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Measure Preserving Interval Maps and Topological Conjugacy

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Abstract

This paper characterizes a continuous map s such that a measure preserving interval map t exists to be topologically conjugate with s. We show that under mild assumptions this problem is reduced to a Markov chain problem. We derive the conditions for such a t to exist and be unique.

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Notes

  1. For t to be a λ-preserving linear Markov map, Ai,j must be the same for all j given i whenever \(A^{*}_{i,j}=1\) and Eq. 7 must be satisfied. Whether {ai,j} exists to meet both conditions depends on A∗. For the first two A∗ in Example 3 t is a linear Markov map, and for the last A∗ t is not a linear or expanding Markov map.

References

  1. Block L, Coven EM. Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval. Trans Amer Math Soc 1987;300(1):297–306. http://www.jstor.org/stable/2000600. Accessed 4 Aug 2022.

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  4. Robinson RC. 2002. Perron Frobenius theorem. https://sites.math.northwestern.edu/~clark/354/2002/perron.pdf. Accessed 4 Aug 2022.

  5. Sericola B. 2013. Markov chains: theory and applications. Applied stochastic methods series. Wiley. https://books.google.com/books?id=tRdwAAAAQBAJ. Accessed 4 Aug 2022.

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Authors and Affiliations

  1. Stanford University, Stanford, CA, 94305, USA

    William Li

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  1. William Li
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Unless explicitly mentioned, all the results presented and proved in this paper are original.

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Correspondence to William Li.

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Li, W. Measure Preserving Interval Maps and Topological Conjugacy. J Dyn Control Syst 29, 569–582 (2023). https://doi.org/10.1007/s10883-022-09607-z

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  • DOI: https://doi.org/10.1007/s10883-022-09607-z

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