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1
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Expanding repellers in limit sets for iterations of holomorphic functions

100%
Przytycki F.
Fundamenta Mathematicae
|
2005
|
tom 186
|
nr 1
85-96
EN
We prove that for Ω being an immediate basin of attraction to an attracting fixed point for a rational mapping of the Riemann sphere, and for an ergodic invariant measure μ on the boundary FrΩ, with positive Lyapunov exponent, there is an invariant subset of FrΩ which is an expanding repeller of Hausdorff dimension arbitrarily close to the Hausdorff dimension of μ. We also prove generalizations and a geometric coding tree abstract version. The paper is a continuation of a paper in Fund. Math. 145 (1994) by the author and Anna Zdunik, where the density of periodic orbits in FrΩ was proved.
2
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On the Hyperbolic Hausdorff Dimension of the Boundary of a Basin of Attraction for a Holomorphic Map and of Quasirepellers

100%
Przytycki F.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2006
|
tom 54
|
nr 1
41-52
EN
e prove that the hyperbolic Hausdorff dimension of Fr Ω, the boundary of the simply connected immediate basin of attraction Ω to an attracting periodic point of a rational mapping of the Riemann sphere, which is not a finite Blaschke product in some holomorphic coordinates, or a 2:1 factor of a Blaschke product, is larger than 1. We prove a "local version" of this theorem, for a boundary repelling to the side of the domain. The results extend an analogous fact for polynomials proved by A. Zdunik and relies on the theory elaborated by M. Urbański, A. Zdunik and the author in the late 80-ties. To prove that the dimension is larger than 1, we use expanding repellers in ∂Ω constructed in [P2]. To reach our results, we deal with a quasi-repeller, i.e. the limit set for a geometric coding tree, and prove that the hyperbolic Hausdorff dimension of the limit set is larger than the Hausdorff dimension of the projection via the tree of any Gibbs measure for a Hölder potential on the shift space, under a non-cohomology assumption. We also consider Gibbs measures for Hölder potentials on Julia sets.
3
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Remarks on the simple connectedness of basins of sinks for iterations of rational maps

100%
Przytycki F.
Banach Center Publications
|
1989
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tom 23
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nr 1
229-235
4
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Iterations of rational functions: which hyperbolic components contain polynomials?

100%
Przytycki F.
Fundamenta Mathematicae
|
1996
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tom 149
|
nr 2
95-118
EN
Let $H^d$ be the set of all rational maps of degree d ≥ 2 on the Riemann sphere, expanding on their Julia set. We prove that if $f ∈ H^d$ and all, or all but one, critical points (or values) are in the basin of immediate attraction to an attracting fixed point then there exists a polynomial in the component H(f) of $H^d$ containing f. If all critical points are in the basin of immediate attraction to an attracting fixed point or a parabolic fixed point then f restricted to the Julia set is conjugate to the shift on the one-sided shift space of d symbols. We give exotic} examples of maps of an arbitrary degree d with a non-simply connected completely invariant basin of attraction and arbitrary number k ≥ 2 of critical points in the basin. For such a map $f ∈ H^d$ with k
5
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Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps

100%
Przytycki F.
Fundamenta Mathematicae
|
1994
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tom 144
|
nr 3
259-278
EN
We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if A is completely invariant (i.e. $f^{-1}(A) = A$), and if μ is an arbitrary f-invariant measure with positive Lyapunov exponents on ∂A, then μ-almost every point q ∈ ∂A is accessible along a curve from A. In fact, we prove the accessibility of every "good" q, i.e. one for which "small neigh bourhoods arrive at large scale" under iteration of f. This generalizes the Douady-Eremenko-Levin-Petersen theorem on the accessibility of periodic sources. We prove a general "tree" version of this theorem. This allows us to deduce that on the limit set of a geometric coding tree (in particular, on the whole Julia set), if the diameters of the edges converge to 0 uniformly as the generation number tends to ∞, then every f-invariant probability ergodic measure with positive Lyapunov exponent is the image, via coding with the help of the tree, of an invariant measure on the full one-sided shift space. The assumption that f is holomorphic on A, or on the domain U of the tree, can be relaxed and one need not assume that f extends beyond A or U. Finally, we prove that if f is polynomial-like on a neighbourhood of ¯ℂ∖ A, then every "good" q ∈ ∂A is accessible along an external ray.
6
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Singularities of k-tuples of vector fields

69%
Jakubczyk B., Przytycki F.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Introduction............................................................................5 1. The main ideas and results................................................6 2. $H^{n,k}$-invariant subsets of $ℋ^{n,k}$.........................22 3. Reduction to germs of differential 1-forms........................35 4. The case k ≥ 2n-3. Proof of Theorem A...........................44 5. The case n = 3, k = 2.......................................................46 Appendix. Connections with control theory...........................59 List of symbols......................................................................61 References..........................................................................63 References to the Appendix.................................................63
7
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Porosity of Collet–Eckmann Julia sets

64%
Przytycki F., Rohde S.
Fundamenta Mathematicae
|
1998
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tom 155
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nr 2
189-199
EN
We prove that the Julia set of a rational map of the Riemann sphere satisfying the Collet-Eckmann condition and having no parabolic periodic point is mean porous, if it is not the whole sphere. It follows that the Minkowski dimension of the Julia set is less than 2.
8
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Topological invariance of the Collet–Eckmann property for S-unimodal maps

64%
Nowicki T., Przytycki F.
Fundamenta Mathematicae
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1998
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tom 155
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nr 1
33-43
EN
We prove that if f, g are smooth unimodal maps of the interval with negative Schwarzian derivative, conjugated by a homeomorphism of the interval, and f is Collet-Eckmann, then so is g.
9
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Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps: geometric coding trees technique

64%
Przytycki F., Zdunik A.
Fundamenta Mathematicae
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1994
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tom 145
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nr 1
65-77
EN
We prove that if A is a basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, then the periodic points in the boundary of A are dense in this boundary. To prove this in the non-simply connected or parabolic situations we prove a more abstract, geometric coding trees version.
10
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Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, II

51%
Przytycki F., Urbański M., Zdunik A.
Studia Mathematica
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1990
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tom 97
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nr 3
189-225
11
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On Ω-stability and structural stability of endomorphisms satisfying Axiom A

44%
Przytycki F.
Studia Mathematica
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1977
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tom 60
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nr 1
61-77
12
Content available remote

Construction of invariant sets for Anosov diffeomorphisms and hyperbolic attractors

44%
Przytycki F.
Studia Mathematica
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1980
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tom 68
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nr 2
199-213
13
Content available remote

On the law of iterated logarithm for Bloch functions

38%
Przytycki F.
Studia Mathematica
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1989
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tom 93
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nr 2
145-154
14
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Periodic points of linked twist mappings

32%
Przytycki F.
Studia Mathematica
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1986
|
tom 83
|
nr 1
19-24
15
Content available remote

Anosov endomorphisms

26%
Przytycki F.
Studia Mathematica
|
1976
|
tom 58
|
nr 3
249-285
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