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Renormalized solutions for stochastic $ p $-Laplace equations with $ L^1 $-initial data: The case of multiplicative noise
A shrinking target theorem for ergodic transformations of the unit interval
Ben Gurion University of the Negev, Departement of Mathematics, Be'er Sheva, 8410501, Israel |
$ f: [0,1) \rightarrow [0,1) $ |
$ \{b_i\}_{i=1}^{\infty} $ |
$ {\underset{n=1}{\overset{\infty}{\cap}} \, {\underset{i=n}{\overset{\infty}{\cup}}}\,} f^{-i}(B (R_{\alpha}^{i} x,b_i)) $ |
$ x \in [0,1) $ |
$ \alpha \in [0,1) $ |
$ B(x,r) $ |
$ r $ |
$ x \in [0,1) $ |
$ R_{\alpha}: [0,1) \rightarrow [0,1) $ |
$ \alpha \in [0,1) $ |
$ 3 $ |
References:
[1] |
J. S. Athreya,
Logarithm laws and shrinking target properties, Proc. Indian Acad. Sci. Math. Sci., 119 (2009), 541-557.
doi: 10.1007/s12044-009-0044-x. |
[2] |
J. Chaika,
Shrinking targets for IETs: Extending a theorem of Kurzweil, Geom. Funct. Anal., 21 (2011), 1020-1042.
doi: 10.1007/s00039-011-0130-y. |
[3] |
N. Chernov and D. Kleinbock,
Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math., 122 (2001), 1-27.
doi: 10.1007/BF02809888. |
[4] |
J. L. Fernández, M. V. Melián and D. Pestana,
Expanding maps, shrinking targets and hitting times, Nonlinearity, 25 (2012), 2443-2471.
doi: 10.1088/0951-7715/25/9/2443. |
[5] |
S. Galatolo,
Hitting time and dimension in axiom A systems, generic interval exchanges and an application to Birkoff sums, J. Stat. Phys., 123 (2006), 111-124.
doi: 10.1007/s10955-006-9041-y. |
[6] |
S. Galatolo and D. H. Kim,
The dynamical Borel-Cantelli lemma and the waiting time problems, Indag. Math. (N.S.), 18 (2007), 421-434.
doi: 10.1016/S0019-3577(07)80031-0. |
[7] |
M. Keane,
Interval exchange transformations, Math. Z., 141 (1975), 25-31.
doi: 10.1007/BF01236981. |
[8] |
S. Kerckhoff, H. Masur and J. Smillie,
Ergodicity of billiard flows and quadratic differentials, Ann. of Math., 124 (1986), 293-311.
doi: 10.2307/1971280. |
[9] |
D. H. Kim,
The shrinking target property of irrational rotations, Nonlinearity, 20 (2007), 1637-1643.
doi: 10.1088/0951-7715/20/7/006. |
[10] |
J. Kurzweil,
On the metric theory of inhomogeneous diophantine approximations, Studia Math., 15 (1955), 84-112.
doi: 10.4064/sm-15-1-84-112. |
[11] |
L. Marchese,
The Khinchin theorem for interval-exchange transformations, J. Mod. Dyn., 5 (2011), 123-183.
doi: 10.3934/jmd.2011.5.123. |
[12] |
F. Maucourant,
Dynamical Borel-Cantelli lemma for hyperbolic spaces, Israel J. Math., 152 (2006), 143-155.
doi: 10.1007/BF02771980. |
[13] |
A. Quas,
Ergodicity and mixing properties, Mathematics of Complexity and Dynamical Systems, 1-3 (2012), 225-240.
doi: 10.1007/978-1-4614-1806-1_15. |
show all references
References:
[1] |
J. S. Athreya,
Logarithm laws and shrinking target properties, Proc. Indian Acad. Sci. Math. Sci., 119 (2009), 541-557.
doi: 10.1007/s12044-009-0044-x. |
[2] |
J. Chaika,
Shrinking targets for IETs: Extending a theorem of Kurzweil, Geom. Funct. Anal., 21 (2011), 1020-1042.
doi: 10.1007/s00039-011-0130-y. |
[3] |
N. Chernov and D. Kleinbock,
Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math., 122 (2001), 1-27.
doi: 10.1007/BF02809888. |
[4] |
J. L. Fernández, M. V. Melián and D. Pestana,
Expanding maps, shrinking targets and hitting times, Nonlinearity, 25 (2012), 2443-2471.
doi: 10.1088/0951-7715/25/9/2443. |
[5] |
S. Galatolo,
Hitting time and dimension in axiom A systems, generic interval exchanges and an application to Birkoff sums, J. Stat. Phys., 123 (2006), 111-124.
doi: 10.1007/s10955-006-9041-y. |
[6] |
S. Galatolo and D. H. Kim,
The dynamical Borel-Cantelli lemma and the waiting time problems, Indag. Math. (N.S.), 18 (2007), 421-434.
doi: 10.1016/S0019-3577(07)80031-0. |
[7] |
M. Keane,
Interval exchange transformations, Math. Z., 141 (1975), 25-31.
doi: 10.1007/BF01236981. |
[8] |
S. Kerckhoff, H. Masur and J. Smillie,
Ergodicity of billiard flows and quadratic differentials, Ann. of Math., 124 (1986), 293-311.
doi: 10.2307/1971280. |
[9] |
D. H. Kim,
The shrinking target property of irrational rotations, Nonlinearity, 20 (2007), 1637-1643.
doi: 10.1088/0951-7715/20/7/006. |
[10] |
J. Kurzweil,
On the metric theory of inhomogeneous diophantine approximations, Studia Math., 15 (1955), 84-112.
doi: 10.4064/sm-15-1-84-112. |
[11] |
L. Marchese,
The Khinchin theorem for interval-exchange transformations, J. Mod. Dyn., 5 (2011), 123-183.
doi: 10.3934/jmd.2011.5.123. |
[12] |
F. Maucourant,
Dynamical Borel-Cantelli lemma for hyperbolic spaces, Israel J. Math., 152 (2006), 143-155.
doi: 10.1007/BF02771980. |
[13] |
A. Quas,
Ergodicity and mixing properties, Mathematics of Complexity and Dynamical Systems, 1-3 (2012), 225-240.
doi: 10.1007/978-1-4614-1806-1_15. |
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