⊃ Vj holds whenever aij = 1. This paper presents two theorems that give sufficient conditions for a strictly A-coupled-expanding map to be chaotic on part of its domain in the sense of, respectively, Auslander and Yorke and Devaney. These results improve on the work of Zhang and Shi [2010]. An example is provided to illustrate that the class of maps the new theorems apply to is significantly wider.

scholarly journals Periodic points and chaos for expanding maps of the interval

1981 ◽  
Vol 24 (1) ◽  
pp. 79-83 ◽  
Author(s):  
Bill Byers
Keyword(s):  
Turning Points ◽  
Periodic Points ◽  
Expanding Maps ◽  
Period 2

Expanding maps of the interval with unique turning points have periodic points of period 2n · 3 for some n and therefore are chaotic.

scholarly journals Extreme-value distributions for some classes of non-uniformly partially hyperbolic dynamical systems

2009 ◽  
Vol 30 (3) ◽  
pp. 757-771 ◽  
Author(s):  
CHINMAYA GUPTA
Keyword(s):  
Sufficient Conditions ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Type I ◽  
Skew Product ◽  
Extreme Value Distributions ◽  
Product System ◽  
Expanding Maps ◽  
Expanding Map ◽  
Product Extension

AbstractIn this note, we obtain verifiable sufficient conditions for the extreme-value distribution for a certain class of skew-product extensions of non-uniformly hyperbolic base maps. We show that these conditions, formulated in terms of the decay of correlations on the product system and the measure of rapidly returning points on the base, lead to a distribution for the maximum of Φ(p)=−log(d(p,p0)) that is of the first type. In particular, we establish the type I distribution for S1 extensions of piecewise C2 uniformly expanding maps of the interval, non-uniformly expanding maps of the interval modeled by a Young tower, and a skew-product extension of a uniformly expanding map with a curve of neutral points.

Invariant rigid geometric structures and expanding maps

2011 ◽  
Vol 32 (3) ◽  
pp. 941-959 ◽  
Author(s):  
YONG FANG
Keyword(s):  
Dynamical Systems ◽  
Geometric Structure ◽  
Closed Manifold ◽  
Expanding Maps ◽  
Geometric Structures ◽  
Global Rigidity ◽  
Chaotic Dynamical Systems ◽  
Expanding Map ◽  
Locally Homogeneous ◽  
Rigid Geometric Structures

AbstractIn the first part of this paper, we consider several natural problems about locally homogeneous rigid geometric structures. In particular, we formulate a notion of topological completeness which is adapted to the study of global rigidity of chaotic dynamical systems. In the second part of the paper, we prove the following result: let φ be a C∞ expanding map of a closed manifold. If φ preserves a topologically complete C∞ rigid geometric structure, then φ is C∞ conjugate to an expanding infra-nilendomorphism.

scholarly journals Explicit coupling argument for non-uniformly hyperbolic transformations

2018 ◽  
Vol 149 (1) ◽  
pp. 101-130 ◽  
Author(s):  
A. Korepanov ◽  
Z. Kosloff ◽  
I. Melbourne
Keyword(s):  
Spectral Properties ◽  
Transfer Operator ◽  
Decay Of Correlations ◽  
Exponential Polynomial ◽  
Expanding Maps ◽  
Stretched Exponential ◽  
Expanding Map ◽  
Mixing Rates ◽  
Rates Of Decay

The transfer operator corresponding to a uniformly expanding map enjoys good spectral properties. We verify that coupling yields explicit estimates that depend continuously on the expansion and distortion constants of the map. For non-uniformly expanding maps with a uniformly expanding induced map, we obtain explicit estimates for mixing rates (exponential, stretched exponential, polynomial) that again depend continuously on the constants for the induced map together with data associated with the inducing time. Finally, for non-uniformly hyperbolic transformations, we obtain the corresponding estimates for rates of decay of correlations.

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