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Piecewise Monotone Maps and the Gauss Endomorphism

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Book cover Transfer Operators, Endomorphisms, and Measurable Partitions

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2217))

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Abstract

The purpose of the next two chapters is to outline applications of our results to a family of examples of dynamics of endomorphisms, and their associated transfer operators.

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References

  1. D. Alpay, P.E.T. Jorgensen, I. Lewkowicz, W-markov measures, transfer operators, wavelets and multiresolutions (2016). arXiv:1606.07692

    Google Scholar 

  2. K. Arslan, V. Milousheva, Meridian surfaces of elliptic or hyperbolic type with pointwise 1-type Gauss map in Minkowski 4-space. Taiwan. J. Math. 20(2), 311–332 (2016)

    Article  MathSciNet  Google Scholar 

  3. B. Bektacs, E.Ö. Canfes, U. Dursun, On rotational surfaces in pseudo-Euclidean space \({\mathbb {E}^4_T}\) with pointwise 1-type Gauss map. Acta Univ. Apulensis Math. Inform. 45, 43–59 (2016)

    Google Scholar 

  4. X. Chao, Y. Lv, On the Gauss map of Weingarten hypersurfaces in hyperbolic spaces. Bull. Braz. Math. Soc. (N.S.) 47(4), 1051–1069 (2016)

    Article  MathSciNet  Google Scholar 

  5. I.P. Cornfeld, S.V. Fomin, Y.G. Sinaı̆, Ergodic Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245 (Springer, New York, 1982). Translated from the Russian by A. B. Sosinskiı̆

    Google Scholar 

  6. G.M. de Freitas, Submanifolds with homothetic Gauss map in codimension two. Geom. Dedicata 180, 151–170 (2016)

    Article  MathSciNet  Google Scholar 

  7. F.H. Ghane, A. Sarizadeh, Some stochastic properties of topological dynamics of semigroup actions. Topol. Appl. 204, 112–120 (2016)

    Article  MathSciNet  Google Scholar 

  8. K. Horbacz, M.Ślȩczka, Law of large numbers for random dynamical systems. J. Stat. Phys. 162(3), 671–684 (2016)

    Article  MathSciNet  Google Scholar 

  9. J.E. Hutchinson, Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)

    Article  MathSciNet  Google Scholar 

  10. P. Jaros, L. Maślanka, F. Strobin, Algorithms generating images of attractors of generalized iterated function systems. Numer. Algorithms 73(2), 477–499 (2016)

    Article  MathSciNet  Google Scholar 

  11. Y.-Q. Ji, Z. Liu, S.-il Ri, Fixed point theorems of the iterated function systems. Commun. Math. Res. 32(2), 142–150 (2016)

    Google Scholar 

  12. P.E.T. Jorgensen, F. Tian, Infinite networks and variation of conductance functions in discrete Laplacians. J. Math. Phys. 56(4), 043506, 27 (2015)

    Article  MathSciNet  Google Scholar 

  13. P.E.T. Jorgensen, S. Pedersen, F. Tian, Spectral theory of multiple intervals. Trans. Am. Math. Soc. 367(3), 1671–1735 (2015)

    Article  MathSciNet  Google Scholar 

  14. S. Kakutani, On equivalence of infinite product measures. Ann. Math. (2) 49, 214–224 (1948)

    Article  MathSciNet  Google Scholar 

  15. M. Keane, Strongly mixing g-measures. Invent. Math. 16, 309–324 (1972)

    Article  MathSciNet  Google Scholar 

  16. J. Llibre, Brief survey on the topological entropy. Discrete Contin. Dyn. Syst. Ser. B 20(10), 3363–3374 (2015)

    Article  MathSciNet  Google Scholar 

  17. C. Radin, Miles of Tiles. Student Mathematical Library, vol. 1 (American Mathematical Society, Providence, RI, 1999)

    MATH  Google Scholar 

  18. A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8, 477–493 (1957)

    Article  MathSciNet  Google Scholar 

  19. H.H. Rugh, The Milnor-Thurston determinant and the Ruelle transfer operator. Commun. Math. Phys. 342(2), 603–614 (2016)

    Article  MathSciNet  Google Scholar 

  20. T. Szarek, M. Urbański, A. Zdunik, Continuity of Hausdorff measure for conformal dynamical systems. Discrete Contin. Dyn. Syst. 33(10), 4647–4692 (2013)

    Article  MathSciNet  Google Scholar 

  21. Y. Yao, W. Li, Generating iterated function systems for the Vicsek snowflake and the Koch curve. Am. Math. Mon. 123(7), 716–721 (2016)

    Article  MathSciNet  Google Scholar 

  22. R. Ye, Y. Zou, J. Lu, Chaotic dynamical systems on fractals and their applications to image encryption, in Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis. Interdisciplinary Mathematical Sciences, vol. 15 (World Scientific, Hackensack, NJ, 2013), pp. 279–304

    Chapter  Google Scholar 

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Bezuglyi, S., Jorgensen, P.E.T. (2018). Piecewise Monotone Maps and the Gauss Endomorphism. In: Transfer Operators, Endomorphisms, and Measurable Partitions. Lecture Notes in Mathematics, vol 2217. Springer, Cham. https://doi.org/10.1007/978-3-319-92417-5_11

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