Porcupine-Like Horseshoes: Topological and Ergodic Aspects

Díaz, Lorenzo J.; Gelfert, Katrin

doi:10.1007/978-3-642-38830-9_12

Cited by 2 publications

(4 citation statements)

References 23 publications

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“…The general overview of the theory of groups acting on the circle and real line may be found in book and article by Navas [34,35]. Moreover, certain examples of skew-products with the Bernoulli shift in the base are investigated as models of partially hyperbolic dynamics (see [14][15][16]29]).…”

Section: Historical Remarksmentioning

confidence: 99%

Alsedà–Misiurewicz systems with place-dependent probabilities*

Czudek

2020

Nonlinearity

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We consider systems of two specific piecewise linear homeomorphisms of the unit interval, so called Alsedà–Misiurewicz systems, and investigate the basic properties of Markov chains which arise when these two transformations are applied randomly with probabilities depending on the point of the interval. Though this iterated function system is not contracting in average and known methods do not apply, stability and the strong law of large numbers are proven.

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Section: Historical Remarksmentioning

confidence: 99%

Alsedà–Misiurewicz systems with place-dependent probabilities*

Czudek

2020

Nonlinearity

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“…For a given a finite set W of words we define the following sets: 6 Condition > t t c 1 implies that the evanescent spine occurs inside a transitive set.…”

Section: Definition 21 (Concatenations) Consider a Set Of Wordsmentioning

confidence: 99%

“…Investigating random iterations of general noncontracting IFS, fractal properties and, in particular, relations between Lyapunov exponents, dimension, and entropy have been studied recently (see, e.g. [9,8] and also the references in [6]). Such approaches focus on properties of measures that are stationary with respect to the IFS and are "essentially" contracting, compare the discussion in the introduction of [13].…”

Section: Introductionmentioning

confidence: 99%

“…Interesting dynamical properties of these skew-products such as occurrence of heterodimensional cycles, transitivity, intermingled contracting and expanding dynamics, and phase transitions associated to the central exponents arise from the reversion of the orientation, the cycle property of f 1 , and minimality-like properties of the iterated function systems (IFS) associated to f 0 and f 1 . See [5,[7][8][9] and the survey [6].…”

Section: Introductionmentioning

confidence: 99%

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Generation of spines in porcupine-like horseshoes

Díaz

Marcarini

2015

Nonlinearity

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We study certain one-parameter families of partially hyperbolic maps Ft : Σ 2 ×R → Σ 2 ×R of skew-product type generating so-called porcupinelike horseshoes. Such sets are topologically transitive and semiconjugate to the shift map in two symbols. They exhibit a very rich fiber structure characterized by the fact that the set Σ 2 is the disjoint union of two dense and uncountable subsets with opposite behavior: corresponding spines (preimage of a sequence by the semiconjugation) are nontrivial and trivial, respectively, that is, the semiconjugation is noninjective and injective, respectively. We will study the bifurcation process of creation and annihilation of nontrivial spines as the parameter t evolves. In particular, we focus on the Hausdorff dimension of these subsets of Σ 2. This study illustrates the richness of the process.

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Porcupine-Like Horseshoes: Topological and Ergodic Aspects

Cited by 2 publications

References 23 publications

Alsedà–Misiurewicz systems with place-dependent probabilities*

Alsedà–Misiurewicz systems with place-dependent probabilities*

Generation of spines in porcupine-like horseshoes

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