MAXIMAL MEASURE AND ENTROPIC CONTINUITY OF LYAPUNOV EXPONENTS FOR $C^r$ SURFACE DIFFEOMORPHISMS WITH LARGE ENTROPY

David Burguet (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

We prove a finite smooth version of the entropic continuity of Lyapunov exponents of Buzzi-Crovisier-Sarig for $C^\infty$ surface diffeomorphisms [9]. As a consequence we show that any $C^r$, $r > 1$, smooth surface diffeomorphism $f$ with $h_{top}(f) > \frac{1}{r} \limsup_n \frac{1}{n} \log^+ \|df^n\|$ admits a measure of maximal entropy. We also prove the $C^r$ continuity of the topological entropy at $f$.

Domains: Dynamical Systems [math.DS]

David Burguet. MAXIMAL MEASURE AND ENTROPIC CONTINUITY OF LYAPUNOV EXPONENTS FOR $C^r$ SURFACE DIFFEOMORPHISMS WITH LARGE ENTROPY. 2023. ⟨hal-03568570v4⟩