We completely describe the functional graph associated to iterations of Chebyshev polynomials over finite fields. Then, we use our structural results to obtain estimates for the average rho length, average number of connected components and the expected value for the period and preperiod of iterating Chebyshev polynomials.

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doi.org/10.1007/s10623-018-0545-7
Designs, Codes and Cryptography
School of Mathematics and Statistics

Qureshi, C. (Claudio), & Panario, D. (2018). The graph structure of Chebyshev polynomials over finite fields and applications. Designs, Codes and Cryptography. doi:10.1007/s10623-018-0545-7