scholarly journals Flexibility of Lyapunov exponents for expanding circle maps

2019 ◽  
Vol 39 (5) ◽  
pp. 2325-2342 ◽  
Author(s):  
Alena Erchenko ◽  
Keyword(s):  
Lyapunov Exponents ◽  
Circle Maps ◽  
Expanding Circle

scholarly journals Circle diffeomorphisms forced by expanding circle maps

2011 ◽  
Vol 32 (6) ◽  
pp. 2011-2024 ◽  
Author(s):  
ALE JAN HOMBURG
Keyword(s):  
Natural Extension ◽  
Skew Product ◽  
Circle Maps ◽  
Expanding Circle ◽  
Topologically Mixing ◽  
Almost All ◽  
Open Class ◽  
Product Maps ◽  
Circle Diffeomorphisms

AbstractWe discuss the dynamics of skew product maps defined by circle diffeomorphisms forced by expanding circle maps. We construct an open class of such systems that are robustly topologically mixing and for which almost all points in the same fiber converge under iteration. This property follows from the construction of an invariant attracting graph in the natural extension, a skew product of circle diffeomorphisms forced by a solenoid homeomorphism.

scholarly journals Eigenfunctions for smooth expanding circle maps

Nonlinearity ◽  
2004 ◽  
Vol 17 (5) ◽  
pp. 1723-1730 ◽  
Author(s):  
Gerhard Keller ◽  
Hans Henrik Rugh
Keyword(s):  
Circle Maps ◽  
Expanding Circle

scholarly journals Mean-Field Coupling of Identical Expanding Circle Maps

2016 ◽  
Vol 164 (4) ◽  
pp. 858-889 ◽  
Author(s):  
Fanni Sélley ◽  
Péter Bálint
Keyword(s):  
Mean Field ◽  
Circle Maps ◽  
Field Coupling ◽  
Expanding Circle

scholarly journals Analytic expanding circle maps with explicit spectra

Nonlinearity ◽  
2013 ◽  
Vol 26 (12) ◽  
pp. 3231-3245 ◽  
Author(s):  
Julia Slipantschuk ◽  
Oscar F Bandtlow ◽  
Wolfram Just
Keyword(s):  
Circle Maps ◽  
Expanding Circle

scholarly journals On stochastic stability of expanding circle maps with neutral fixed points

2013 ◽  
Vol 28 (3) ◽  
pp. 423-452 ◽  
Author(s):  
Weixiao Shen ◽  
Sebastian van Strien
Keyword(s):  
Fixed Points ◽  
Stochastic Stability ◽  
Circle Maps ◽  
Expanding Circle

Chaos after Accumulation of Torus Doublings

2021 ◽  
Vol 31 (01) ◽  
pp. 2150009
Author(s):  
Munehisa Sekikawa ◽  
Naohiko Inaba
Keyword(s):  
Lyapunov Exponents ◽  
Coupling Parameter ◽  
Piecewise Linear ◽  
Discrete Dynamical System ◽  
Absolute Value ◽  
Circle Maps ◽  
Lyapunov Spectra ◽  
The Absolute ◽  
Approach Zero ◽  
Fundamental Mechanism

A recent paper investigates the bifurcation diagrams involved with torus doubling and asserts that the chaotic attractors observed after torus doubling have two Lyapunov exponents that are exactly zero. Against this assertion, we claim that the absolute value of one of the calculated zero Lyapunov exponents is not exactly zero but is instead slightly positive, because successive torus doubling is constrained by a very small underlying parameter. We justify our position by calculating Lyapunov spectra precisely using an autonomous piecewise-linear dynamical circuit. Our numerical results show that one of the Lyapunov exponents is close to, but not exactly, zero. In addition, we consider coupled logistic and sine-circle maps whose dynamics express the fundamental mechanism that causes torus doubling, and we confirm that torus doubling occurs fewer times when the coupling parameter of this discrete dynamical system is relatively larger. Consequently, the absolute value of the second Lyapunov exponent of this discrete system does not approach zero after the accumulation of torus doubling when the coupling parameter is set to larger values.

scholarly journals Lower bounds for the Ruelle spectrum of analytic expanding circle maps

2017 ◽  
Vol 39 (2) ◽  
pp. 289-310 ◽  
Author(s):  
OSCAR F. BANDTLOW ◽  
FRÉDÉRIC NAUD
Keyword(s):  
Lower Bounds ◽  
Blaschke Products ◽  
Expanding Maps ◽  
Circle Maps ◽  
Expanding Circle ◽  
Dense Set

We prove that there exists a dense set of analytic expanding maps of the circle for which the Ruelle eigenvalues enjoy exponential lower bounds. The proof combines potential theoretic techniques and explicit calculations for the spectrum of expanding Blaschke products.

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