Simon, K., Solomyak, B. and Urba nski, M. (1998). Parabolic iterated function systems with overlaps II: invariant measures, preprint. Home/Search Document Details and Download
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Singularity of Some Random Continued Fractions - Lyons (1999) (8 citations) (Correct)
....1995, Peres and Solomyak 1996) However, based on some numerical evidence, we believe that there is also an interval where is absolutely continuous: Conjecture 1.2. For all suciently small, is absolutely continuous with respect to Lebesgue measure. After seeing a preprint of this work, Simon, Solomyak, and Urba nksi (1998) made a great deal of progress on this conjecture. They proved that for Lebesgue a.e. 2 (0:215; c ) the measure is absolutely continuous. In particular, this shows that the threshold c in Theorem 1.1 is sharp. Other work on random continued fractions includes Bernadac (1993, 1995) ....
....; 2:2) which by (2.1) is at most lim n 1 log 2 n log D n (C n M D n ) M = log 2 2 ; 2:3) where is the (top) Lyapunov exponent of the random matrix 1 X 1 1 X ; see (2.4) for the de nition of and Bougerol and Lacroix (1985) Cor. VI.2.3, for this property of . [Simon, Solomyak and Urba nski (1998) have now proved that the quantity in (2.2) is in fact equal to that in (2.3) for Lebesgue a.e. 2 ( c ; 1=2) In particular, is singular if 1 2 log 2. Note that is strictly increasing in since 1 X 1 1 X is stochastically increasing in . Thus, if c , then is ....
Simon, K., Solomyak, B. and Urba nski, M. (1998). Parabolic iterated function systems with overlaps II: invariant measures, preprint.
Invariant Measures For Parabolic Ifs With Overlaps And.. - Simon, Solomyak.. (1999) (1 citation) Self-citation (Simon Solomyak) (Correct)
....Section 6 we prove the results on random continued fractions; the main difficulty is checking transversality. Section 7 contains concluding remarks; in particular, we present the (much easier) hyperbolic analog of our main theorem. A preliminary version of this paper was circulated as a preprint [SSU2]. 4 K. SIMON, B. SOLOMYAK, AND M. URBA NSKI 2. Definitions and statement of main result Let X ae R be a closed interval and 2 (0; 1] A C 1 map OE : X X is hyperbolic if 0 jOE 0 (x)j 1 for all x 2 X. We say that a C 1 map OE : X X is parabolic if the following ....
K. Simon, B. Solomyak, and M. Urba'nski, Parabolic iterated function systems with overlaps II:
Absolutely Continuous Measures With L^q Density Arising.. - Solomyak, Urbanski Self-citation (Solomyak) (Correct)
....overlaps on the real line and invariant measures associated with them. A shift invariant Borel probability measure on the coding space projects into a measure on the limit set of the IFS. We consider families of IFS depending on parameter and satisfying a certain transversality condition. In [SSU2] sufficient conditions were found for the measure to be absolutely continuous for a.e. parameter value. Here we investigate when this measure has a density in L q , for q 2 (1; 2] We prove that if the q dimension of the measure , with respect to a certain metric, is greater than one, then has ....
....threshold is sharp and establish the existence of L 2 density for a.e. parameter in some interval below the threshold. 1. Introduction We continue to study parabolic iterated function systems with overlaps on the real line. In [SSU1] we investigated the Hausdorff dimension of the limit set. In [SSU2] we focused on the properties of invariant measures. An ergodic shift invariant measure with positive entropy on the symbolic space induces an invariant (stationary) measure on the limit set of the iterated function system. The Hausdorff dimension of this measure equals the ratio of entropy over ....
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K. Simon, B. Solomyak, and M. Urba'nski, Parabolic iterated function systems with overlaps, invariant measures Preprint, 1998.
Absolutely Continuous Measures With L^q Density Arising.. - Solomyak, Urbanski Self-citation (Solomyak) (Correct)
....have a density in L 2 above a certain threshold; we show that this threshold is sharp and establish the existence of L 2 density for a.e. parameter in some interval below the threshold. 1. Introduction We continue to study parabolic iterated function systems with overlaps on the real line. In [SSU1] we investigated the Hausdorff dimension of the limit set. In [SSU2] we focused on the properties of invariant measures. An ergodic shift invariant measure with positive entropy on the symbolic space induces an invariant (stationary) measure on the limit set of the iterated function system. The ....
....PARABOLIC IFS 3 Lemma 2.1. For every neighborhood V of v there exists a constant L(V ) 1 such that for all x 2 X n V and all n 1, L(V ) Gamma1 jOE n (x) Gamma vj Delta (n 1) 1 fi L(V ) 2.2) L(V ) Gamma1 j(OE n ) x)j Delta (n 1) fi 1 fi L(V ) 2. 3) Now, following [SSU1, SSU2], we define the class of parabolic iterated function systems (IFS) under investigation. Definition 2.2. Let Phi = fOE 1 ; OE k g be a collection of C 1 functions on a closed interval X ae R such that OE k is parabolic with the fixed point v and the other functions are hyperbolic. We ....
[Article contains additional citation context not shown here]
K. Simon, B. Solomyak, and M. Urba'nski, Parabolic iterated function systems with overlaps, dimension of the limit set, Preprint, 1998.
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