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April
2020, 40(4): 1989-2009.
doi: 10.3934/dcds.2020102
Isentropes and Lyapunov exponents
Department of Analysis, ELTE Eötvös Loránd University, Pázmány Péter Sétány 1/c, 1117, Budapest, Hungary, ZB's ORCID ID: 0000-0001-5481-8797 |
We consider skew tent maps $ T_{ { \alpha }, { \beta }}(x) $ such that $ ( \alpha, \beta)\in[0, 1]^{2} $ is the turning point of $ T _{\alpha, \beta} $, that is, $ T_{ { \alpha }, { \beta }} = \frac{ { \beta }}{ { \alpha }}x $ for $ 0\leq x \leq { \alpha } $ and $ T_{ { \alpha }, { \beta }}(x) = \frac{ { \beta }}{1- { \alpha }}(1-x) $ for $ { \alpha }<x\leq 1 $. We denote by $ \underline{M} = K( \alpha, \beta) $ the kneading sequence of $ T _{\alpha, \beta} $, by $ h( \alpha, \beta) $ its topological entropy and $ \Lambda = \Lambda_{\alpha, \beta} $ denotes its Lyapunov exponent. For a given kneading squence $ \underline{M} $ we consider isentropes (or equi-topological entropy, or equi-kneading curves), $ ( \alpha, \Psi_{ \underline{M}}( \alpha)) $ such that $ K( \alpha, \Psi_{ \underline{M}}( \alpha)) = \underline{M} $. On these curves the topological entropy $ h( \alpha, \Psi_{ \underline{M}}( \alpha)) $ is constant.
We show that $ \Psi_{ \underline{M}}'( \alpha) $ exists and the Lyapunov exponent $ \Lambda_{\alpha, \beta} $ can be expressed by using the slope of the tangent to the isentrope. Since this latter can be computed by considering partial derivatives of an auxiliary function $ \Theta_{ \underline{M}} $, a series depending on the kneading sequence which converges at an exponential rate, this provides an efficient new method of finding the value of the Lyapunov exponent of these maps.
Keywords:
Skew tent map,
topological entropy,
isentrope,
invariant measure,
Lyapunov exponent,
Markov partition.
Mathematics Subject Classification:
Primary: 37B25; Secondary: 28D20, 37B40, 37E05.
Citation:
Zoltán Buczolich, Gabriella Keszthelyi. Isentropes and Lyapunov exponents. Discrete and Continuous Dynamical Systems,
2020, 40
(4)
: 1989-2009.
doi: 10.3934/dcds.2020102
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Smooth deformations of piecewise expanding unimodal maps, Discrete Contin. Dyn. Syst., 23 (2009), 685-703.
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L. Billings and E. M. Bollt,
Probability density functions of some skew tent maps, Chaos Solitons Fractals, 12 (2001), 365-376.
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Z. Buczolich and G. Keszthelyi, Tangents of Isentropes of skew tent maps in the square, (in preparation). |
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M. C. Mackey and M. Tyran-Kamińska,
Central limit theorem behavior in the skew tent map, Chaos Solitons Fractals, 38 (2008), 789-805.
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J. Milnor and W. Thurston,
On iterated maps of the interval, Dynamical Systems (College Park, MD, 1986–87), Lecture Notes in Math., Springer, Berlin, 1342 (1988), 465-563.
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J. Milnor and C. Tresser,
On entropy and monotonicity for real cubic maps, With an appendix by Adrien Douady and Pierrette Sentenac. Comm. Math. Phys., 209 (2000), 123-178.
