What are some applications of arithmetic dynamics? In classical real or complex dynamics, we iterate over the reals or complex numbers. One application of this, among many, is the discrete logistic map for population growth.
In arithmetic dynamics, we iterate polynomial or rational maps over, for instance, finite fields, the rationals or a $p$-adic field. (Not an exhaustive list.)
Franco Vivaldi has investigated round-off errors in computer arithmetic using $p$-adic numbers. (See http://www.maths.qmul.ac.uk/~fvivaldi/research/ for more information.)
What are some other applications of arithmetic dynamics?
 A: There are many applications, in particular to cryptography. There is a book Applied Algebraic Dynamics;
see also the articles T-functions revisited: new criteria for bijectivity/transitivity and Secure cloud computations: Description of (fully)homomorphic ciphers within the P-adic model of encryption.
Biology: Automaton model of protein: Dynamics of conformational and functional states
Cognition and psychology:
A.Yu. Khrennikov, Human subconscious as the $p$-adic dynamical system. Journal of Theoretical Biology, 193, 179-196 (1998).
D. Dubischar, M. Gundlach, O. Steinkamp, A.Yu. Khrennikov,
A $p$-adic model for the process of thinking disturbed by physiological and information noise. Journal of
Theoretical Biology, 197, 451-467 (1999).
A.Yu. Khrennikov, Information Dynamics in Cognitive, Psychological, Social and Anomalous
Phenomena, Springer-Science + Business Media, B.Y., Dordrecht, NL, 2004.
Albeverio S, Khrennikov A and Kloeden P E  Memory retrieval as a $p$-adic dynamical system
BioSystems 49 105--115 (1999).
Khrennikov, A. (2002). Classical and Quantum Mental Models and Freud's Theory of Unconscious Mind.
Växjö, SWE: Växjö University Press.
A.Yu. Khrennikov, Modelling of psychological behavior on the basis of ultrametric mental space:
Encoding of categories by balls. P-Adic Numbers, Ultrametric Analysis, and Applications, 2, 1-20 (2010).
A: Pollard's Rho algorithm (and its variations) for factoring an integer $N$ essentially rely on the structure revealed by repeated iteration of a polynomial mod $N$. As far as I recall it remains one of the fastest algorithms for finding small factors of composite $N$.
A: Monomial dynamical systems over finite fields by Colón-Reyes, Jarrah, Laubenbacher and Sturmfels mentions some applications of dynamics over finite fields in the introduction:

Finite dynamical systems are time-discrete dynamical systems on finite state sets. Well-known
examples include cellular automata and Boolean networks, which have found broad applications in
engineering, computer science, and, more recently, computational biology. (See, e.g., [15; 1; 7; 19]
for biological applications.) More general multi-state systems have been used in control theory
[11; 20; 22; 23], the design and analysis of computer simulations [4; 2; 3; 18].

