We show that for every $C^\infty$ diffeomorphism of a closed Riemannian manifold, if there exists a positive volume set of points which admit some expansion with a positive Lyapunov exponent (in a weak sense) then there exists an invariant …
We define the topological multiplicity of an invertible topological system (X, T ) as the minimal number $k$ of real continuous functions $f_1, \cdots , f_k$ such that the functions $f_i\circ T^n$, $n \in \mathbb Z$, $1 \leq i \leq k$, span a dense …
For a topological system with positive topological entropy, we show that the induced transformation on the set of probability measures endowed with the weak-$*$ topology has infinite topological mean dimension. We also estimate the rate of divergence …
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 $C^r$, $r1$, smooth surface diffeomorphism $f$ with …
A $C^\infty$ surface diffeomorphism admits a SRB measure if and only if the set $\{x, \ \limsup_n\frac{1}{n}\log \|d_xf^n\|0 \}$ has positive Lebesgue measure. Moreover the basins of the ergodic SRB measures are covering this set Lebesgue almost …
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