Local limit theorem and distribution of periodic orbits of Lasota-Yorke transformations with infinite Markov partition

Takehiko MORITA

doi:10.2969/jmsj/04620309

Takehiko MORITA

Author information

JOURNAL FREE ACCESS

1994 Volume 46 Issue 2 Pages 309-343

DOI https://doi.org/10.2969/jmsj/04620309

Details

Published: 1994 Received: September 04, 1991 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
Correction information
Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : Dedicated to Professor Takesi Watanabe on his sixtieth birthday

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) V. Baradi and G. Keller, Zeta functions and transfer operators for piecewise monotonic transformations, Comm. Math. Phys., 127 (1990), 459-47
2) R. Bowen, Bernoulli maps of the interval, Israel J. Math., 28 (1977), 161-168.
3) R. Bowen and C. Series, Markov maps associated to Fuchsian groups, Publ. Math. IHES, 50 (1980), 401-418.
4) I. P. Cornfeld, S. V. Formin and Ya. G. Sinai, Ergodic theory, Springer, 1982.
5) N. Dunford and J. T. Schwartz, Linear operators I, Interscience, 1955.
6) N. Haydn, Meromorphic extension of the zeta function for Axiom A flows, Ergodic Theory Dynamical Systems, 10 (1990), 347-360.
7) F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z., 180 (1982), 119-140.
8) I. A. lbragimov and Y. V. Linnik, Independent and stationary sequences of random variables, Wolter-Norolhoff, 1971.
9) C. Ionescu Tulcea and G. Marinescu, Théorie ergodique pour des classes d'operations non complètement continues, Ann. of Math., 52 (1950), 141-147.
10) H. Ishitani, A central limit theorem of mixed type for a class of 1-dimensional transformations, Hiroshima Math. J., 16 (1986), 161-188.
11) T. Kato, Perturbation theory for linear operators, Springer, 1976.
12) A. Lasota and J. Yorke, On the existence of invariant measures for piecewise monotonec transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488.
13) D. Mayer, On a ζ function related to the continued fraction transformation, Bull. Soc. Math. France, 104 (1976), 195-203.
14) D. Mayer, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys., 130 (1990), 311-333.
15) T. Morita, A generalized local limit theorem for Lasota-Yorke transformations, Osaka J. Math., 26 (1989), 579-595.
16) W. Parry, Bowen's equidistribution theory and Dirichlet density theorem, Ergodic Theory Dynamical System, 4 (1984), 117-134.
17) W. Parry and M. Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flow, Ann. of Math., 118 (1983), 573-592.
18) W. Philipp, The remainder in the central limit theorem for mixing stochastic processes, Ann. Math. Statist., 40 (1969), 601-609.
19) W. Philipp, Mixing sequence of random variables and probabilistic number theory, Mem. Amer. Math. Soc., 114 (1971).
20) M. Pollicott, Distributions of closed geodesics on the modular surface and quadratic irrationals, Bull. Soc. Math. France, 114 (1986), 431-446.
21) M. Pollicott, Meromorphic extensions of generalized zeta functions, Invent. Math., 85 (1986), 147-164.
22) J. Rousseau-Egele, Un thèoréme de la limite locale pour une classes de transformations dilatantes et monotones par marceaux, Ann. Prob., 11 (1983), 772-788.
23) D. Ruelle, Thermodynamic formalism, Addison-Wesley, 1978.
24) M. Rychlik, Bounded variation and invariant measures, Studia Math., LXXVI (1983), 69-80.
25) L. Schwartz, Théorie de distributions, Hermann, 1966.
26) C. Series, The modular surface and continued fractions, J. London Math. Soc., 31 (1985), 69-80.

Right : [1] V. Baradi and G. Keller, Zeta functions and transfer operators for piecewise monotonic transformations, Comm. Math. Phys., 127 (1990), 459-477.
[2] R. Bowen, Bernoulli maps of the interval, Israel J. Math., 28 (1977), 161-168.
[3] R. Bowen and C. Series, Markov maps associated to Fuchsian groups, Publ. Math. IHES, 50 (1980), 401-418.
[4] I. P. Cornfeld, S. V. Formin and Ya. G. Sinai, Ergodic theory, Springer, 1982.
[5] N. Dunford and J. T. Schwartz, Linear operators I, Interscience, 1955.
[6] N. Haydn, Meromorphic extension of the zeta function for Axiom A flows, Ergodic Theory Dynamical Systems, 10 (1990), 347-360.
[7] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z., 180 (1982), 119-140.
[8] I. A. lbragimov and Y. V. Linnik, Independent and stationary sequences of random variables, Wolter-Norolhoff, 1971.
[9] C. Ionescu Tulcea and G. Marinescu, Théorie ergodique pour des classes d'operations non complètement continues, Ann. of Math., 52 (1950), 141-147.
[10] H. Ishitani, A central limit theorem of mixed type for a class of 1-dimensional transformations, Hiroshima Math. J., 16 (1986), 161-188.
[11] T. Kato, Perturbation theory for linear operators, Springer, 1976.
[12] A. Lasota and J. Yorke, On the existence of invariant measures for piecewise monotonec transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488.
[13] D. Mayer, On a ζ function related to the continued fraction transformation, Bull. Soc. Math. France, 104 (1976), 195-203.
[14] D. Mayer, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys., 130 (1990), 311-333.
[15] T. Morita, A generalized local limit theorem for Lasota-Yorke transformations, Osaka J. Math., 26 (1989), 579-595.
[16] W. Parry, Bowen's equidistribution theory and Dirichlet density theorem, Ergodic Theory Dynamical System, 4 (1984), 117-134.
[17] W. Parry and M. Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flow, Ann. of Math., 118 (1983), 573-592.
[18] W. Philipp, The remainder in the central limit theorem for mixing stochastic processes, Ann. Math. Statist., 40 (1969), 601-609.
[19] W. Philipp, Mixing sequence of random variables and probabilistic number theory, Mem. Amer. Math. Soc., 114 (1971).
[20] M. Pollicott, Distributions of closed geodesics on the modular surface and quadratic irrationals, Bull. Soc. Math. France, 114 (1986), 431-446.
[21] M. Pollicott, Meromorphic extensions of generalized zeta functions, Invent. Math., 85 (1986), 147-164.
[22] J. Rousseau-Egele, Un thèoréme de la limite locale pour une classes de transformations dilatantes et monotones par marceaux, Ann. Prob., 11 (1983), 772-788.
[23] D. Ruelle, Thermodynamic formalism, Addison-Wesley, 1978.
[24] M. Rychlik, Bounded variation and invariant measures, Studia Math., LXXVI (1983), 69-80.
[25] L. Schwartz, Théorie de distributions, Hermann, 1966.
[26] C. Series, The modular surface and continued fractions, J. London Math. Soc., 31 (1985), 69-80.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

Corresponding author

Correction information

Register with J-STAGE for free!