Countable Partition

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Stratmann, Bernd O. - One of the best experts on this subject based on the ideXlab platform.

  • Strong renewal theorems and Lyapunov spectra for alpha-Farey and alpha-Luroth systems
    'Cambridge University Press (CUP)', 2013
    Co-Authors: Kesseboehmer Marc, Munday Sara, Stratmann, Bernd O.
    Abstract:

    In this paper, we introduce and study the alpha-Farey map and its associated jump transformation, the alpha-Luroth map, for an arbitrary Countable Partition alpha of the unit interval with atoms which accumulate only at the origin. These maps represent linearized generalizations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic theoretical properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called alpha-sum-level sets for the alpha-Luroth map. Similar results have previously been obtained for the Farey map and the Gauss map by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of the alpha-Farey map and the alpha-Luroth map in terms of the thermodynamical formalism. We show how to derive these spectra and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen Partition alpha.Publisher PDFPeer reviewe

  • Strong renewal theorems and Lyapunov spectra for $\alpha$-Farey and $\alpha$-L\"uroth systems
    'Cambridge University Press (CUP)', 2011
    Co-Authors: Kesseböhmer Marc, Munday Sara, Stratmann, Bernd O.
    Abstract:

    In this paper we introduce and study the $\alpha$-Farey map and its associated jump transformation, the $\alpha$-L\"uroth map, for an arbitrary Countable Partition $\alpha$ of the unit interval with atoms which accumulate only at the origin. These maps represent linearised generalisations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic-theoretic properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called $\alpha$-sum-level sets for the $\alpha$-L\"uroth map. Similar results have previously been obtained for the Farey map and the Gauss map, by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of the $\alpha$-Farey map and the $\alpha$-L\"uroth map in terms of the thermodynamical formalism. We show how to derive these spectra, and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen Partition $\alpha$.Comment: 29 pages, 16 figure

Munday Sara - One of the best experts on this subject based on the ideXlab platform.

  • Strong renewal theorems and Lyapunov spectra for alpha-Farey and alpha-Luroth systems
    'Cambridge University Press (CUP)', 2013
    Co-Authors: Kesseboehmer Marc, Munday Sara, Stratmann, Bernd O.
    Abstract:

    In this paper, we introduce and study the alpha-Farey map and its associated jump transformation, the alpha-Luroth map, for an arbitrary Countable Partition alpha of the unit interval with atoms which accumulate only at the origin. These maps represent linearized generalizations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic theoretical properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called alpha-sum-level sets for the alpha-Luroth map. Similar results have previously been obtained for the Farey map and the Gauss map by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of the alpha-Farey map and the alpha-Luroth map in terms of the thermodynamical formalism. We show how to derive these spectra and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen Partition alpha.Publisher PDFPeer reviewe

  • On the derivative of the \alpha-Farey-Minkowski function
    2012
    Co-Authors: Munday Sara
    Abstract:

    In this paper we study the family of $\alpha$-Farey-Minkowski functions $\theta_\alpha$, for an arbitrary Countable Partition $\alpha$ of the unit interval with atoms which accumulate only at the origin, which are the conjugating homeomorphisms between each of the $\alpha$-Farey systems and the tent map. We first show that each function $\theta_\alpha$ is singular with respect to the Lebesgue measure and then demonstrate that the unit interval can be written as the disjoint union of the following three sets: $\Theta_0:={x\in\U:\theta_\alpha'(x)=0}, \Theta_\infty:={x\in\U:\theta_\alpha'(x)=\infty} and \Theta_\sim:=\U\setminus(\Theta_0\cup\Theta_\infty)$. The main result is that [\dim_{\mathrm{H}}(\Theta_\infty)=\dim_{\mathrm{H}}(\Theta_\sim)=\sigma_\alpha(\log2)

  • Strong renewal theorems and Lyapunov spectra for $\alpha$-Farey and $\alpha$-L\"uroth systems
    'Cambridge University Press (CUP)', 2011
    Co-Authors: Kesseböhmer Marc, Munday Sara, Stratmann, Bernd O.
    Abstract:

