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Description
We investigate Lyapunov exponents as dynamical probes of black hole phase transitions in dilatonic Reissner-Nordstrom-AdS black holes within Einstein-Maxwell-dilaton theory. Thermodynamic quantities and the Lyapunov exponent of charged probe particles are analyzed in both the Einstein and string frames, providing a direct comparison between the thermodynamic phase structure of the black hole and that captured by the Lyapunov exponent. Thermodynamic quantities, including the Hawking temperature and Wald entropy, are invariant under conformal frame transformations, yielding identical phase structures in the two frames. By contrast, the Lyapunov exponent exhibits a nontrivial frame dependence for massive probe particles due to the dilaton coupling, while the frame dependence disappears in the massless limit. Numerical analysis shows that the phase structure captured by the Lyapunov exponent, including characteristic cusp behavior and transition points, is independent of the choice of frame, despite the frame-dependent value of the Lyapunov exponent itself. Therefore, the Lyapunov exponent exhibits frame-dependent values, whereas the critical structure it captures is preserved across conformal frames.
