Linear response in the intermittent family: Differentiation in a weighted $C^0$-norm
Wael Bahsoun Benoît Saussol
Discrete & Continuous Dynamical Systems - A 2016, 36(12): 6657-6668 doi: 10.3934/dcds.2016089
We provide a general framework to study differentiability of SRB measures for one dimensional non-uniformly expanding maps. Our technique is based on inducing the non-uniformly expanding system to a uniformly expanding one, and on showing how the linear response formula of the non-uniformly expanding system is inherited from the linear response formula of the induced one. We apply this general technique to interval maps with a neutral fixed point (Pomeau-Manneville maps) to prove differentiability of the corresponding SRB measure. Our work covers systems that admit a finite SRB measure and it also covers systems that admit an infinite SRB measure. In particular, we obtain a linear response formula for both finite and infinite SRB measures. To the best of our knowledge, this is the first work that contains a linear response result for infinite measure preserving systems.
keywords: intermittent maps. Linear response
SRB measures for certain Markov processes
Wael Bahsoun Paweł Góra
Discrete & Continuous Dynamical Systems - A 2011, 30(1): 17-37 doi: 10.3934/dcds.2011.30.17
We study Markov processes generated by iterated function systems (IFS). The constituent maps of the IFS are monotonic transformations of the interval. We first obtain an upper bound on the number of SRB (Sinai-Ruelle-Bowen) measures for the IFS. Then, when all the constituent maps have common fixed points at 0 and 1, theorems are given to analyze properties of the ergodic invariant measures $\delta_0$ and $\delta_1$. In particular, sufficient conditions for $\delta_0$ and/or $\delta_1$ to be, or not to be, SRB measures are given. We apply some of our results to asset market games.
keywords: SRB-Measures. Iterated Function System
Quasi-invariant measures, escape rates and the effect of the hole
Wael Bahsoun Christopher Bose
Discrete & Continuous Dynamical Systems - A 2010, 27(3): 1107-1121 doi: 10.3934/dcds.2010.27.1107
Let $T$ be a piecewise expanding interval map and $T_H$ be an abstract perturbation of $T$ into an interval map with a hole. Given a number , 0 < < l, we compute an upper-bound on the size of a hole needed for the existence of an absolutely continuous conditionally invariant measure (accim) with escape rate not greater than -ln(1-). The two main ingredients of our approach are Ulam's method and an abstract perturbation result of Keller and Liverani.
keywords: Transfer Operator Interval Maps Ulam's Method. Escape Rates
Deterministic representation for position dependent random maps
Wael Bahsoun Christopher Bose Anthony Quas
Discrete & Continuous Dynamical Systems - A 2008, 22(3): 529-540 doi: 10.3934/dcds.2008.22.529
We give a deterministic representation for position dependent random maps and describe the structure of its set of invariant measures. Our construction generalizes the skew product representation of random maps with constant probabilities. In particular, we establish one-to-one correspondence between eigenfunctions corresponding to eigenvalues of unit modulus for the Frobenius-Perron (transfer) operator of the random map and for those of the skew. An immediate consequence is one-to-one correspondence between absolutely continuous invariant measures (acims) for the position dependent random map and acims for its deterministic representation.
keywords: Random map skew product absolutely continuous invariant measure.

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