The recurrence dimension for piecewise monotonic maps of the interval
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 3, pp. 439-449.

We investigate a weighted version of Hausdorff dimension introduced by V. Afraimovich, where the weights are determined by recurrence times. We do this for an ergodic invariant measure with positive entropy of a piecewise monotonic transformation on the interval [0,1], giving first a local result and proving then a formula for the dimension of the measure in terms of entropy and characteristic exponent. This is later used to give a relation between the dimension of a closed invariant subset and a pressure function.

Classification : 37E05, 37C45, 28A80, 37B40, 28A78
Hofbauer, Franz 1

1 Fakultät für Mathematik Universität Wien Nordbergstraße 15 A 1090 Wien, Austria
@article{ASNSP_2005_5_4_3_439_0,
     author = {Hofbauer, Franz},
     title = {The recurrence dimension for piecewise monotonic maps of the interval},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {439--449},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 4},
     number = {3},
     year = {2005},
     mrnumber = {2185864},
     zbl = {1170.37316},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2005_5_4_3_439_0/}
}
TY  - JOUR
AU  - Hofbauer, Franz
TI  - The recurrence dimension for piecewise monotonic maps of the interval
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2005
SP  - 439
EP  - 449
VL  - 4
IS  - 3
PB  - Scuola Normale Superiore, Pisa
UR  - http://www.numdam.org/item/ASNSP_2005_5_4_3_439_0/
LA  - en
ID  - ASNSP_2005_5_4_3_439_0
ER  - 
%0 Journal Article
%A Hofbauer, Franz
%T The recurrence dimension for piecewise monotonic maps of the interval
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2005
%P 439-449
%V 4
%N 3
%I Scuola Normale Superiore, Pisa
%U http://www.numdam.org/item/ASNSP_2005_5_4_3_439_0/
%G en
%F ASNSP_2005_5_4_3_439_0
Hofbauer, Franz. The recurrence dimension for piecewise monotonic maps of the interval. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 3, pp. 439-449. http://www.numdam.org/item/ASNSP_2005_5_4_3_439_0/

[1] V. Afraimovich, Pesins dimension for Poincaré recurrences, Chaos 7 (1997), 12-20. | MR | Zbl

[2] V. Afraimovich and J. Urias, Dimension-like characteristics of invariant sets in dynamical systems, In: “Dynamics and randomness”, A. Maass, S. Martinez and J. San Martin (eds.), Kluwer Academic Publishers, Dordrecht, 2002, pp. 1-30. | MR | Zbl

[3] V. Afraimovich, J.-R. Chazottes and B. Saussol, Local dimensions for Poincaré recurrence, Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 64-74. | MR | Zbl

[4] J.-R. Chazottes and B. Saussol, On pointwise dimensions and spectra of measures, C. R. Acad. Sci. Paris Ser. I Math. 333 (2001), 719-723. | MR | Zbl

[5] F. Hofbauer, Piecewise invertible dynamical systems, Probab. Theory Related Fields 72 (1986), 359-386. | MR | Zbl

[6] F. Hofbauer, An inequality for the Ljapunov exponent of an ergodic invariant measure for a piecewise monotonic map on the interval, In: “Lyapunov exponents”, Proceedings, Oberwolfach, 1990, Lecture Notes in Mathematics 1486, L. Arnold, H. Crauel and J.-P. Eckmann (eds.), Springer, Berlin, 1991, pp. 227-231. | MR | Zbl

[7] F. Hofbauer, Local dimension for piecewise monotonic maps on the interval, Ergod. Theory Dynam. Systems 15 (1995), 1119-1142. | MR | Zbl

[8] F. Hofbauer and P. Raith, The Hausdorff dimension of an ergodic invariant measure for a piecewise monotonic map of the interval, Canad. J. Math. 35 (1992), 84-98. | MR | Zbl

[9] F. Hofbauer, P. Raith and T. Steinberger, Multifractal dimensions for invariant subsets of piecewise monotonic interval maps, Fund. Math. 176 (2003), 209-232. | MR | Zbl

[10] G. Keller, Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems, Trans. Amer. Math. Soc. 314 (1989), 433-497. | MR | Zbl

[11] G. Keller, Lifting measures to Markov extensions, Monatsh. Math. 108 (1989), 183-200. | MR | Zbl

[12] W. De Melo and S. Van Strien, “One-dimensional dynamics”, Springer-Verlag Berlin-Heidelberg-New York, 1993. | MR | Zbl

[13] L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), 82-196. | MR | Zbl

[14] Y. Pesin, “Dimension theory in dynamical systems: Contemporary views and applications”, The University of Chicago Press Chicago and London, 1997. | MR | Zbl

[15] B. Saussol, S. Troubetzkoy and S. Vaienti, Recurrence, dimensions and Lyapunov exponents, J. Statist. Phys. 106 (2002), 623-634. | MR | Zbl

[16] B. Saussol, S. Troubetzkoy and S. Vaienti, Recurrence and Lyapunov exponents, Moscow Math. J. 3 (2003), 189-203. | MR | Zbl

[17] P. Walters, “An introduction to ergodic theory”, Springer-Verlag Berlin-Heidelberg-New York, 1982. | MR | Zbl