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Invariant measure for cubic Fibonacci-like polynomials

  • School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

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https://doi.org/10.52396/JUSTC-2023-0036
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  • Author Bio:

    Wenxiu Ma is currently a graduate student of University of Science and Technology of China. Her research mainly focuses on one dimensional dynamics and complex dynamics

  • Corresponding author: E-mail: mwx@mail.ustc.edu.cn
  • Received Date: 07 March 2023
  • Accepted Date: 05 June 2023
    Abstract

  • Abstract

    A special class of cubic polynomials possessing decay of geometry property is studied. This class of cubic bimodal maps has generalized Fibonacci combinatorics. For maps with bounded combinatorics, we show that they have an absolutely continuous invariant measure.

    Graphical abstract

    Abstract

    A special class of cubic polynomials possessing decay of geometry property is studied. This class of cubic bimodal maps has generalized Fibonacci combinatorics. For maps with bounded combinatorics, we show that they have an absolutely continuous invariant measure.

    • Public Summary

    • We study the combinatorial properties of (r, t)-Fibonacci bimodal maps.
    • We construct an induced map G and show that G admits an acip.
    • We prove that for any f ∈B, f has an acip.

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    • References(15)
  • [1]
    Bruin H, Shen W, van Strien S. Invariant measure exists without a growth condition. Communications in Mathematical Physics, 2003, 241: 287–306. doi: 10.1007/s00220-003-0928-z
    [2]
    Graczyk J, Swiatiek G. The Real Fatou Conjecture. Princeton, USA: Princeton University Press, 1998.
    [3]
    Jakobson M, Swiatiek G. Metric properties of non-renormalizable S-unimodal maps. Part I. Induced expansion and invariant measures. Ergodic Theory and Dynamical Systems, 1994, 14: 721–755. doi: 10.1017/S0143385700008130
    [4]
    Ji H, Li S. On the combinatorics of Fibonacci-like non-renormalizable maps. Commun. Math. Stat., 2020, 8: 473–496. doi: 10.1007/s40304-020-00210-x
    [5]
    Ji H, Ma W. Decay of geometry for a class of cubic polynomials. arXiv: 2304.10689, 2023.
    [6]
    Keller G, Nowicki T. Fibonacci maps re(al)-visited. Ergodic Theory and Dynamical Systems, 1995, 15: 99–120. doi: 10.1017/S0143385700008269
    [7]
    Kozlovski O, Shen W, van Strien S. Rigidity for real polynomials. Annals of Mathematics, 2007, 165 (3): 749–841. doi: 10.4007/annals.2007.165.749
    [8]
    Lyubich M, Milnor J. The Fibonacci unimodal map. J. Amer. Math. Soc., 1993, 6 (2): 425–457. doi: 10.1090/S0894-0347-1993-1182670-0
    [9]
    Lyubich M. Combinatorics, geometry and attractors of quasi-quadratic maps. Annals of Mathematics, 1994, 140: 345–404. doi: 10.2307/2118604
    [10]
    Mañé R. Hyperbolicity, sinks and measure in one-dimensional dynamics. Communications in Mathematical Physics, 1985, 100: 495–524. doi: 10.1007/BF01217727
    [11]
    de Melo W, van Strien S. One-Dimensional Dynamics. Berlin: Springer-Verlag, 1993.
    [12]
    Shen W. Decay of geometry for unimodal maps: An elementary proof. Annals of Mathematics, 2006, 163: 383–404. doi: 10.4007/annals.2006.163.383
    [13]
    Straube E. On the existence of invariant absolutely continuous measures. Communications in Mathematical Physics, 1981, 81: 27–30. doi: 10.1007/BF01941798
    [14]
    Swiatek G, Vargas E. Decay of geometry in the cubic family. Ergodic Theory and Dynamical Systems, 1998, 18: 1311–1329. doi: 10.1017/S0143385798117558
    [15]
    Vargas E. Fibonacci bimodal maps. Discrete and Continuous Dynamical Systems, 2008, 22 (3): 807–815. doi: 10.3934/dcds.2008.22.807
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    Figure  1.  Examples of types $ {\cal{A}} $ $ {\cal{B}} $ $ {\cal{C}} $.

    [1]
    Bruin H, Shen W, van Strien S. Invariant measure exists without a growth condition. Communications in Mathematical Physics, 2003, 241: 287–306. doi: 10.1007/s00220-003-0928-z
    [2]
    Graczyk J, Swiatiek G. The Real Fatou Conjecture. Princeton, USA: Princeton University Press, 1998.
    [3]
    Jakobson M, Swiatiek G. Metric properties of non-renormalizable S-unimodal maps. Part I. Induced expansion and invariant measures. Ergodic Theory and Dynamical Systems, 1994, 14: 721–755. doi: 10.1017/S0143385700008130
    [4]
    Ji H, Li S. On the combinatorics of Fibonacci-like non-renormalizable maps. Commun. Math. Stat., 2020, 8: 473–496. doi: 10.1007/s40304-020-00210-x
    [5]
    Ji H, Ma W. Decay of geometry for a class of cubic polynomials. arXiv: 2304.10689, 2023.
    [6]
    Keller G, Nowicki T. Fibonacci maps re(al)-visited. Ergodic Theory and Dynamical Systems, 1995, 15: 99–120. doi: 10.1017/S0143385700008269
    [7]
    Kozlovski O, Shen W, van Strien S. Rigidity for real polynomials. Annals of Mathematics, 2007, 165 (3): 749–841. doi: 10.4007/annals.2007.165.749
    [8]
    Lyubich M, Milnor J. The Fibonacci unimodal map. J. Amer. Math. Soc., 1993, 6 (2): 425–457. doi: 10.1090/S0894-0347-1993-1182670-0
    [9]
    Lyubich M. Combinatorics, geometry and attractors of quasi-quadratic maps. Annals of Mathematics, 1994, 140: 345–404. doi: 10.2307/2118604
    [10]
    Mañé R. Hyperbolicity, sinks and measure in one-dimensional dynamics. Communications in Mathematical Physics, 1985, 100: 495–524. doi: 10.1007/BF01217727
    [11]
    de Melo W, van Strien S. One-Dimensional Dynamics. Berlin: Springer-Verlag, 1993.
    [12]
    Shen W. Decay of geometry for unimodal maps: An elementary proof. Annals of Mathematics, 2006, 163: 383–404. doi: 10.4007/annals.2006.163.383
    [13]
    Straube E. On the existence of invariant absolutely continuous measures. Communications in Mathematical Physics, 1981, 81: 27–30. doi: 10.1007/BF01941798
    [14]
    Swiatek G, Vargas E. Decay of geometry in the cubic family. Ergodic Theory and Dynamical Systems, 1998, 18: 1311–1329. doi: 10.1017/S0143385798117558
    [15]
    Vargas E. Fibonacci bimodal maps. Discrete and Continuous Dynamical Systems, 2008, 22 (3): 807–815. doi: 10.3934/dcds.2008.22.807
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