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Persistent Non-statistical Dynamics in One-Dimensional Maps

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Abstract

We study a class \(\widehat{{\mathfrak {F}}}\) of one-dimensional full branch maps introduced in Coates et al. (Commun Math Phys 402(2):1845–1878, 2023), admitting two indifferent fixed points as well as critical points and/or singularities with unbounded derivative. We show that \(\widehat{{\mathfrak {F}}}\) can be partitioned into 3 pairwise disjoint subfamilies \(\widehat{{\mathfrak {F}}} = {\mathfrak {F}} \cup {\mathfrak {F}}_\pm \cup {\mathfrak {F}}_*\) such that all \(g \in {\mathfrak {F}}\) have a unique physical measure equivalent to Lebesgue, all \(g \in {\mathfrak {F}}_{\pm }\) have a physical measure which is a Dirac-\(\delta \) measure on one of the (repelling) fixed points, and all \(g \in {\mathfrak {F}}_{*}\) are non-statistical and in particular have no physical measure. Moreover we show that these subfamilies are intermingled: they can all be approximated by maps in the other subfamilies in natural topologies.

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Notes

  1. Notice that we can “normalize” this extended metric to give a standard bounded metric on \( \widehat{{\mathfrak {F}}} \) by defining \( {\tilde{d}}_r ( f, g ) :={ {\tilde{d}}_{r} ( f,g ) }/{(1 + {\tilde{d}}_{r} (f, g ))} \) when \( d_{r} ( f,g )< \infty \) and \( {\tilde{d}}_{r} ( f,g ) :=1 \) otherwise. The metrics \( d_{r}\) and \( \tilde{d}_{r}\) lead to equivalent topologies and so for our purposes it does not really matter which one we use.

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  1. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo (USP), São Carlos, Brasil

    Douglas Coates

  2. Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy

    Stefano Luzzatto

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  1. Douglas Coates
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D.C. was partially supported by the ERC project 692925 NUHGD and by the Abdus Salam ICTP visitors program.

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Coates, D., Luzzatto, S. Persistent Non-statistical Dynamics in One-Dimensional Maps. Commun. Math. Phys. 405, 102 (2024). https://doi.org/10.1007/s00220-024-04957-0

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