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| 2. |
- Färm, David, et al.
(författare)
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Dimension of Countable Intersections of Some Sets Arising in Expansions in Non-Integer Bases
- 2010
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Ingår i: Fundamenta Mathematicae. - : Institute of Mathematics, Polish Academy of Sciences. - 0016-2736 .- 1730-6329. ; 209, s. 157-176
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Tidskriftsartikel (refereegranskat)abstract
- Abstract in Undetermined We consider expansions of real numbers in non-integer bases. These expansions are generated by beta-shifts. We prove that some sets arising in metric number theory have the countable intersection property. This allows us to consider sets of reals that have common properties in a countable number of different (non-integer) bases. Some of the results are new even for integer bases.
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| 3. |
- Färm, David, et al.
(författare)
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Large intersection classes on fractals
- 2011
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Ingår i: Nonlinearity. - : IOP Publishing. - 0951-7715 .- 1361-6544. ; 24:4, s. 1291-1309
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Tidskriftsartikel (refereegranskat)abstract
- Abstract in Undetermined We consider limit sets of conformal iterated function systems, and introduce classes of subsets of these limit sets, with the property that the classes are closed under countable intersections and that all sets in the classes have a large Hausdorff dimension. Using these classes we determine the Hausdorff dimension and large intersection properties of some sets occurring in ergodic theory, Diophantine approximation and complex dynamics.
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| 4. |
- Färm, David, et al.
(författare)
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Non-typical points for β-shifts
- 2013
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Ingår i: Bulletin of the Polish Academy of Sciences: Mathematics. - : Institute of Mathematics, Polish Academy of Sciences. - 0239-7269 .- 1732-8985. ; 61:2, s. 123-132
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Tidskriftsartikel (refereegranskat)abstract
- We study sets of non-typical points under the map fβ↦βx mod 1 for non-integer β and extend our results from [Fund. Math. 209 (2010)] in several directions. In particular, we prove that sets of points whose forward orbit avoid certain Cantor sets, and the set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition β>1.541 found in the above paper can be removed.
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| 5. |
- Färm, David
(författare)
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Nontypical Behaviour of Orbits and Birkhoff Averages for Expanding Maps
- 2011
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of an introductory chapter followed by five papers. In the first paper, expanding maps on the unit interval are considered. The set of points for which the forward orbit is bounded away from a given point is studied. It is shown that this set has full Hausdorff dimension and that it has large intersection properties. In the second paper, expanding maps on the unit interval with bounded distortion are considered. The set of points for which certain Birkhoff averages accumulate at a given value is studied. It is shown that this set has large intersection properties. As an application, it is shown that the set of points for which these Birkhoff averages do not converge has full Hausdorff dimension, even for countably many different maps simultaneously. In the third paper, non-integer base expansions of numbers on the unit interval, are studied. For a dense set of bases, these expansions are generated by expanding maps for which the results from the first two papers of this thesis apply. In this paper, approximation arguments are used to extend these results to all bases. In the fourth paper, attractors of conformal iterated function systems are considered. A family of classes of sets with large intersection properties, introduced by K. Falconer, is extended from Euclidean spaces without holes to this new fractal setting. As an application, the results from the second paper of this thesis are generalized. In the fifth paper, a family of hyperbolic maps, similar to the fat baker's transformation, is studied. Depending on the parameters, these maps either expand or shrink area. It is shown, using the SRB-measures of the maps, that in the expanding case the attractor has positive Lebesgue measure for typical values of the parameters, while in the contracting case it tends to have Hausdorff dimension according to a certain formula. A key part of the proofs is the transversality of certain power series with coefficients from non-integer base expansions of real numbers.
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| 6. |
- Färm, David
(författare)
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Simultaneously non-convergent frequencies of words in different expansions
- 2011
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Ingår i: Monatshefte für Mathematik. - : Springer Science and Business Media LLC. - 0026-9255 .- 1436-5081. ; 162:4, s. 409-427
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Tidskriftsartikel (refereegranskat)abstract
- We consider expanding maps such that the unit interval can be represented as a full symbolic shift space with bounded distortion. There are already theorems about the Hausdorff dimension for sets defined by the set of accumulation points for the frequencies of words in one symbolic space at a time. We show that the dimension is preserved when such sets defined using different maps are intersected. More precisely, it is proven that the dimension of any countable intersection of sets defined by their sets of accumulation for frequencies of words in different expansions, has dimension equal to the infimum of the dimensions of the sets that are intersected. As a consequence, the set of numbers for which the frequencies do not exist has full dimension even after countable intersections. We also prove that this holds for a dense set of non-integer base expansions.
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| 7. |
- Färm, David
(författare)
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Simultaneously non-dense orbits under different expanding maps
- 2010
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Ingår i: Dynamical Systems. - : Informa UK Limited. - 1468-9367 .- 1468-9375. ; 25:4, s. 531-545
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Tidskriftsartikel (refereegranskat)abstract
- Given a point and an expanding map on the unit interval, we consider the set of points for which the forward orbit under this map is bounded away from the given point. It is well-known that in many cases such sets have full Hausdorff dimension. We prove that such sets have a large intersection property, i.e. countable intersections of such sets also have full Hausdorff dimension. This result applies to a class of maps including multiplication by integers modulo 1 and x -> 1/x modulo 1. We prove that the same properties hold for multiplication modulo 1 by a dense set of non-integer numbers between 1 and 2.
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