A shrinking target theorem for ergodic transformations of the unit interval

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August 2022, 42(8): 4003-4011. doi: 10.3934/dcds.2022042

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

Received May 2021 Revised January 2022 Published August 2022 Early access April 2022

We show that for any ergodic Lebesgue measure preserving transformation

$ f: [0,1) \rightarrow [0,1) $

and any decreasing sequence

$ \{b_i\}_{i=1}^{\infty} $

of positive real numbers with divergent sum, the set

$ {\underset{n=1}{\overset{\infty}{\cap}} \, {\underset{i=n}{\overset{\infty}{\cup}}}\,} f^{-i}(B (R_{\alpha}^{i} x,b_i)) $

has full Lebesgue measure for almost every

$ x \in [0,1) $

and almost every

$ \alpha \in [0,1) $

. Here

$ B(x,r) $

is the ball of radius

$ r $

centered at

$ x \in [0,1) $

and

$ R_{\alpha}: [0,1) \rightarrow [0,1) $

is rotation by

$ \alpha \in [0,1) $

. As a corollary, we provide partial answer to a question asked by Chaika (Question

$ 3 $

, [2]) in the context of interval exchange transformations.

Keywords: Shrinking target problem, interval exchange transformations.

Mathematics Subject Classification: Primary: 37A05, 37A10, 37E10.

Citation: Shrey Sanadhya. A shrinking target theorem for ergodic transformations of the unit interval. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4003-4011. doi: 10.3934/dcds.2022042

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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