Logarithm laws and shrinking target properties

Athreya, Jayadev S.

doi:10.1007/s12044-009-0044-x

Cited by 28 publications

(51 citation statements)

References 31 publications

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“…Typical hole locations Typical behaviour is given by Borel-Cantelli results, which as discussed in Ref. [1] apply to the doubling map, and more generally.…”

Section: Dependence On Hole Positionmentioning

confidence: 94%

“…For the escape problem, we expect that for strongly chaotic systems (eg with exponential decay of correlations), the survival probability P (n) = µ(M n ) decays exponentially with time n, so that an escape rate can be defined: γ = lim n→∞ − 1 n ln P (n) (1) Here µ is the measure of initial conditions (assumed to be invariant under the dynamics) and M n the subset of the phase space M that survives for at least n iterations before reaching the hole H ⊂ M . In general the limit might not exist, and we need upper and lower escape rates, possibly infinite.…”

Section: Introductionmentioning

confidence: 99%

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Open circle maps: small hole asymptotics

Dettmann

2012

Nonlinearity

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We consider escape from chaotic maps through a subset of phase space, the hole. Escape rates are known to be locally constant functions of the hole position and size. In spite of this, for the doubling map we can extend the current best result for small holes, a linear dependence on hole size h, to include a smooth h 2 ln h term and explicit fractal terms to h 2 and higher orders, confirmed by numerical simulations. For more general hole locations the asymptotic form depends on a dynamical Diophantine condition using periodic orbits ordered by stability.

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“…Typical hole locations Typical behaviour is given by Borel-Cantelli results, which as discussed in Ref. [1] apply to the doubling map, and more generally.…”

Section: Dependence On Hole Positionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Open circle maps: small hole asymptotics

Dettmann

2012

Nonlinearity

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“…We say that (X, µ, T ) has the monotone shrinking target property (MSTP) if for any x 0 ∈ X, every nested sequence of balls B m centered at x 0 satisfying µ(B m ) = ∞ is a BC-sequence. Many interesting systems are known to have either the STP or MSTP property, see [1], [22], and references therein for examples. A more comprehensive introduction to dynamical Borel-Cantelli lemmas, including examples, can be found in [3].…”

Section: Ifmentioning

confidence: 99%

Shrinking targets and eventually always hitting points for interval maps

2020

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We study shrinking target problems and the set E ah of eventually always hitting points. These are the points whose first n iterates will never have empty intersection with the n-th target for sufficiently large n. We derive necessary and sufficient conditions on the shrinking rate of the targets for E ah to be of full or zero measure especially for some interval maps including the doubling map, some quadratic maps and the Manneville-Pomeau map. We also obtain results for the Gauß map and correspondingly for the maximal digits in continued fractions expansions. In the case of the doubling map we also compute the packing dimension of E ah complementing already known results on the Hausdorff dimension of E ah .

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“…Let B be a family of measurable subsets of Υ and let F = {f n } denote a sequence of µ-preserving transformations of Υ Definition 1.3. (Borel-Cantelli families) 1 We say that B is Borel-Cantelli for F if for every sequence {A n : n ∈ N} of sets from B,…”

Section: Introductionmentioning

confidence: 99%

“…The functions ∆ and d(x 0 , ·) are UDL. 1 The definition of Borel-Cantelli families makes sense in the setting of any probability space, and indeed this is the definition in [35]. See also [13] for even more general formulations and applications.…”

Section: Introductionmentioning

confidence: 99%