    In this paper we introduce and study the $\alpha$-Farey map and its associated jump transformation, the $\alpha$-L\"uroth map, for an arbitrary Countable Partition $\alpha$ of the unit interval with atoms which accumulate only at the origin. These maps represent linearised generalisations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic-theoretic properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called $\alpha$-sum-level sets for the $\alpha$-L\"uroth map. Similar results have previously been obtained for the Farey map and the Gauss map, by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of the $\alpha$-Farey map and the $\alpha$-L\"uroth map in terms of the thermodynamical formalism. We show how to derive these spectra, and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen Partition $\alpha$.Comment: 29 pages, 16 figure

Yi Shi - One of the best experts on this subject based on the ideXlab platform.

  • a Countable Partition for singular flows and its application on the entropy theory
    arXiv: Dynamical Systems, 2019
    Co-Authors: Yi Shi, Fan Yang, Jiagang Yang
    Abstract:

    In this paper, we construct a Countable Partition $\mathscr{A}$ for flows with hyperbolic singularities by introducing a new cross section at each singularity. Such Partition forms a Kakutani tower in a neighborhood of the singularity, and turns out to have finite metric entropy for every invariant probability measure. Moreover, each element of $\mathscr{A}^\infty$ will stay in a scaled tubular neighborhood for arbitrarily long time. This new construction enables us to study entropy theory for singular flows away from homoclinic tangencies, and show that the entropy function is upper semi-continuous with respect to both invariant measures and the flows.

  • A Countable Partition for singular flows, and its application on the entropy theory
    2019
    Co-Authors: Yi Shi, Yang Fan, Yang Jiagang
    Abstract:

    In this paper, we construct a Countable Partition $\mathscr{A}$ for flows with hyperbolic singularities by introducing a new cross section at each singularity. Such Partition forms a Kakutani tower in a neighborhood of the singularity, and turns out to have finite metric entropy for every invariant probability measure. Moreover, each element of $\mathscr{A}^\infty$ will stay in a scaled tubular neighborhood for arbitrarily long time. This new construction enables us to study entropy theory for singular flows away from homoclinic tangencies, and show that the entropy function is upper semi-continuous with respect to both invariant measures and the flows.Comment: 4 figure

Jiagang Yang - One of the best experts on this subject based on the ideXlab platform.

  • a Countable Partition for singular flows and its application on the entropy theory
    arXiv: Dynamical Systems, 2019
    Co-Authors: Yi Shi, Fan Yang, Jiagang Yang
    Abstract:

    In this paper, we construct a Countable Partition $\mathscr{A}$ for flows with hyperbolic singularities by introducing a new cross section at each singularity. Such Partition forms a Kakutani tower in a neighborhood of the singularity, and turns out to have finite metric entropy for every invariant probability measure. Moreover, each element of $\mathscr{A}^\infty$ will stay in a scaled tubular neighborhood for arbitrarily long time. This new construction enables us to study entropy theory for singular flows away from homoclinic tangencies, and show that the entropy function is upper semi-continuous with respect to both invariant measures and the flows.

Yang Jiagang - One of the best experts on this subject based on the ideXlab platform.

  • A Countable Partition for singular flows, and its application on the entropy theory
    2019
    Co-Authors: Yi Shi, Yang Fan, Yang Jiagang
    Abstract:

    In this paper, we construct a Countable Partition $\mathscr{A}$ for flows with hyperbolic singularities by introducing a new cross section at each singularity. Such Partition forms a Kakutani tower in a neighborhood of the singularity, and turns out to have finite metric entropy for every invariant probability measure. Moreover, each element of $\mathscr{A}^\infty$ will stay in a scaled tubular neighborhood for arbitrarily long time. This new construction enables us to study entropy theory for singular flows away from homoclinic tangencies, and show that the entropy function is upper semi-continuous with respect to both invariant measures and the flows.Comment: 4 figure

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