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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For a $d-$dimensional cellular automaton with $d\\geq1$ we introduce a rescaled entropy which estimates the growth rate of the entropy at small scales by generalizing previous approaches [1, 9]. We also define a notion of Lyapunov exponent and …