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The recurrence dimension for piecewise monotonic maps of the interval. (English) Zbl 1170.37316
Summary: 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.
##### MSC:
 37E05 Dynamical systems involving maps of the interval 37C45 Dimension theory of smooth dynamical systems 28A80 Fractals 37B40 Topological entropy 28A78 Hausdorff and packing measures
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##### References:
 [1] V. Afraimovich, Pesins dimension for Poincaré recurrences, Chaos 7 (1997), 12-20. Zbl0933.37019 MR1439803 · Zbl 0933.37019 [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. Zbl1031.37024 MR1975573 · Zbl 1031.37024 [3] V. Afraimovich, J.-R. Chazottes and B. Saussol, Local dimensions for Poincaré recurrence, Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 64-74. Zbl0955.37013 MR1777857 · Zbl 0955.37013 [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. Zbl1012.37017 MR1868941 · Zbl 1012.37017 [5] F. Hofbauer, Piecewise invertible dynamical systems, Probab. Theory Related Fields 72 (1986), 359-386. Zbl0578.60069 MR843500 · Zbl 0578.60069 [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. Zbl0744.58042 MR1178961 · Zbl 0744.58042 [7] F. Hofbauer, Local dimension for piecewise monotonic maps on the interval, Ergod. Theory Dynam. Systems 15 (1995), 1119-1142. Zbl0842.58019 MR1366311 · Zbl 0842.58019 [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. Zbl0701.28005 MR1157469 · Zbl 0701.28005 [9] F. Hofbauer, P. Raith and T. Steinberger, Multifractal dimensions for invariant subsets of piecewise monotonic interval maps, Fund. Math. 176 (2003), 209-232. Zbl1051.37011 MR1992820 · Zbl 1051.37011 [10] G. Keller, Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems, Trans. Amer. Math. Soc. 314 (1989), 433-497. Zbl0686.58027 MR1005524 · Zbl 0686.58027 [11] G. Keller, Lifting measures to Markov extensions, Monatsh. Math. 108 (1989), 183-200. Zbl0712.28008 MR1026617 · Zbl 0712.28008 [12] W. de Melo and S. van Strien, “One-dimensional dynamics”, Springer-Verlag Berlin-Heidelberg-New York, 1993. Zbl0791.58003 MR1239171 · Zbl 0791.58003 [13] L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), 82-196. Zbl0841.28012 MR1361481 · Zbl 0841.28012 [14] Y. Pesin, “Dimension theory in dynamical systems: Contemporary views and applications”, The University of Chicago Press Chicago and London, 1997. Zbl0895.58033 MR1489237 · Zbl 0895.58033 [15] B. Saussol, S. Troubetzkoy and S. Vaienti, Recurrence, dimensions and Lyapunov exponents, J. Statist. Phys. 106 (2002), 623-634. Zbl1138.37300 MR1884547 · Zbl 1138.37300 [16] B. Saussol, S. Troubetzkoy and S. Vaienti, Recurrence and Lyapunov exponents, Moscow Math. J. 3 (2003), 189-203. Zbl1083.37504 MR1996808 · Zbl 1083.37504 [17] P. Walters, “An introduction to ergodic theory”, Springer-Verlag Berlin-Heidelberg-New York, 1982. Zbl0475.28009 MR648108 · Zbl 0475.28009
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