scholarly journals Contact unimodal map germs from the plane to the plane

2020 ◽  
Vol 358 (8) ◽  
pp. 923-930
Author(s):  
Muhammad Ahsan Binyamin ◽  
Saima Aslam ◽  
Khawar Mehmood
Keyword(s):  
Unimodal Map

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.

scholarly journals Effective high-temperature estimates for intermittent maps

2017 ◽  
Vol 39 (8) ◽  
pp. 2159-2175
Author(s):  
BENOÎT R. KLOECKNER
Keyword(s):  
Perturbation Theory ◽  
High Temperature ◽  
Limit Theorem ◽  
Spectral Gap ◽  
Lipschitz Constant ◽  
Transfer Operator ◽  
Linear Operators ◽  
Suitable Assumption ◽  
Unimodal Map ◽  
Definition Of

Using quantitative perturbation theory for linear operators, we prove a spectral gap for transfer operators of various families of intermittent maps with almost constant potentials (‘high-temperature’ regime). Hölder and bounded $p$-variation potentials are treated, in each case under a suitable assumption on the map, but the method should apply more generally. It is notably proved that for any Pommeau–Manneville map, any potential with Lipschitz constant less than 0.0014 has a transfer operator acting on $\operatorname{Lip}([0,1])$ with a spectral gap; and that for any two-to-one unimodal map, any potential with total variation less than 0.0069 has a transfer operator acting on $\operatorname{BV}([0,1])$ with a spectral gap. We also prove under quite general hypotheses that the classical definition of spectral gap coincides with the formally stronger one used in Giulietti et al [The calculus of thermodynamical formalism. J. Eur. Math. Soc., to appear. Preprint, 2015, arXiv:1508.01297], allowing all results there to be applied under the high-temperature bounds proved here: analyticity of pressure and equilibrium states, central limit theorem, etc.

AN ALGORITHM TO COMPUTE THE TOPOLOGICAL ENTROPY OF A UNIMODAL MAP

1999 ◽  
Vol 09 (09) ◽  
pp. 1881-1882 ◽  
Author(s):  
HENK BRUIN
Keyword(s):  
Topological Entropy ◽  
Unimodal Map ◽  
Interval Map

We describe an algorithm, due to F. Hofbauer, to compute the topological entropy of a unimodal interval map.

Quasisymmetric invariance of the Collet–Eckmann condition

1998 ◽  
Vol 18 (3) ◽  
pp. 703-715
Author(s):  
DUNCAN SANDS ◽  
TOMASZ NOWICKI
Keyword(s):  
Unimodal Map

The Collet–Eckmann condition is quasisymmetrically invariant: if an S-unimodal map $f$ is quasisymmetrically conjugate to an S-unimodal map $g$ then $f$ satisfies the Collet–Eckmann condition if and only if $g$ satisfies the Collet–Eckmann condition.

scholarly journals $mathcal{A}$-unimodal map-germs into the plane

2004 ◽  
Vol 33 (1) ◽  
pp. 47-64 ◽  
Author(s):  
J.H. RIEGER
Keyword(s):  
Unimodal Map

scholarly journals The Fibonacci unimodal map

1993 ◽  
Vol 6 (2) ◽  
pp. 425-425 ◽  
Author(s):  
Mikhail Lyubich ◽  
John Milnor
Keyword(s):  
Unimodal Map

scholarly journals The conjugacy of Collet–Eckmann's map of the interval with the tent map is Holder continuous

1989 ◽  
Vol 9 (2) ◽  
pp. 379-388 ◽  
Author(s):  
T. Nowicki ◽  
F. Przytycki
Keyword(s):  
Lyapunov Exponent ◽  
Schwarzian Derivative ◽  
Critical Value ◽  
Positive Lyapunov Exponent ◽  
Hölder Continuous ◽  
Tent Map ◽  
Unimodal Map

AbstractIt is proved that a homeomorphism h, which conjugates a smooth unimodal map of the interval with negative Schwarzian derivative and positive Lyapunov exponent along the forward trajectory of the critical value with a tent map, and its inverse h−1 are Hölder continuous.

scholarly journals A Cr unimodal map with an arbitrary fast growth of the number of periodic points

2011 ◽  
Vol 32 (1) ◽  
pp. 159-165 ◽  
Author(s):  
V. KALOSHIN ◽  
O. S. KOZLOVSKI
Keyword(s):  
Critical Points ◽  
Acta Math ◽  
Periodic Points ◽  
Fast Growth ◽  
Unimodal Map ◽  
One Dimensional

AbstractIn this paper we present a surprising example of a Cr unimodal map of an interval f:I→I whose number of periodic points Pn(f)=∣{x∈I:fnx=x}∣ grows faster than any ahead given sequence along a subsequence nk=3k. This example also shows that ‘non-flatness’ of critical points is necessary for the Martens–de Melo–van Strien theorem [M. Martens, W. de Melo and S. van Strien. Julia–Fatou–Sullivan theory for real one-dimensional dynamics. Acta Math.168(3–4) (1992), 273–318] to hold.

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.

Recognition of unimodal map germs from the plane to the plane by invariants

2018 ◽  
Vol 28 (07) ◽  
pp. 1199-1208
Author(s):  
Saima Aslam ◽  
Muhammad Ahsan Binyamin ◽  
Gerhard Pfister
Keyword(s):  
Normal Form ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Closed Field ◽  
Unimodal Maps ◽  
Unimodal Map ◽  
Algebraically Closed Field

In this paper, we characterize the classification of unimodal maps from the plane to the plane with respect to [Formula: see text]-equivalence given by Rieger in terms of invariants. We recall the classification over an algebraically closed field of characteristic [Formula: see text]. On the basis of this characterization, we present an algorithm to compute the type of the unimodal maps from the plane to the plane without computing the normal form and also give its implementation in the computer algebra system Singular.

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