Results 1  10
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12
Recurrence rate in rapidly mixing dynamical systems, preprint
 Dyn. Syst
"... Abstract. For measure preserving dynamical systems on metric spaces we study the time needed by a typical orbit to return back close to its starting point. We prove that when the decay of correlation is superpolynomial the recurrence rates and the pointwise dimensions are equal. This gives a broad ..."
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Cited by 18 (7 self)
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Abstract. For measure preserving dynamical systems on metric spaces we study the time needed by a typical orbit to return back close to its starting point. We prove that when the decay of correlation is superpolynomial the recurrence rates and the pointwise dimensions are equal. This gives a broad class of systems for which the recurrence rate equals the Hausdorff dimension of the invariant measure. 1.
Return Time Statistics For Unimodal Maps
"... We prove that a nonat Sunimodal map satisfying a weak summability condition has exponential return time statistics on intervals around a.e. point. Moreover we prove that the convergence to the entropy in the OrnsteinWeiss formula enjoys normal uctuations. 1. ..."
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Cited by 14 (4 self)
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We prove that a nonat Sunimodal map satisfying a weak summability condition has exponential return time statistics on intervals around a.e. point. Moreover we prove that the convergence to the entropy in the OrnsteinWeiss formula enjoys normal uctuations. 1.
THE FIRST RETURN TIME PROPERTIES OF AN IRRATIONAL ROTATION
, 2008
"... If an ergodic system has positive entropy, then the ShannonMcMillanBreiman theorem provides a relationship between the entropy and the size of an atom of the iterated partition. The system also has OrnsteinWeiss’ first return time property, which offers a method of computing the entropy via an or ..."
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Cited by 7 (3 self)
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If an ergodic system has positive entropy, then the ShannonMcMillanBreiman theorem provides a relationship between the entropy and the size of an atom of the iterated partition. The system also has OrnsteinWeiss’ first return time property, which offers a method of computing the entropy via an orbit. We consider irrational rotations which are the simplest model of zero entropy. We prove that almost every irrational rotation has the analogous properties if properly normalized. However there are some irrational rotations that exhibit different behavior.
ENTROPY AND POINCARÉ RECURRENCE FROM A GEOMETRICAL VIEWPOINT
, 809
"... Abstract. We study Poincaré recurrence from a purely geometrical viewpoint. In [8] it was proven that the metric entropy is given by the exponential growth rate of return times to dynamical balls. Here we use combinatorial arguments to provide an alternative and more direct proof of this result and ..."
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Cited by 5 (0 self)
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Abstract. We study Poincaré recurrence from a purely geometrical viewpoint. In [8] it was proven that the metric entropy is given by the exponential growth rate of return times to dynamical balls. Here we use combinatorial arguments to provide an alternative and more direct proof of this result and to prove that minimal return times to dynamical balls grow linearly with respect to its length. Some relations using weighted versions of recurrence times are also obtained for equilibrium states. Then we establish some interesting relations between recurrence, dimension, entropy and Lyapunov exponents of ergodic measures. 1.
Quantitative recurrence properties of expanding maps
"... Under a map T, a point x recurs at rate given by a sequence {rn} near a point x0 if d(T n (x), x0) < rn infinitely often. Let us fix x0, and consider the set of those x’s. In this paper, we study the size of this set for expanding maps and obtain its measure and sharp lower bounds on its dimensio ..."
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Under a map T, a point x recurs at rate given by a sequence {rn} near a point x0 if d(T n (x), x0) < rn infinitely often. Let us fix x0, and consider the set of those x’s. In this paper, we study the size of this set for expanding maps and obtain its measure and sharp lower bounds on its dimension involving the entropy of T, the local dimension near x0 and the upper limit of 1 1 log. We apply our results in several concrete examples including subshifts of finite type, n rn Gauss transformation and inner functions.
Chaotic Generator in Digital Secure Communication
"... Abstract—A chaotic orbit generated by a nonlinear system is irregular, aperiodic, unpredictable and has sensitive dependence on initial conditions. However, the chaotic trajectory is still not well enough to be a crypto system in digital secure communication. Therefore, we propose a Modified Logisti ..."
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Abstract—A chaotic orbit generated by a nonlinear system is irregular, aperiodic, unpredictable and has sensitive dependence on initial conditions. However, the chaotic trajectory is still not well enough to be a crypto system in digital secure communication. Therefore, we propose a Modified Logistic Map (MLM) and give a theoretical proof to show that the MLM is a chaotic map according to Devaney’s definition. Based on the MLMs, we establish a Modified
FIRST POINCARÉ RETURNS, NATURAL MEASURE, UPOS AND KOLMOGOROVSINAI ENTROPY
, 908
"... PACS: 05.45.–a Nonlinear dynamics and chaos; 65.40.gd Entropy Abstract. It is known that unstable periodic orbits of a given map give information about the natural measure of a chaotic attractor. In this work we show how these orbits can be used to calculate the density function of the first Poincar ..."
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PACS: 05.45.–a Nonlinear dynamics and chaos; 65.40.gd Entropy Abstract. It is known that unstable periodic orbits of a given map give information about the natural measure of a chaotic attractor. In this work we show how these orbits can be used to calculate the density function of the first Poincaré returns. The close relation between periodic orbits and the Poincaré returns allows for analytical and semianalytical estimations of relevant quantities in dynamical systems, as the decay of correlation and the KolmogorovSinai entropy, in terms of this density function. Since return times can be trivially observed and measured, our approach is highly oriented to the treatment of experimental systems. 1.
The spectrum of Poincaré recurrence
, 2008
"... We investigate the relationship between Poincaré recurrence and topological entropy of a dynamical system (X, f). For 0 ≤ α ≤ β ≤ ∞, let D(α, β) be the set of x with lower and upper recurrence rates α and β, respectively. Under the assumptions that the system is not minimal and that the map f is p ..."
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We investigate the relationship between Poincaré recurrence and topological entropy of a dynamical system (X, f). For 0 ≤ α ≤ β ≤ ∞, let D(α, β) be the set of x with lower and upper recurrence rates α and β, respectively. Under the assumptions that the system is not minimal and that the map f is positively expansive and satisfies the specification condition, we show that for any open subset ∅ ̸ = U ⊆ X, D(α, β) ∩ U has the full topological entropy of X. This extends a result of Feng and Wu [The Hausdorff dimension of recurrence sets in symbolic spaces. Nonlinearity 14 (2001), 81–85] for symbolic spaces. 1