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M. Misiurewicz and E. Visinescu,
Kneading sequences of skew tent maps, Ann. Inst. Henri Poincaré, Probab. Stat., 27 (1991), 125-140.
|
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A. Radulescu,
The connected isentropes conjecture in a space of quartic polynomials, Discrete Contin. Dyn. Syst., 19 (2007), 139-175.
doi: 10.3934/dcds.2007.19.139. |
show all references
References:
| [1] |
V. Baladi and D. Smania,
Linear response formula for piecewise expanding unimodal maps, Nonlinearity, 21 (2008), 677-711.
doi: 10.1088/0951-7715/21/4/003. |
| [2] |
V. Baladi and D. Smania,
Smooth deformations of piecewise expanding unimodal maps, Discrete Contin. Dyn. Syst., 23 (2009), 685-703.
doi: 10.3934/dcds.2009.23.685. |
| [3] |
L. Billings and E. M. Bollt,
Probability density functions of some skew tent maps, Chaos Solitons Fractals, 12 (2001), 365-376.
doi: 10.1016/S0960-0779(99)00204-0. |
| [4] |
A. Boyarsky and P. Góra, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension, Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1997.
doi: 10.1007/978-1-4612-2024-4. |
| [5] |
H. Bruin and S. van Strien,
On the structure of isentropes of polynomial maps, Dyn. Syst., 28 (2013), 381-392.
doi: 10.1080/14689367.2013.822458. |
| [6] |
Z. Buczolich and G. Keszthelyi,
Equi-topological entropy curves for skew tent maps in the square, Math. Slovaca, 67 (2017), 1577-1594.
doi: 10.1515/ms-2017-0072. |
| [7] |
Z. Buczolich and G. Keszthelyi, Tangents of Isentropes of skew tent maps in the square, (in preparation). |
| [8] |
P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Progress in Physics, 1. Birkhäuser, Boston, Mass., 1980. |
| [9] |
G. Keller,
Stochastic stability in some chaotic dynamical systems, Monatsh. Math., 94 (1982), 313-333.
doi: 10.1007/BF01667385. |
| [10] |
D. J. Lai and G. R. Chen,
On statistical properties of the Lyapunov exponent of the generalized skew tent map, Stochastic Anal. Appl., 20 (2002), 375-388.
doi: 10.1081/SAP-120003440. |
| [11] |
A. Lasota and J. A. Yorke,
On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1974), 481-488.
doi: 10.1090/S0002-9947-1973-0335758-1. |
| [12] |
M. C. Mackey and M. Tyran-Kamińska,
Central limit theorem behavior in the skew tent map, Chaos Solitons Fractals, 38 (2008), 789-805.
doi: 10.1016/j.chaos.2007.01.013. |
| [13] |
J. Milnor and W. Thurston,
On iterated maps of the interval, Dynamical Systems (College Park, MD, 1986–87), Lecture Notes in Math., Springer, Berlin, 1342 (1988), 465-563.
doi: 10.1007/BFb0082847. |
| [14] |
J. Milnor and C. Tresser,
On entropy and monotonicity for real cubic maps, With an appendix by Adrien Douady and Pierrette Sentenac. Comm. Math. Phys., 209 (2000), 123-178.
doi: 10.1007/s002200050018. |
| [15] |
M. Misiurewicz and E. Visinescu,
Kneading sequences of skew tent maps, Ann. Inst. Henri Poincaré, Probab. Stat., 27 (1991), 125-140.
|
| [16] |
A. Radulescu,
The connected isentropes conjecture in a space of quartic polynomials, Discrete Contin. Dyn. Syst., 19 (2007), 139-175.
doi: 10.3934/dcds.2007.19.139. |
Figure 3.
Isentropes and tangents computed from $ \gamma $
Figure 4.
Illustration for the proofs of Proposition 12 and Theorem 2
| .3 | .8 | .20444 | -.36406 | -.36452 |
| .49 | .56 | .30996 | -.40344 | -.4244 |
| .5 | .7 | .27034 | -.64303 | -.64064 |
| .5 | .8 | .26918 | -.73861 | -.73739 |
| .6 | .75 | .35597 | -.76258 | -.76132 |
| .6 | .9 | .47736 | -.4599 | -.45991 |
| .3 | .8 | .20444 | -.36406 | -.36452 |
| .49 | .56 | .30996 | -.40344 | -.4244 |
| .5 | .7 | .27034 | -.64303 | -.64064 |
| .5 | .8 | .26918 | -.73861 | -.73739 |
| .6 | .75 | .35597 | -.76258 | -.76132 |
| .6 | .9 | .47736 | -.4599 | -.45991 |
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