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Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
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Absolutely Continuous Invariant Measures for Piecewise Convex Maps
If $f$ is piecewise $C^2$, since your map is piecewise expanding it has an ACIM. See Lasota and York (Trans AMS, 1973).
See also the Chaos, Fractals, and Noise by A. Lasota and M. Mackey. One of the …